C5320 Physical Principles of NMR
Luk ´aˇs ˇZ´ıdek
December 24, 2015
ii
Contents
1 No spin 1
1.1 Wave function and state of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Superposition and localization in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Operators and possible results of measurement . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Matrix representation and expected result of measurement . . . . . . . . . . . . . . . . . 4
1.5 Operators of position and momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5.1 Operator of momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5.2 Operator of position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5.3 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Operator of energy and equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6.1 Schr¨odinger equation in matrix representation and stationary states . . . . . . . 8
1.7 Operator of angular momentum and rotation in space . . . . . . . . . . . . . . . . . . . 8
1.7.1 Operator of angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7.2 Eigenvalues of angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7.3 Angular momentum and rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.8 Operator of orbital magnetic moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.9 Hamiltonian of orbital magnetic moment in magnetic field . . . . . . . . . . . . . . . . . 11
2 Single spin 13
2.1 Relativistic quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Relation to Schr¨odinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Operators of spin angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Eigenfunctions and eigenvalues of ˆIz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Eigenfunctions of ˆIx and ˆIy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 Operators of spin magnetic moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.8 Hamiltonian of spin magnetic moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.9 Spin and magnetogyric ratio of real particles . . . . . . . . . . . . . . . . . . . . . . . . 22
2.10 Stationary states and energy level diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.11 Oscillatory states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Ensembles of spins not interacting with other spins 27
3.1 Mixed state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 Equation of motion: Liouville-von Neumann equation . . . . . . . . . . . . . . . . . . . 29
3.5 Rotation in operator space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 General strategy of analyzing NMR experiments . . . . . . . . . . . . . . . . . . . . . . 31
iii
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3.7 Operator of the observed quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.8 Static field B0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.9 Radio-frequency field B1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.10 Phenomenology of chemical shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.11 Hamiltonian of chemical shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.12 Secular approximation and averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.13 Thermal equilibrium as the initial state . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.14 Relaxation due to chemical shift anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.14.1 Classical analysis: fluctuations B0 and loss of coherence . . . . . . . . . . . . . 38
3.14.2 Rigid molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.14.3 Internal motions changing orientation of chemical shift tensor . . . . . . . . . . . 41
3.14.4 Chemical/conformational exchange . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.14.5 Quantum description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.14.6 Relaxation of Mz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.14.7 Relaxation of M+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.15 The one-pulse experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.15.1 Excitation by radio wave pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.15.2 Evolution of chemical shift after excitation . . . . . . . . . . . . . . . . . . . . . 47
4 Ensembles of spins interacting through space 49
4.1 Product operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Liouville-von Neumann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Through-space dipole-dipole interaction (dipolar coupling) . . . . . . . . . . . . . . . . . 50
4.4 Hamiltonian of dipolar coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5 Secular approximation and averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6 Relaxation due to the dipole-dipole interactions . . . . . . . . . . . . . . . . . . . . . . . 53
4.7 2D spectroscopy based on dipole-dipole interactions . . . . . . . . . . . . . . . . . . . . 56
4.7.1 Two-dimensional spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.7.2 Nuclear Overhauser efect spectroscopy (NOESY) . . . . . . . . . . . . . . . . . . 58
5 Ensembles of spins interacting through bonds 61
5.1 Secular approximation and averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Relaxation due to the J-coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 2D spectroscopy based on scalar coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3.1 Evolution in the presence of the scalar coupling . . . . . . . . . . . . . . . . . . . 62
5.4 Spin echoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4.1 Free evolution (Figure 5.1A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4.2 Refocusing echo (Figure 5.1B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4.3 Decoupling echo (Figure 5.1C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4.4 Recoupling echo (Figure 5.1D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.5 INEPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.6 Heteronuclear Single-Quantum Correlation (HSQC) . . . . . . . . . . . . . . . . . . . . 68
5.6.1 Decoupling trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.6.2 Benefits of HSQC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.7 Systems with multiple protons - attached proton test (APT . . . . . . . . . . . . . . . . 69
5.8 Homonucler correlation based on scalar coupling (COSY) . . . . . . . . . . . . . . . . . 70
Chapter 1
No spin
1.1 Wave function and state of the system
We postulate that the state of the system is completely described by a wave function.
• Newton mechanics: coordinates and moments of all particles describe all properties of the current
state and all future states
• Quantum mechanics: wave function describes all properties of the current state and all future
states
Quantum mechanics is postulated, not derived. It can be only tested experimentally. Introduced
because Newton mechanics did not described experiments correctly.
Example – two-slit (Young) experiment:
• Question: Particles or waves?
• Answer: Particles, but with probabilities added like waves
(Complex) probability amplitude: Ψ = Ceiφ
(Real) probability density: ρ = Ψ∗
Ψ = |Ψ|2
= |C|2
Probability of finding single particle in volume L3
:
L
0
L
0
L
0
Ψ∗
Ψdxdydz
Wave function of a free particle moving in direction x (coordinate frame can be always chosen so
that x is the direction of motion of a free particle):
Ψ = Cei2π( x
λ − t
T )
= Ce
i
¯h (px−Et)
, (1.1)
where h = 2π¯h is the Planck’s constant, p = mv is momentum (along x), and E is (kinetic) energy.
Note that Ψ corresponds to a monochromatic wave with period equal to h/E, wavelength equal to
h/p, and a complex amplitude (it may contain a phase factor eiφ
).
Calculating ”square”: real number c2
= cc, complex number |c|2
= cc∗
, real vector |v|2
= v · v =
v1v1 + v2v2 + · · ·, complex vector |v|2
= v†
· v = v∗
1v1 + v∗
2v2 + · · ·, (continuous) function
b
a
f∗
(x)f(x)dx
(function can be viewed as a vector of infinite number of infinitely ”dense” elements – summation →
integration).
Dirac’s notation: |v , |f is a vector v or function f, respectively:
1
2 CHAPTER 1. NO SPIN
v|v = v†
· v =
N
j=1
v∗
j vj (1.2)
f|f =
∞
−∞
f∗
(x)f(x)dx (1.3)
1.2 Superposition and localization in space
Note that a monochromatic wave function describes exactly what is p of the particle, but does not say
anything about position of the particle because ρ = Ψ∗
Ψ = |C| is the same for any x (distribution of
probability is constant from x = −∞ to x = ∞). Wave function describing a particle (more) localized
in space can be obtained by superposition of monochromatic waves.
Ψ(x, t) = c1Ae
i
¯h (p1x−E1t)
+ c2Ae
i
¯h (p2x−E2t)
+ · · · (1.4)
We postulate that if possible states of our system are described by wave functions ψ1, ψ2, . . ., their
linear combination also describes a possible state of the system.
Note that monochromatic waves are orthogonal:
∞
−∞
A∗
e− i
¯h (p1x−E1t)
Ae
i
¯h (p2x−E2t)
dx = |A|2
e
i
¯h (E1−E2)t
∞
−∞
e
i
¯h (p1−p2)x
dx =
|A|2
e
i
¯h (E1−E2)t
∞
−∞
cos
(p1 − p2)x
¯h
dx + i|A|2
e
i
¯h (E1−E2)t
∞
−∞
sin
(p1 − p2)x
¯h
dx = 0 (1.5)
unless p1 = p2 (positive and negative parts of sine and cosine functions cancel each other during
integration, with the exception of cos 0 = 1).
Values of A can be also normalized to give the result of Eq. 1.5 equal to 1 if p1 = p2 and E1 = E2.
It follows from the property of the Fourier transform that in such a case |A|2
= 1/h if we integrate over
a single coordinate (or |A|2
= 1/h3
if we integrate over three coordinates etc.).
In the language of algebra, the complete set of normalized monochromatic waves constitutes orthonormal
basis for wave functions, in a similar way as unit vectors ı, , k are the orthonormal basis for
all vectors in the Cartesian coordinate system x, y, z.
Also, Ψ can be normalized based on the condition
∞
−∞
Ψ∗
Ψdx = P = 1 (1.6)
(if a particle exists, it must be somewhere). It requires
∞
−∞
(c∗
1c1 + c∗
2c2 + · · ·)dx = 1. (1.7)
1.3. OPERATORS AND POSSIBLE RESULTS OF MEASUREMENT 3
1.3 Operators and possible results of measurement
We postulated that the wave function contains a complete information about the system, but how
can we extract this information from the wave function? For example, how can we get the value of a
momentum of a free particle described by Eq. 1.4? Calculation of ∂Ψ/∂x gives us a clue:
∂Ψ
∂x
= c1
∂
∂x
e
i
¯h (p1x−E1t)
+ c2
∂
∂x
e
i
¯h (p2x−E2t)
+ · · · =
i
¯h
p1c1e
i
¯h (p1x−E1t)
+
i
¯h
p2c2e
i
¯h (p2x−E2t)
+ · · · (1.8)
It implies that
− i¯h
∂
∂x
e
i
¯h (p1x−E1t)
= p1e
i
¯h (p1x−E1t)
, −i¯h
∂
∂x
e
i
¯h (p2x−E2t)
= p2e
i
¯h (p2x−E2t)
, . . . (1.9)
We see that
1. calculation of the partial derivative of any monochromatic wave and multiplying the result by
−i¯h gives us the same wave just multiplied by a constant. In general, the instruction to calculate
the partial derivative and multiply the result by −i¯h is known as operator. If application of the
operator to a function gives the same function, only multiplied by a constant, the function is
called eigenfunction of the operator and the constant is called eigenvalue of the operator.
2. the eigenvalues are well-defined, measurable physical quantities – possible values of the momentum
along x.
3. the eigenvalues can be obtained by applying the operator to the eigenfunction and multiplying
the result by the complex conjugate of the eigenfunction:
p1 = e− i
¯h (p1x−E1t)
−i¯h
∂
∂x
e
i
¯h (p1x−E1t)
= e− i
¯h (p1x−E1t)
p1e
i
¯h (p1x−E1t)
= p1 e− i
¯h (p1x−E1t)
e
i
¯h (p1x−E1t)
=1
(1.10)
We postulate that any measurable property is represented by an operator (acting on the wave
function) and that result of a measurement must be one of eigenvalues of the operator.
Here, we usually write operators with ”hats”, like ˆA. Writing ˆAΨ means ”take function Ψ and
modify it as described by ˆA. It is not a multiplication: ˆAΨ = ˆA · Ψ, ˆA is not a number but an
instruction what to do with Ψ!
Recipe to calculate possible results of a measurement:
1. Identify the operator representing what you measure ( ˆA)
2. Find all eigenfunctions |ψ1 , |ψ2 , . . . of the operator and use them as an orthonormal basis for Ψ:
Ψ = c1|ψ1 + c2|ψ2 , . . .
3. Calculate individual eigenvalues Aj as
ψj| ˆAψj = ψj|Aj · ψj = Aj ψj|ψj
=1
= Aj. (1.11)
The first equality in step 3 follows from the definition of eigenfunctions, then Aj is just a (real)
number and can be factored out of the brackets (representing integration or summation) as described
by the second equality, and the last equality reflects orthonormality of |ψj .
4 CHAPTER 1. NO SPIN
1.4 Matrix representation and expected result of measurement
Eq. 1.11 tells us what are the possible results of a measurement, but it does not say which value is
actually measured. We can only calculate probabilities of getting individual eigenvalues and predict the
expected result of the measurement.
We postulate that the expected result of measuring a quantity A represented by an operator ˆA in
a state of the system described by a wave function Ψ is
A = Ψ| ˆA|Ψ . (1.12)
There are three ways how to do the calculation described by Eq. 1.12:
1. Express Ψ, calculate its complex conjugate Ψ∗
≡ Ψ|, calculate ˆAΨ ≡ | ˆAΨ , and in the manner
of Eq. 1.3
A = Ψ| ˆA|Ψ ≡ Ψ|( ˆAΨ) =
∞
−∞
· · · Ψ∗
(x, . . .) ˆAΨ(x, . . .)dx . . . . (1.13)
Three dots in Eq. 1.13 tell us that for anything else that a single free particle (with zero spin) we
integrate over all degrees of freedom, not just over x.
2. Find eigenfunctions ψ1, ψ2, . . . of ˆA and write Ψ as their linear combination Ψ = c1ψ1 +c2ψ2 +· · ·
(use the eigenfunctions as an orthonormal basis for Ψ). Due to the orthonormality of the basis
functions, the result of Eq. 1.13 is A = c∗
1c1A1 +c∗
2c2A2 +· · · , where A1, A2, . . . are eigenvalues of
ˆA. We see that A is a weighted average of eigenvalues Aj with the weights equal to the squares
of the coefficients (c∗
j cj = |cj|2
). The same result is obtained if we calculate
A = c∗
1 c∗
2 · · ·



A1 0 · · ·
0 A2 · · ·
...
...
...






c1
c2
...


 (1.14)
We see that we can replace (i) operators by two-dimensional diagonal matrices, with eigenvalues
forming the diagonal, and (ii) wave functions by one-dimensional matrices (known as state vectors)
composed of the coefficients cj. Eq. 1.14 shows calculation of the expected results of the
measurement of A using matrix representation of operators and wave functions. Matrix representation
is a big simplification because it allows us to calculate A without knowing how the
operator ˆA and its eigenfunctions look like! We just need the eigenvalues and coefficients cj. This
simplification is paid by the fact that the right coefficients are defined by the right choice of the
basis.
3. Write Ψ as a linear combination of basis functions ψ1, ψ1, . . . (not necessarily eigenfunctions of ˆA)
Ψ = c1ψ1 + c2ψ2 + · · · (1.15)
Build a two-dimensional matrix ˆP from the products of coefficients c ∗
j ck:
ˆP =



c1c ∗
1 c1c ∗
2 · · ·
c2c ∗
1 c2c ∗
2 · · ·
...
...
...


 . (1.16)
1.5. OPERATORS OF POSITION AND MOMENTUM 5
Multiply the matrix ˆP by a matrix1 ˆA representing the operator ˆA in the basis ψ1, ψ1, . . .. The
sum of the diagonal elements (called trace) of the resulting matrix ˆP ˆA is equal to the expected
value A
A = Tr{ ˆP ˆA } (1.17)
Why should we use such a bizarre way of calculating the expected value of A when it can be
calculated easily from Eq. 1.14? The answer is that Eq. 1.17 is more general. We can use the
same basis for operators with different sets of eigenfunctions.
1.5 Operators of position and momentum
We need to find operators in order to describe measurable quantities. Let’s start with the most fundamental
quantities, position of a particle x and momentum p = mv.
1.5.1 Operator of momentum
We have already obtained the operator of momentum of a particle moving in the x direction when
calculating ∂Ψ/∂x (Eq. 1.9). If a particle moves in a general direction, operators of components of the
momentum tensor are derived in the same manner.
ˆpx ≡
∂
∂x
(1.18)
ˆpy ≡
∂
∂y
(1.19)
ˆpz ≡
∂
∂z
(1.20)
1.5.2 Operator of position
The wave function Ψ(x, t) defined by Eq. 1.4 is a function of the position of the particle, not of the
momentum (it is a sum of contributions of all possible momenta). If we define basis as a set of functions
ψj = Ψ(xj, t) for all possible positions xj, operator of position is simply multiplication by the value of
the coordinate describing the given position. Operators of the y and z are defined in the same manner.
ˆx ≡ x · (1.21)
To see how the operator acts, write Ψ∗
(x, t) and xΨ(x, t) as the set of functions Ψ(xj, t) for all
possible positions xj:
xΨ(x, t) =





x1c1e
i
¯h (p1x1−E1t)
+ x1c2e
i
¯h (p2x1−E2t)
+ x1c3e
i
¯h (p3x1−E3t)
+ · · ·
x2c1e
i
¯h (p1x2−E1t)
+ x2c2e
i
¯h (p2x2−E2t)
+ x2c3e
i
¯h (p3x2−E3t)
+ · · ·
x3c1e
i
¯h (p1x3−E1t)
+ x3c2e
i
¯h (p2x3−E2t)
+ x3c3e
i
¯h (p3x3−E3t)
+ · · ·
...





=





ψ1
ψ2
ψ3
...





(1.22)
1How can we get a matrix representation of an operator with eigenfunctions different from the basis? The complete set
of N functions defines an abstract N-dimensional space (N = ∞ for free particles!). The wave function Ψ is represented
by a vector in this space built from coefficients c1, c2, . . ., as described by Eq. 1.15, and a change of the basis is described as
a rotation in this space. The same rotation describes how the matrix representing the operator ˆA changes upon changing
the basis. Note that the matrix is not diagonal if the basis functions are not eigenfunctions of ˆA.
6 CHAPTER 1. NO SPIN
If the position of the particle is e.g. x2,
Ψ(x2, t) =





0
c1e
i
¯h (p1x2−E1t)
+ c2e
i
¯h (p2x2−E2t)
+ c3e
i
¯h (p3x2−E3t)
+ · · ·
0
...





=





0
ψ2
0
...





(1.23)
and x · Ψ(x, t) for x = x2 is
x2Ψ(x2, t) =






0
x2 c1e
i
¯h (p1x2−E1t)
+ c2e
i
¯h (p2x2−E2t)
+ c3e
i
¯h (p3x2−E3t)
+ · · ·
0
...






=





0
x2ψ2
0
...





. (1.24)
We see that multiplication of Ψ(x2, t) = ψ2 by x2 results in x2ψ2, i.e., ψ2 is an eigenfunction of the
operator ˆx = x· and x2 is the corresponding eigenvalue.
Note that multiplication by pj does not work in the same way! We could multiply ψ2 by x2 because
ψ2 does not depend on any other value of the x coordinate. However, ψ2 depends on all possible values
of p. On the other hand, partial derivative gave us each monochromatic wave multiplied by its value of
p and ensured that the monochromatic waves acted as eigenfunctions.
1.5.3 Commutators
If we apply two operators subsequently to the same wave function, order of the operators sometimes
does not matter. E.g., ˆxˆpyΨ = ˆpy ˆxΨ (ˆx and ˆpy commute). It means that x and py can be measured
independently at the same time. However, sometimes the order of operators makes a difference. For
example
ˆxˆpxΨ = −i¯hx
∂Ψ
∂x
(1.25)
but
ˆpx ˆxΨ = −i¯h
∂(xΨ)
∂x
= −i¯hΨ − i¯hx
∂Ψ
∂x
(1.26)
The difference is known as the commutator and is written as ˆxˆpx − ˆpx ˆx = [ˆx, ˆpx]. A non-zero
commutator tells us that ˆx and ˆpx are not independent and cannot be measured exactly at the same
time. Analysis of the action of the operators shows that
• commutators of operators of a coordinate and the momentum component in the same direction
are equal to −i¯h (i.e., multiplication of Ψ by the factor −i¯h)
• all other position and coordinate operators commute.
Written in a mathematically compact form,
[ˆrj, ˆpk] = −i¯hδj,k [ˆrj, ˆrk] = [ˆrj, ˆpk] = 0, (1.27)
where j and k are x, y, or z, rj is the x, y, or z component of the position vector r = (rx, ry, rz) ≡
(x, y, z), pk is the x, y, or z component of the momentum vector p = (px, py, pz), and δj,k = 1 for j = k
and δj,k = 0 for j = k.
1.6. OPERATOR OF ENERGY AND EQUATION OF MOTION 7
The described commutator relations follow from the way how we defined Ψ in Eq. 1.4. However, we
can also use Eq. 1.27 as the fundamental definition and Eq. 1.4 as its consequence:
We postulate that operators of position and momentum obey the relations
[ˆrj, ˆpk] = −i¯hδj,k [ˆrj, ˆrk] = [ˆpj, ˆpk] = 0. (1.28)
Note that we only postulate relations between operators. Other choices are possible and correct as
long as Eq. 1.27 holds.
1.6 Operator of energy and equation of motion
We obtained the operator of momentum by calculating ∂Ψ/∂x. What happens if we calculate ∂Ψ/∂t?
∂Ψ
∂t
= c1
∂
∂t
e
i
¯h (p1x−E1t)
+c2
∂
∂t
e
i
¯h (p2x−E2t)
+· · · = −
i
¯h
E1c1e
i
¯h (p1x−E1t)
−
i
¯h
E2c2e
i
¯h (p2x−E2t)
−· · · (1.29)
and consequently
i¯h
∂
∂t
e
i
¯h (p1x−E1t)
= E1e
i
¯h (p1x−E1t)
, i¯h
∂
∂t
e
i
¯h (p2x−E2t)
= E2e
i
¯h (p2x−E2t)
, . . . (1.30)
1. First, we obtain the operator of energy from Eq. 1.30, in analogy to Eq. 1.9.
2. The second achievement is Eq. 1.29 itself. Energy of free particles is just the kinetic energy (by
definition). Therefore, all energies Ej in the right-hand side of Eq. 1.29 can be written as
Ej =
mv2
j
2
=
p2
j
2m
, (1.31)
resulting in
∂Ψ
∂t
= −
i
¯h
p2
1
2m
c1e
i
¯h (p1x−E1t)
+
p2
2
2m
c2e
i
¯h (p2x−E2t)
+ · · · (1.32)
But an equation with the p2
j terms can be also obtained by calculating
1
2m
∂2
Ψ
∂x2
=
1
2m
∂
∂x
∂Ψ
∂x
= −
1
¯h2
p2
1
2m
c1e
i
¯h (p1x−E1t)
+
p2
2
2m
c2e
i
¯h (p2x−E2t)
+ · · · (1.33)
Comparison of Eqs. 1.32 and 1.33 gives us the equation of motion
i¯h
∂Ψ
∂t
= −
¯h2
2m
∂2
Ψ
∂x2
(1.34)
If we extend our analysis to particles experiencing a time-independent potential energy V (x, y, z),
the energy will be given by
Ej =
p2
j
2m
+ V, (1.35)
where pj is now the absolute value of a momentum vector pj (we have to consider all three
direction x, y, z because particles change direction of motion in the presence of a potential). The
time derivative of Ψ is now
8 CHAPTER 1. NO SPIN
∂Ψ
∂t
= −
i
¯h
p2
1
2m
c1e
i
¯h (p1r−E1t)
+
p2
2
2m
c2e
i
¯h (p2r−E2t)
+ · · · −
i
¯h
V (r)Ψ (1.36)
and
p2
1
2m
c1e
i
¯h (p1r−E1t)
+
p2
2
2m
c2e
i
¯h (p2r−E2t)
+ · · · = −
¯h2
2m
∂2
Ψ
∂x2
+
∂2
Ψ
∂x2
+
∂2
Ψ
∂x2
(1.37)
Substituting Eq. 1.37 into Eq. 1.36 gives us the famous Schr¨odinger equation
i¯h
∂Ψ
∂t
= −
¯h2
2m
∂2
∂x2
+
∂2
∂x2
+
∂2
∂x2
+ V (x, y, z)
ˆH
Ψ (1.38)
The sum of kinetic and potential energy is known as Hamiltonian in the classical mechanics and
the same term is used for the operator ˆH.
The association of Hamiltonian (energy operator) with the time derivative makes it essential for
analysis of dynamics of systems in quantum mechanics:
We postulate that evolution of a system in time is given by the Hamiltonian:
i¯h
∂Ψ
∂t
= ˆHΨ (1.39)
1.6.1 Schr¨odinger equation in matrix representation and stationary states
Eq.1.39 can be also written for matrix representation of Ψ and ˆH. If eigenfunctions of ˆH are used as a
basis
i¯h
d
dt



c∗
1
c∗
2
...


 =



E1 0 · · ·
0 E2 · · ·
...
...
...






c∗
1
c∗
2
...


 , (1.40)
which is simply a set of independent differential equations
dcj
dt
= −i
Ej
¯h
cj ⇒ cj = aje−i
Ej
¯h t
, (1.41)
where the (possibly complex) integration constant aj is given by the value of cj at t = 0.
Note that the coefficients cj evolve, but the products c∗
j cj = |aj|2
do not change in time. Each
product c∗
j cj describes the probability that the system is in the state with the energy equal to the
eigenvalue Ej, described by an eigenfunction ψj. We see that states corresponding to the eigenfunctions
of the Hamiltonian are stationary (do not vary in time). Only such states can be described by the energy
level diagram.
1.7 Operator of angular momentum and rotation in space
In a search for operators needed to describe NMR experiment, we start from what we know, position
and momentum operators. We use classical physics and just replace the values of coordinates and
momentum components by their operators.
1.7. OPERATOR OF ANGULAR MOMENTUM AND ROTATION IN SPACE 9
1.7.1 Operator of angular momentum
Classical definition of the vector of angular momentum L is
L = r × p (1.42)
The sign ”×” denotes the vector product:
Lx = rypz − rzpy (1.43)
Ly = rzpx − rxpz (1.44)
Lz = rxpy − rypx (1.45)
Going to the operators
ˆLx = ˆry ˆpz − ˆrz ˆpy = −i¯hy
∂
∂z
+ i¯hz
∂
∂y
(1.46)
ˆLy = ˆrz ˆpx − ˆrx ˆpz = −i¯hz
∂
∂x
+ i¯hx
∂
∂z
(1.47)
ˆLz = ˆrx ˆpy − ˆry ˆpx = −i¯hx
∂
∂y
+ i¯hy
∂
∂x
(1.48)
ˆL2
= ˆL2
x + ˆL2
y + ˆL2
z (1.49)
It follows from Eq. 1.27 that
[ˆLx, ˆLy] = i¯hˆLz (1.50)
[ˆLy, ˆLz] = i¯hˆLx (1.51)
[ˆLz, ˆLx] = i¯hˆLy (1.52)
but
[ˆL2
, ˆLx] = [ˆL2
, ˆLy] = [ˆL2
, ˆLz] = 0 (1.53)
• Two components of angular momentum cannot be measured exactly at the same time
• Eqs. 1.50–1.53 can be used as a definition of angular momentum operators if the position and
momentum operators are not available.
1.7.2 Eigenvalues of angular momentum
Let’s find eigenvalues Lz,j and eigenfunctions ψj of ˆLz. In spherical coordinates (r, ϑ, ϕ), ψj =
Q(r, ϑ)Rj(ϕ) and ˆLz = −i¯h ∂
∂ϕ
Eigenvalues and eigenfunctions are defined by
ˆLzψj = Lz,jψj (1.54)
−i¯h
∂(QRj)
∂ϕ
= Lz,j(QRj) (1.55)
−i¯hQ
dRj
dϕ
= Lz,jQRj (1.56)
10 CHAPTER 1. NO SPIN
−i¯h
d ln Rj
dϕ
= Lz,j (1.57)
Rj = ei
Lz,j
¯h ϕ
(1.58)
Since ψj(ϕ) = ψj(ϕ + 2π),
• Value of the z-component of the angular momentum must be an integer multiple of ¯h
1.7.3 Angular momentum and rotation
Rotation about an axis given by the angular frequency vector ω
dr
dt
= ω × r (1.59)
drx
dt
= ωyrz − ωzry (1.60)
dry
dt
= ωzrx − ωxrz (1.61)
drz
dt
= ωxry − ωyrx (1.62)
If a coordinate frame is chosen so that ω = (0, 0, ω)
drx
dt
= −ωry (1.63)
dry
dt
= ωrx (1.64)
drz
dt
= 0 (1.65)
Solution: multiply the second equation by i and add it to the first equation or subtract it from the
first equation.
d(rx + iry)
dt
= ω(−ry + irx) = +iω(rx + iry) (1.66)
d(rx − iry)
dt
= ω(−ry − irx) = −iω(rx − iry) (1.67)
rx + iry = C+e+iωt
(1.68)
rx − iry = C−e−iωt
(1.69)
where the integration constants C+ = rx(0) + iry(0) = reφ0
and C+ = rx(0) − iry(0) = re−φ0
are
given by the initial phase φ0 of r in the coordinate system:
rx + iry = re+(iωt+φ0)
= r(cos(ωt + φ0) + i(sin(ωt + φ0)) (1.70)
rx − iry = re−(iωt+φ0)
= r(cos(ωt + φ0) − i(sin(ωt + φ0)), (1.71)
1.8. OPERATOR OF ORBITAL MAGNETIC MOMENT 11
• Comparison with Eq. 1.58 shows that the eigenfunction of ˆLz describes rotation about z.
For zero initial phase, rx(t = 0) = r, and evolution of rx and ry is obtained by adding and subtracting
Eqs. 1.70 and 1.71:
rx = r cos(ωt) (1.72)
ry = r sin(ωt) (1.73)
1.8 Operator of orbital magnetic moment
A moving charged particle can be viewed as an electric current. Classical definition of the magnetic
moment of a charged particle travelling in a circular path (orbit) is
µ =
Q
2
(r × v) =
Q
2m
(r × p) =
Q
2m
L = γL, (1.74)
where Q is the charge of the particle, m is the mass of the particle, v is the velocity of the particle,
and γ is known as the magnetogyric ratio (constant).2
Therefore, we can write the operators
ˆµx = γ ˆLx ˆµy = γ ˆLy ˆµz = γ ˆLz ˆµ2
= γ2 ˆL2
. (1.75)
1.9 Hamiltonian of orbital magnetic moment in magnetic field
Classically, the energy of a magnetic moment µ in a magnetic field of induction B is E = −µ · B.
Accordingly, the Hamiltonian of the interactions of an orbital magnetic moment with a magnetic field
is
ˆH = −Bx ˆµx − By ˆµy − Bz ˆµz = −γ (Bx ˆµx + By ˆµy + Bz ˆµz) = −
Q
2m
Bx
ˆIx + By
ˆIy + Bz
ˆIz . (1.76)
2The term gyromagnetic ratio is also used.
12 CHAPTER 1. NO SPIN
Chapter 2
Single spin
2.1 Relativistic quantum mechanics
The angular momentum discussed in Section 1.7.1 is associated with the change of direction of a moving
particle. However, the theory discussed so-far does not explain the experimental observation that even
point-like particles moving along straight lines posses a well defined angular momentum, so-called spin.
The origin of the spin is relativistic. The Schr¨odinger equation is not relativistic and does not
describe the spin. According to the special theory of relativity, time is slower and mass increases at a
speed v close to the speed of light (in vacuum) c, and energy is closely related to the mass:
t =
t0
1 − v2/c2
m =
m0
1 − v2/c2
Et = mc2
=
m0c2
1 − v2/c2
, (2.1)
where m0 is the rest mass, m0 is the rest energy, t0 is the proper time (i.e., mass, energy, and time
in the coordinate frame moving with the particle), and Et is the total energy. The metrical properties
of space and time are given by
c2
dt2
0 = c2
dt2
− dx2
− dy2
− dz2
. (2.2)
Multiplied by m2
and divided by dt2
,
mc2 dt0
dt
2
= (mc2
)2
− mc
dx
dt
2
− mc
dy
dt
2
− mc
dz
dt
2
. (2.3)
Using Eqs. 2.2,
(m0c2
)2
= (mc2
) − (mcvx)2
− (mcvy)2
− (mcvz)2
(2.4)
m2
0c4
= E2
t − c2
p2
x − c2
p2
y − c2
p2
z (2.5)
Let us look for an equation of motion that fulfills Eq. 2.5 for a monochromatic wave function. As
the answer is not intuitive, we will proceed step by step.
Monochromatic wave function ψ can be viewed as a continuous series of values of ψ(x, y, z, t) for
each time and place:
ψ = e
i
¯h (pxx+pyy+pzz−Ett)
(2.6)
Partial derivatives of ψ serve as operators of energy and momentum:
13
14 CHAPTER 2. SINGLE SPIN
i¯h
∂ψ
∂x
= −pxψ i¯h
∂ψ
∂y
= −pyψ i¯h
∂ψ
∂z
= −pzψ i¯h
∂ψ
∂t
= Etψ (2.7)
Zero spin, zero mass
If a free particle does not have spin, it has only momentum in the direction of motion (px = p if the
direction of motion defines the x axis). If the particle has a zero rest mass (m0 = 0), we can write the
following equations:
i¯h
∂ψ
∂t
= −ic¯h
∂ψ
∂x
(2.8)
i¯h
∂ψ
∂t
= +ic¯h
∂ψ
∂x
(2.9)
and write them in an operator form
i¯h
∂
∂t
+ ic¯h
∂
∂x
ψ = ˆO+
ψ = 0 (2.10)
i¯h
∂
∂t
− ic¯h
∂
∂x
ψ = ˆO−
ψ = 0. (2.11)
Expressing the partial derivatives for a monochromatic wave function,
ˆO+
ψ = (Et − cp)ψ (2.12)
Acting by ˆO−
on the result
ˆO− ˆO+
ψ = ˆO2
ψ = ˆO−
((Et − cp)ψ) = (Et + cp)(Et − cp)ψ = (E2
t − c2
p2
)ψ = 0 (2.13)
We see that the Eqs. 2.10–2.11 satisfy Eq. 2.5, the desired value of E2
t − c2
p2
s an eigenvalue of the
operator ˆO2
, and ψ is its eigenfuction. The operators ˆO−
and ˆO+
can be viewed as ”square roots” of
ˆO2
:
¯h2 ∂2
∂t2
− c2
¯h2 ∂2
∂x2
Ψ = − i¯h
∂
∂t
− ic¯h
∂
∂x
i¯h
∂
∂t
+ ic¯h
∂
∂x
Ψ = 0. (2.14)
In general, the operator ˆO2
should look like
(m0c2
)2
Ψ + ¯h2 ∂2
Ψ
∂t2
− c2
¯h2 ∂2
Ψ
∂z2
− c2
¯h2 ∂2
Ψ
∂x2
− c2
¯h2 ∂2
Ψ
∂y2
(2.15)
Such an operator cannot be decomposed into ”sqare roots” as in the case of zero spin and zero
rest mass. If we try to calculate product of some (more complex) operators ˆO+
and ˆO−
, we never
get ˆO2
from Eq. 2.15: we always obtain some additional terms that do not cancel each other. No
monochromatic wave function can serve as an eigenfuction for such operator if the particle has a spin
or mass.
However, the solution can be found if we write the equation of motions for more monochromatic
functions, coupled in such a way that they cancel unwanted terms of the product ˆO− ˆO+
.
2.1. RELATIVISTIC QUANTUM MECHANICS 15
Non-zero spin, zero mass
Let us try to solve the problem for a particle with a spin, but with a zero rest mass. This is a good
approximation of neutrinos. We can write the following equations of motions:
i¯h
∂(u1ψ)
∂t
= −ic¯h
∂(u1ψ)
∂z
− ic¯h
∂(u2ψ)
∂x
− ic¯h
∂(−iu2ψ)
∂y
(2.16)
i¯h
∂(u2ψ)
∂t
= +ic¯h
∂(u2ψ)
∂z
− ic¯h
∂(u1ψ)
∂x
− ic¯h
∂(iu1ψ)
∂y
, (2.17)
where u1ψ and u2ψ are monochromatic functions. We can group them into vectors
i¯h
∂
∂t
u1ψ
u2ψ
+ ic¯h
∂
∂z
u1ψ
−u2ψ
+ ic¯h
∂
∂x
u2ψ
u1ψ
+ ic¯h
∂
∂y
−iu2ψ
iu1ψ
= 0 (2.18)
and write the equations in an operator form
i¯h
∂
∂t
1 0
0 1
+ ic¯h
∂
∂z
0 1
−1 0
+ ic¯h
∂
∂x
0 1
1 0
+ ic¯h
∂
∂y
0 −i
i 0
u1ψ
u2ψ
= ˆO+
Ψ = 0 (2.19)
i¯h
∂
∂t
1 0
0 1
− ic¯h
∂
∂z
1 0
0 −1
− ic¯h
∂
∂x
0 1
1 0
− ic¯h
∂
∂y
0 −i
i 0
u1ψ
u2ψ
= ˆO−
Ψ = 0. (2.20)
Now,
ˆO+
Ψ =
Etu1ψ − cpzu1ψ − cpxu2ψ + icpyu2ψ
Etu2ψ + cpzu2ψ − cpxu1ψ − icpyu1ψ
(2.21)
and
ˆO2
Ψ = ˆO− ˆO+
Ψ =
E2
t − c2
p2
z − c2
p2
x − c2
p2
y
E2
t − c2
p2
z − c2
p2
x − c2
p2
y
u1ψ
u2ψ
= (E2
t − c2
p2
z − c2
p2
x − c2
p2
y)Ψ (2.22)
E2
t − c2
p2
z − c2
p2
x − c2
p2
y is an eigenvalue of ˆO2
, and the vector Ψ is an eigenfunction. This success is
paid by the fact that we need two functions u1ψ, u2ψ instead of one. The series of values constituting
the wave function Ψ is twice as long compared to ψ of the spin-less particle because we have two values
for each x, y, z, t. It means that the wave function is not unambiguously defined by x, y, z, t – it has one
more degree of freedom, represented by the new ”coordinate” u.
Zero spin, non-zero mass
To test the effect of mass, we now find a solution for a particle without spin but with a non-zero rest
mass. The following equations of motion work in this case:
i¯h
∂(uψ)
∂t
= −ic¯h
∂(vψ∗
)
∂z
− ic¯h
∂(vψ∗
)
∂x
−ic¯h
∂(−ivψ∗
)
∂y
+ m0c2
uψ (2.23)
i¯h
∂(vψ∗
)
∂t
= −ic¯h
∂(uψ)
∂z
− ic¯h
∂(uψ)
∂x
−ic¯h
∂(−iuψ)
∂y
− m0c2
uψ∗
. (2.24)
The red partial derivatives are equal to zero if the particle moves in the x direction and py = pz = 0
where u1ψ and u2ψ are monochromatic functions. We can group them into vectors
16 CHAPTER 2. SINGLE SPIN
i¯h
∂
∂t
uψ
−vψ∗ +ic¯h
∂
∂z
uψ
−vψ∗ + ic¯h
∂
∂x
vψ∗
−uψ
+ic¯h
∂
∂y
−ivψ∗
iuψ
− m0c2 uψ
vψ∗ = 0 (2.25)
and write the equations in an operator form
i¯h
∂
∂t
1 0
0 −1
+ic¯h
∂
∂z
0 1
−1 0
+ ic¯h
∂
∂x
0 1
−1 0
+ic¯h
∂
∂y
0 −i
i 0
− m0c2 1 0
0 1
uψ
vψ∗ = ˆO+
Ψ = 0
(2.26)
i¯h
∂
∂t
1 0
0 −1
+ic¯h
∂
∂z
0 1
−1 0
+ ic¯h
∂
∂x
0 1
−1 0
+ic¯h
∂
∂y
0 −i
i 0
+ m0c2 1 0
0 1
u1ψ
u2ψ
= ˆO−
Ψ = 0.
(2.27)
ˆO+
Ψ =
Etuψ + cpvψ∗
− m0c2
uψ
Etvψ∗
+ cpuψ − m0c2
vψ∗ (2.28)
and
ˆO2
Ψ = ˆO− ˆO+
Ψ =
E2
t − c2
p2
− (m0c2
)2
E2
t − c2
p2
− (m0c2
)2
uψ
vψ∗ = (E2
t − c2
p2
− (m0c2
)2
)Ψ (2.29)
Again, we achieved the desired result using two monochromatic functions. This time, one contained
complex conjugate to ψ – it represents an antiparticle.
Non-zero spin, non-zero mass
Finally we describe the solution for the most interesting particles as electron or quarks. If the particle
has a spin and a non-zero rest mass, effects of discussed in the previous sections combine. We need four
equations of motions with four components of the wave function:
i¯h
∂(u1ψ)
∂t
= −ic¯h
∂(v1ψ∗
)
∂z
− ic¯h
∂(v2ψ∗
)
∂x
− ic¯h
∂(−iv2ψ∗
)
∂y
+ m0c2
u1ψ (2.30)
i¯h
∂(u2ψ)
∂t
= +ic¯h
∂(v2ψ∗
)
∂z
− ic¯h
∂(v1ψ∗
)
∂x
+ ic¯h
∂(−iv1ψ∗
)
∂y
+ m0c2
u2ψ (2.31)
i¯h
∂(v1ψ∗
)
∂t
= −ic¯h
∂(u1ψ)
∂z
− ic¯h
∂(u2ψ)
∂x
− ic¯h
∂(iu2ψ)
∂y
− m0c2
v1ψ∗
(2.32)
i¯h
∂(v2ψ∗
)
∂t
= +ic¯h
∂(u2ψ)
∂z
− ic¯h
∂(u1ψ)
∂x
+ ic¯h
∂(iu1ψ)
∂y
− m0c2
v2ψ∗
. (2.33)
We can group the monochromatic functions into vectors
i¯h
∂
∂t




u1ψ
u2ψ
−v1ψ∗
−v2ψ∗



 + ic¯h
∂
∂z




v1ψ∗
−v2ψ∗
−u1ψ
u2ψ



 + ic¯h
∂
∂x




v2ψ∗
v1ψ∗
−u2ψ
−u1ψ



 + ic¯h
∂
∂y




−iv2ψ∗
iv1ψ∗
iu2ψ
−iu1ψ



 − m0c2




u1ψ
u2ψ
v1ψ∗
v2ψ∗



 = 0
(2.34)
and write the equations in an operator form
2.1. RELATIVISTIC QUANTUM MECHANICS 17



i¯h
∂
∂t




1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 −1



 + ic¯h
∂
∂z




0 0 1 0
0 0 0 −1
−1 0 0 0
0 1 0 0



 + ic¯h
∂
∂x




0 0 0 1
0 0 1 0
0 −1 0 0
−1 0 0 0



 + ic¯h
∂
∂y




0 0 0 −i
0 0 i 0
0 i 0 0
−i 0 0 0




−m0c2




1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1












u1ψ
u2ψ
v1ψ∗
v2ψ∗



 = ˆO+
Ψ = 0
(2.35)



−i¯h
∂
∂t




1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 −1



 − ic¯h
∂
∂z




0 0 1 0
0 0 0 −1
−1 0 0 0
0 1 0 0



 − ic¯h
∂
∂x




0 0 0 1
0 0 1 0
0 −1 0 0
−1 0 0 0



 − ic¯h
∂
∂y




0 0 0 −i
0 0 i 0
0 i 0 0
−i 0 0 0




−m0c2




1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1












u1ψ
u2ψ
v1ψ∗
v2ψ∗



 = ˆO−
Ψ = 0.
(2.36)
or shortly
i¯h
∂
∂t
ˆγ0
+ ic¯h
∂
∂x
ˆγ1
+ ic¯h
∂
∂y
ˆγ2
+ ic¯h
∂
∂z
ˆγ3
− m0c2ˆ1




u1ψ
u2ψ
v1ψ∗
v2ψ∗



 = ˆO+
Ψ = 0
(2.37)
−i¯h
∂
∂t
ˆγ0
− ic¯h
∂
∂x
ˆγ1
− ic¯h
∂
∂y
ˆγ2
− ic¯h
∂
∂z
ˆγ3
− m0c2ˆ1




u1ψ
u2ψ
v1ψ∗
v2ψ∗



 = ˆO−
Ψ = 0.
(2.38)
ˆO+
Ψ =




Etu1ψ + cpxv2ψ∗
− icpyv2ψ∗
+ cpzv1ψ∗
− m0c2
u1ψ
Etu2ψ + cpxv1ψ∗
+ icpyv1ψ∗
− cpzv2ψ∗
− m0c2
u2ψ
Etv1ψ∗
+ cpxu2ψ∗
− icpyu2ψ∗
+ cpzu1ψ∗
− m0c2
v1ψ∗
Etv2ψ∗
+ cpxu1ψ∗
+ icpyu1ψ∗
− cpzu2ψ∗
− m0c2
v2ψ∗



 (2.39)
and
ˆO2
Ψ = ˆO− ˆO+
Ψ =




E2
t − c2
p2
− (m0c2
)2
E2
t − c2
p2
− (m0c2
)2
E2
t − c2
p2
− (m0c2
)2
E2
t − c2
p2
− (m0c2
)2








u1ψ
u2ψ
v1ψ∗
v2ψ∗



 = (E2
t − c2
p2
− (m0c2
)2
)Ψ (2.40)
18 CHAPTER 2. SINGLE SPIN
2.2 Dirac equation
Eqs. 2.37 and 2.38 are known as the Dirac equation. When postulated by Dirac, they naturally explained
the behavior of particles with spin number 1/2 and predicted existence of antiparticles, discovered a
few years later.
Dirac equations are valid generally, not just for monochromatic ψ describing free particles. ˆO− ˆO+
always gives ˆO2
with the eigenvalue (E2
t − c2
p2
− (m0c2
)2
) and eigenfunction Ψ. The unwanted matrix
products of ˆO− ˆO+
cancel due to the properties of the 4 × 4 matrices ˆγj
:
ˆγ0
· ˆγ0
= 1 ˆγ1
· ˆγ1
= −1 ˆγ2
· ˆγ2
= −1 ˆγ3
· ˆγ3
= −1 (2.41)
and
ˆγj
· ˆγk
+ ˆγk
· ˆγj
= 0 (2.42)
for j = k.
2.3 Relation to Schr¨odinger equation
We came to the Schr¨odinger equation using the relation E = p2
/2m (energy of a free particle, i.e.,
kinetic energy), which is only an approximation for low speeds, obtained by neglecting the E2
term
(E2
(m0c2
)2
for v2
c2
) in Eq. 2.5:
(m0c2
)2
= (m0c2
+ E)2
− c2
p2
= (m0c2
)2
+ 2E(m0c2
) + E2
− c2
p2
≈ (m0c2
)2
+ 2E(m0c2
) − c2
p2
⇒ E =
p2
2m0
(2.43)
2.4 Operators of spin angular momentum
The 2 × 2 matrices in the operator Eqs. 2.20 and 2.20 and constituting the 4 × 4 matrices in Eqs. 2.37
and 2.38 are known as Pauli matrices. When we calculate their commutators, we find that
0 1
1 0
,
0 −i
i 0
= i2
1 0
0 −1
(2.44)
0 −i
i 0
,
1 0
0 −1
= i2
0 1
1 0
(2.45)
1 0
0 −1
,
0 1
1 0
= i2
0 −i
i 0
(2.46)
If we multiply the Pauli matrices by ¯h/2, we obtain the relations presented in Eqs. 1.50–1.53 as
a definition of angular momentum operators. Therefore, Pauli matrices provide a basis for operators
of spin angular momentum, a strange physical quantity describing intrinsic angular momentum of a
point-like particle and not associated with its motion:
ˆIx =
¯h
2
0 1
1 0
ˆIy =
¯h
2
0 −i
i 0
ˆIz =
¯h
2
1 0
0 −1
ˆI2
=
3¯h2
4
1 0
0 1
(2.47)
2.5. EIGENFUNCTIONS AND EIGENVALUES OF ˆIZ 19
2.5 Eigenfunctions and eigenvalues of ˆIz
The fact that ˆIz is diagonal tells us that we have written the matrix representations of the operators of
the spin angular momentum in the basis formed by the eigenfunctions of ˆIz:
ˆIx =
¯h
2
0 1
1 0
ˆIy =
¯h
2
0 −i
i 0
ˆIz =
¯h
2
1 0
0 −1
ˆI2
=
3¯h2
4
1 0
0 1
(2.48)
This basis is a good choice if the matrix representing Hamiltonian is also diagonal in this basis and
eigenfunctions of ˆIz are the same as eigenfunctions of the Hamiltonian, representing stationary states.
Traditionally, eigenfunctions of ˆIz are written as |α or | ↑ and |β or | ↓ .
ˆIz|α = +
¯h
2
|α ˆIz| ↑ = +
¯h
2
| ↑
¯h
2
1 0
0 −1
1
0
= +
¯h
2
1
0
(2.49)
ˆIz|β = −
¯h
2
|β ˆIz| ↓ = −
¯h
2
| ↓
¯h
2
1 0
0 −1
0
1
= −
¯h
2
0
1
(2.50)
Note that the vectors used to represent |α and |β in Eqs. 2.49 and 2.50 are not the only choice.
Vectors in Eqs. 2.49 and 2.50 have a phase set to zero (they are made of real numbers). Any other
phase φ would work as well, e.g.
1
0
→
eiφ
0
. (2.51)
• If the particle is in state |α , the result of measuring Iz is always +¯h/2. The expected value is
Iz = α|Iz|α = 1 0
¯h
2
1 0
0 −1
1
0
= +
¯h
2
. (2.52)
• If the particle is in state |β , the result of measuring Iz is always −¯h/2. The expected value is
Iz = β|Iz|β = 0 1
¯h
2
1 0
0 −1
0
1
= −
¯h
2
. (2.53)
• Any state cα|α + cβ|β is possible, but the result of a single measurement of Iz is always +¯h/2
or −¯h/2. However, the expected value of Iz is
Iz = α|Iz|β = c∗
α c∗
β
¯h
2
1 0
0 −1
cα
cβ
= (|cα|2
− |cβ|2
)
¯h
2
. (2.54)
Wave functions |α and |β are not eigenfunctions of ˆIx or ˆIy.
The eigenvalues ±¯h/2 are closely related to the fact that spin is a relativistic effect. Special relativity
requires that the Dirac equation must not change if we rotate the coordinate frame or if it moves with
a constant speed (Lorentz transformation). This requirement allows us to determine eigenvalues of the
operators represented by the Pauli matrices:
• We know that the matrices in the Dirac equation do not change if we rotate the coordinate system.
• We know how ∂/∂t, ∂/∂x, ∂/∂y, and ∂/∂z change if we change the coordinate system by a
rotation (or by a boost to a different speed).
• We can calculate how the set of eigenfunctions Ψ change by the rotation (and boost)
20 CHAPTER 2. SINGLE SPIN
• We obtain the following function describing rotation about z:
Rj = ei
Iz,j
¯h
ϕ
2 . (2.55)
This looks very similar to Eq. 1.58, but with one important difference: rotation by 2π (360 ◦
) does
not give the same eigenfunction Rj as no rotation (ϕ = 0), but changes its sign. Only rotation by
4π (720 ◦
) reverts the system to the initial state!
• Eq. 1.58 tells us that the eigenvalues of the operator of the spin angular momentum are half-integer
multiples of ¯h:
Iz,1 =
¯h
2
Iz,2 = −
¯h
2
. (2.56)
2.6 Eigenfunctions of ˆIx and ˆIy
Eigenfunctions of ˆIx are the following linear combinations of |α and |β :
1
√
2
|α +
1
√
2
|β =
1
√
2
1
1
≡ | → (2.57)
−
i
√
2
|α +
i
√
2
|β =
1
√
2
−i
i
≡ | ← (2.58)
or these linear combinations multiplied by a phase factor eiφ
. E.g., | ← can be represented by
eiπ/2 1
√
2
−i
i
= i
1
√
2
−i
i
=
1
√
2
1
−1
. (2.59)
Eigenvalues are again ¯h/2 and −¯h/2:
ˆIx| → = +
¯h
2
| →
¯h
2
0 1
1 0
1
√
2
1
1
= −
¯h
2
·
1
√
2
1
1
(2.60)
ˆIx| ← = +
¯h
2
| ←
¯h
2
0 1
1 0
1
√
2
−i
i
= −
¯h
2
·
1
√
2
−i
i
(2.61)
Eigenfunctions of ˆIy are the following linear combinations of |α and |β :
1 − i
2
|α +
1 + i
2
|β =
1
2
1 − i
1 + i
≡ |⊗ (2.62)
−
1 + i
2
|α +
1 − i
2
|β =
1
2
1 + i
1 − i
≡ | (2.63)
or these linear combinations multiplied by a phase factor eiφ
. E.g., |⊗ can be represented by
eiπ/4 1
2
1 − i
1 + i
=
1 + i
√
2
1
2
1 − i
1 + i
=
1
√
2
1
i
. (2.64)
Eigenvalues are again ¯h/2 and −¯h/2:
ˆIy|⊗ = +
¯h
2
|⊗
¯h
2
0 −i
i 0
1
2
1 − i
1 + i
= +
¯h
2
·
1
2
1 − i
1 + i
(2.65)
ˆIy| = −
¯h
2
|
¯h
2
0 −i
i 0
1
2
1 + i
1 − i
= −
¯h
2
·
1
2
1 + i
1 − i
(2.66)
2.7. OPERATORS OF SPIN MAGNETIC MOMENT 21
2.7 Operators of spin magnetic moment
Similarly to the orbital magnetic moment, the magnetic moment associated with the spin is directly
proportional to the spin angular momentum µ = γI. Therefore, we can write the operators
ˆµx = γ ˆIx ˆµy = γ ˆIy ˆµz = γ ˆIz ˆµ2
= γ2 ˆI2
. (2.67)
However, the value of γ = Q/2m derived for the orbital magnetic moment gives wrong values of
the spin magnetic moment. The correct γ for spin magnetic moment must be derived from relativistic
quantum mechanics (more precisely, from quantum electrodynamics), as shown in the next section.
2.8 Hamiltonian of spin magnetic moment
The classical theory of electromagnetism (Maxwell equations) show that energy and momentum of a
particle in en electromagnetic field must be transformed as follows
E → E − QV p → p − QA, (2.68)
where V is the electric potential and A is a so-called vector potential, related to the magnetic
induction B:
B = × A, (2.69)
= (∂/∂x, ∂/∂y, ∂/∂z).
Accordingly, the operators of energy and momentum change to
i¯h
∂
∂t
→ i¯h
∂
∂t
−QV −i¯h
∂
∂x
→ −i¯h
∂
∂x
−QAx −i¯h
∂
∂y
→ −i¯h
∂
∂y
−QAy −i¯h
∂
∂z
→ −i¯h
∂
∂z
−QAz
(2.70)
This modifies the ˆO+
and ˆO−
in the Dirac equation so that the first two rows of the operator ˆO2
become
i¯h
∂
∂t
− QV
2
− c2
i¯h
∂
∂x
+ QAx
2
− c2
i¯h
∂
∂y
+ QAy
2
− c2
i¯h
∂
∂z
+ QAz
2
− m2
0c4 1 0
0 1
−Qc2
¯hBx
0 1
1 0
− Qc2
¯hBy
0 −i
i 0
− Qc2
¯hBz
1 0
0 −1
(2.71)
i¯h∂/∂t is the operator of the total energy Et = E + m0c2
. Therefore, we can express it as a sum
of the Hamiltonian of the Schr¨odinger equation and the mass term: i¯h∂/∂t = ˆH + m0c2
. Also, we can
replace the Pauli matrices in the second line by the operators of the spin angular momentum:
ˆH + m0c2
− QV
2
− c2
i¯h
∂
∂x
+ QAx
2
− c2
i¯h
∂
∂y
+ QAy
2
− c2
i¯h
∂
∂z
+ QAz
2
− m2
0c4 ˆ1
−2Qc2
¯hBx
ˆIx − 2Qc2
¯hBy
ˆIy − 2Qc2
¯hBz
ˆIz (2.72)
The ˆH + m0c2
− QV
2
term can be expressed as
22 CHAPTER 2. SINGLE SPIN
ˆH + m0c2
− QV
2
= m2
0c4
+ c2 ( ˆH − QV )2
c2
+ 2m0( ˆH − QV ) . (2.73)
If the speed of the particle is much lower than the speed of light, m0c2 ˆH − QV , and the term
divided by c2
can be neglected. Then, the m2
0c4
terms in the expression 2.74 cancel each other and c2
can be factored out:
c2
2m0( ˆH − QV ) − i¯h
∂
∂x
+ QAx
2
− i¯h
∂
∂y
+ QAy
2
− i¯h
∂
∂z
+ QAz
2
ˆ1
−2Qc2
Bx
ˆIx − 2Qc2
By
ˆIy − 2Qc2
Bz
ˆIz (2.74)
Since ˆO2
Ψ = 0, the Hamiltonian ˆH is equal to
ˆH = −
1
2m
i¯h
∂
∂x
+ QAx
2
+ i¯h
∂
∂y
+ QAy
2
+ i¯h
∂
∂z
+ QAz
2
ˆ1 + QV ˆ1
+ 2
Q
2m
Bx
ˆIx + 2
Q
2m
By
ˆIy + 2
Q
2m
Bz
ˆIz
ˆHI
(2.75)
The second line describes the contribution to the Hamiltonian due to the interactions of the spin
magnetic moment with the magnetic field. Comparison with 1.76 shows that
ˆHI = −γBx
ˆIx + By
ˆIy + Bz
ˆIz, (2.76)
where
γ = 2
Q
2m
. (2.77)
2.9 Spin and magnetogyric ratio of real particles
Eq. 2.75, used to derive the value of γ, describes interaction of a particle with an external electromagnetic
field. However, charged particles are themselves sources of magnetic fields. Therefore, γ is not exactly
twice Q/2m. In general, the value of γ is
γ = g
Q
2m
, (2.78)
where the constant g include corrections for interactions of the particle with its own field (and other
effects). For electron, the corrections are small and easy to calculate. The current theoretical prediction
of g = 2.0023318361(10), compared to a recent experimental measured value of g = 2.0023318416(13).
On the other hand, ”corrections” for the constituents of atomic nuclei, quarks, are two orders of magnitude
higher than the basic value of 2! It is because quarks are not ”naked” as electrons, they are
confined in protons and nucleons, ”dressed” by interactions, not only electromagnetic, but mostly strong
nuclear with gluon. Therefore, the magnetogyric ratio of proton is difficult to calculate and we rely on
its experimental value. Everything is even more complicated when we go to higher nuclei, consisting of
multiple protons and neutrons. In such cases, adding spin angular momenta represents another level of
complexity. Fortunately, all equations derived for electron also apply to nuclei with the same eigenvalues
2.10. STATIONARY STATES AND ENERGY LEVEL DIAGRAM 23
of spin magnetic moments (spin-1/2 nuclei), if the value of γ is replaced by the correct value for the
given nucleus.1
2.10 Stationary states and energy level diagram
In the presence of a homogeneous magnetic field B0 = (0, 0, B0), the evolution of the system is given
by the Hamiltonian ˆH = −γB0
ˆIz. The Schr¨odinger equation is then
i¯h
∂
∂t
cα
cβ
= −γB0
¯h
2
1 0
0 −1
cα
cβ
, (2.79)
which is a set of two equations with separated variables
dcα
dt
= +i
γB0
2
cα (2.80)
dcβ
dt
= −i
γB0
2
cβ (2.81)
with the solution
cα = cα(t = 0)e+i
γB0
2 t
= cα(t = 0)e−i
ω0
2 t
(2.82)
cβ = cβ(t = 0)e−i
γB0
2 t
= cβ(t = 0)e+i
ω0
2 t
. (2.83)
If the initial state is |α , cα(t = 0) = 1, cβ(t = 0) = 0, and
cα = e−i
ω0
2 t
(2.84)
cβ = 0. (2.85)
Note that the evolution changes only the phase factor, but the system stays in state |α (all vectors
described by Eq. 2.51 correspond to state |α ). It can be shown by calculating the probability that the
system is in the |α or |β state.
Pα = c∗
αcα = e+i
ω0
2 t
e−i
ω0
2 t
= 1 (2.86)
Pβ = c∗
βcβ = 0 (2.87)
If the initial state is |β , cα(t = 0) = 0, cβ(t = 0) = 1, and
cα = 0 (2.88)
cβ = e+i
ω0
2 t
. (2.89)
Again, the evolution changes only the phase factor, but the system stays in state |β . The probability
that the system is in the |α or |β state is
Pα = c∗
αcα = 0 (2.90)
Pβ = c∗
βcβ = e−i
ω0
2 t
e+i
ω0
2 t
= 1 (2.91)
1NMR in organic chemistry and biochemistry is usually limited to spin-1/2 nuclei because signal decays too fast if the
spin number is grater than 1/2.
24 CHAPTER 2. SINGLE SPIN
• The states described by basis functions which are eigenfunctions of the Hamiltonian do not evolve
(are stationary). It makes sense to draw energy level diagram for such states, with energy of
each state given by the corresponding eigenvalue of the Hamiltonian. Energy of the |α state
is −¯hω0/2 = and energy of the |β state is +¯hω0/2 =. The measurable quantity is the energy
difference ¯hω0, corresponding to the angular frequency ω0.
2.11 Oscillatory states
In the presence of a homogeneous magnetic field B1 = (B1, 0, 0), the evolution of the system is given
by the Hamiltonian ˆH = −γB0
ˆIx. The Schr¨odinger equation is then
i¯h
∂
∂t
cα
cβ
= −γB1
¯h
2
0 1
1 0
cα
cβ
, (2.92)
which is a set of two equations
dcα
dt
= i
γB1
2
cβ (2.93)
dcβ
dt
= i
γB1
2
cα (2.94)
These equations have similar structure as Eqs. 1.63 and 1.64. Adding and subtracting them leads
to the solution
cα + cβ = C+e+i
γB1
2 t
= C+e−i
ω1
2 t
(2.95)
cα − cβ = C−e−i
γB1
2 t
= C−e+i
ω1
2 t
. (2.96)
If the initial state is |α , cα(t = 0) = 1, cβ(t = 0) = 0, C+ = C− = 1, and
cα = cos
ω1
2
t (2.97)
cβ = −i sin
ω1
2
t . (2.98)
Probability that the system is in the |α or |β state is calculated as
Pα = c∗
αcα = cos2 ω1
2
t =
1
2
+
1
2
cos(ω1t) (2.99)
Pβ = c∗
βcβ = sin2 ω1
2
t =
1
2
−
1
2
cos(ω1t) (2.100)
If the initial state is |β , cα(t = 0) = 0, cβ(t = 0) = 1, C+ = 1, C− = −1, and
cα = −i sin
ω1
2
t (2.101)
cβ = cos
ω1
2
t . (2.102)
Probability that the system is in the |α or |β state is calculated as
2.11. OSCILLATORY STATES 25
Pα = c∗
αcα = sin2 ω1
2
t =
1
2
+
1
2
cos(ω1t) (2.103)
Pβ = c∗
βcβ = cos2 ω1
2
t =
1
2
−
1
2
cos(ω1t) (2.104)
In both cases, the system oscillates between the |α and |β states.
• The states described by basis functions different from eigenfunctions of the Hamiltonian are not
stationary but oscillate between |α and |β with the angular frequency ω1, given by the difference
of the eigenvalues of the Hamiltonian (−¯hω1/2 and ¯hω1/2).
26 CHAPTER 2. SINGLE SPIN
Chapter 3
Ensembles of spins not interacting
with other spins
3.1 Mixed state
So far, we worked with systems in so-called pure states, when we described the whole studied system
by its complete wave function. It is fine if the system consists of one particle or a small number of
particles. However, the complete wave function of whole molecules (or ensembles of whole molecules)
is very complicated, represented by multidimensional vectors and in properties described by operators
represented by multidimensional matrices. In NMR spectroscopy, we are interested only with properties
of molecules associated with spins of the observed nuclei. If we assume motions of the whole molecule,
of its atoms, and of electrons and nuclei in the atoms, do not depend on the spin, we can divide the
complete wave function into spin wave functions and wave function describing all the other degrees
of freedom. The result of this division is that spin wave functions for different molecules are not
identical. Therefore, the spin wave function describing the whole set of nuclei in different molecules
is represented by multidimensional vectors and with properties described by operators represented by
multidimensional matrices. This can be simplified dramatically if
1. the measured quantity does not depend on other coordinates that spin coordinates α or β –
true for magnetization in homogeneous magnetic fields (contributions of individual nuclei to the
magnetization then do not depend on their positions is space)
2. the interactions of the observed magnetic moments change only eigenvalues, not eigenfunctions –
true for interactions with fields which can be described without using spin eigenfunctions
Using the same basis for different nuclei ⇒ multidimensional operator matrices → two-dimensional
operator matrices (for spin-1/2 nuclei).
Expected value A of a quantity A for a single nucleus can be calculated using Eq.1.17 as a trace
of the following product of matrices:
A =
cαc∗
α cαc∗
β
cβc∗
α cβc∗
β
A11 A12
A21 A22
(3.1)
Expected value A of a quantity A for multiple nuclei with the same basis is
A =
cα,1c∗
α,1 cα,1c∗
β,1
cβ,1c∗
α,1 cβ,1c∗
β,1
A11 A12
A21 A22
+
cα,2c∗
α,2 cα,2c∗
β,2
cβ,2c∗
α,2 cβ,2c∗
β,2
A11 A12
A21 A22
+ · · ·
27
28 CHAPTER 3. ENSEMBLES OF SPINS NOT INTERACTING WITH OTHER SPINS
=
cα,1c∗
α,1 cα,1c∗
β,1
cβ,1c∗
α,1 cβ,1c∗
β,1
+
cα,2c∗
α,2 cα,2c∗
β,2
cβ,2c∗
α,2 cβ,2c∗
β,2
+ · · ·
A11 A12
A21 A22
= N
cαc∗
α cαc∗
β
cβc∗
α cβc∗
β
ˆρ
A11 A12
A21 A22
ˆA
= N ˆρ ˆA (3.2)
The matrix ˆρ is the (probability) density matrix, the horizontal bar indicates average over the whole
ensemble of nuclei in the sample.
Why probability density? Because the probability P = Ψ|Ψ ⇒ the operator of probability is the
unit matrix ˆ1: Ψ|Ψ ≡ Ψ|ˆ1|Ψ . Therefore, the expectation value of probability can be also calculated
using Eq.1.17 as Tr{ˆρˆ1} = Tr{ˆρ}.
• Two-dimensional basis is sufficient for the whole set of N nuclei (if they do not interact with each
other).
• Statistical approach: the possibility to use a 2D basis is paid by loosing the information about
the microscopic state. The same density matrix can describe an astronomic number of possible
combinations of individual angular momenta which give the same macroscopic result. What is
described by the density matrix is called the mixed state.
• Choice of the basis is encoded in the definition of ˆρ (eigenfunctions of ˆIz).
• The state is described not by a vector, but by a matrix, ˆρ is a matrix like operators.
• Any 2×2 matrix can be written as a linear combination of four 2×2 matrices. Such four matrices
can be used as a basis of all 2 × 2 matrices, including operators (in the same manner as two
selected 2-component vectors serve as a basis for all 2-component vectors).
• Good choice of a basis is a set of orthonormal matrices.1
• Diagonal elements of ˆρ (or matrices with diagonal elements only) are known as populations. They
tell what populations of pure α and β states would give the same polarization along z.
• Off-diagonal elements (or matrices with diagonal elements only) are known as coherences. They
tell what combinations of coefficients cα and cβ would give the same coherence of phases of the
rotation about z.
3.2 Coherence
• Coherence is a very important issue in NMR
• In a pure state, cαc∗
β is given by amplitudes and by the difference of phases of cα and cβ: cαc∗
β =
|cα||cβ|e−i(φα−φβ )
.
• In a mixed state, cα,j and cβ,j is different for the observed nucleus in each molecule j. If cα,j and
cβ,j describe stationary states, only phases of cα,j and cβ,j change as the system evolves. Therefore,
cαc∗
β = |cα||cβ|·e−i(φα−φβ ). The phase of cαc∗
β is given by e−i(φα−φβ ). If the evolution of phases is
1Orthonormality for a set of four matrices ˆA1, ˆA2, ˆA3, ˆA4 can be defined as Tr{ ˆA†
j
ˆAk} = δj,k, where j and k ∈
{1, 2, 3, 4}, δj,k = 1 for j = k and δj,k = 0 for j = k, and ˆA†
j is an adjoint matrix of ˆAj, i.e., matrix obtained from ˆAj by
exchanging rows and columns and replacing all numbers with their complex conjugates.
3.3. BASIS SETS 29
coherent, φα,j and φβ,j vary but φα,j − φβ,j is constant. In such a case, cαc∗
β = |cα||cβ|ei(φα−φβ )
.
However, if the phases φα,j and φβ,j evolve independently, e−i(φα−φβ ) = e−iφα ·eiφβ = 0·0 (because
φα,j and φβ,j can be anywhere between 0 and 2π and the average value of both real component
cos(φα,j) and imaginary component sin(φα,j) of eiφα,j
in the interval (0, 2π) is zero). Obviously,
cαc∗
β = 0 in such a case.
3.3 Basis sets
Usual choices of basis matrices are:
• Cartesian operators, equal to the operators of spin angular momentum divided by ¯h. In this text,
these matrices are written as Ix, Iy, etc. In a similar fashion, we write H = ˆH/¯h for Hamiltonians
with eigenvalues expressed in units of (angular) frequency, not energy. The normalization factor√
2 is often omitted (then the basis is still orthogonal, but not orthonormal).
√
2It =
1
√
2
1 0
0 1
√
2Iz =
1
√
2
1 0
0 −1
√
2Ix =
1
√
2
0 1
1 0
√
2Iy =
1
√
2
0 −i
i 0
(3.3)
• Single-element population
Iα = It + Iz =
1 0
0 0
Iβ = It − Iz =
0 0
0 1
(3.4)
and transition operators
I+ = Ix + iIy =
0 1
0 0
I− = Ix − iIy =
0 0
1 0
(3.5)
•
√
2It =
1
√
2
1 0
0 1
√
2Iz =
1
√
2
1 0
0 −1
I+ =
0 1
0 0
I− =
0 0
1 0
(3.6)
3.4 Equation of motion: Liouville-von Neumann equation
In order to describe the evolution of mixed states in time, we must find an equation describing how
elements of the density matrix change in time.
We start with the Schr¨odinger equation for a single spin in matrix representation:
i¯h
d
dt
cα
cβ
=
Hα,α Hα,β
Hβ,α Hβ,β
cα
cβ
=
Hα,αcα + Hα,βcβ
Hβ,αcα + Hβ,βcβ
. (3.7)
Note that the Hamiltonian matrix is written in a general form, the basis functions are not necessarily
eigenfunctions of the operator. However, the matrix must be Hermitian, i.e., Hj,k = H∗
k,j:
Hα,β = H∗
β,α Hβ,α = H∗
α,β. (3.8)
If we multiply Eq. 3.7 by the basis functions from left, we obtained the differential equations for cα
and cβ (because the basis functions are orthonormal):
30 CHAPTER 3. ENSEMBLES OF SPINS NOT INTERACTING WITH OTHER SPINS
( 1 0 )i¯h
d
dt
cα
cβ
= i¯h
dcα
dt
= Hα,αcα + Hα,βcβ (3.9)
( 0 1 )i¯h
d
dt
cα
cβ
= i¯h
dcβ
dt
= Hβ,αcα + Hβ,βcβ. (3.10)
In general,
dck
dt
= −
i
¯h
l
Hk,lcl (3.11)
and its complex conjugate (using Eq. 3.8) is
dc∗
k
dt
= +
i
¯h
l
H∗
k,lc∗
l = +
i
¯h
l
Hl,kc∗
l . (3.12)
Elements of the density matrix consist of the products cjc∗
k. Therefore, we must calculate
dcjc∗
k
dt
= cj
dc∗
k
dt
+ c∗
k
dcj
dt
=
i
¯h
l
Hl,kcjc∗
l −
i
¯h
l
Hj,lclc∗
k (3.13)
For multiple nuclei with the same basis,
d(cj,1c∗
k,1 + cj,2c∗
k,2 + · · ·)
dt
= cj,1
dc∗
k,1
dt
+ c∗
k,1
dcj,1
dt
+ cj,2
dc∗
k,2
dt
+ c∗
k,2
dcj,2
dt
+ · · · (3.14)
=
i
¯h
l
Hl,k(cj,1c∗
l,1 + cj,2c∗
l,2 + · · ·) −
i
¯h
l
Hj,l(cl,1c∗
k,1 + cl,2c∗
k,2 + · · ·) (3.15)
Note that
l
(cj,1c∗
l,1 + cj,2c∗
l,2 + · · ·)Hl,k = N
l
ρj,lHl,k (3.16)
is the j, k element of the product N ˆρ ˆH, and
l
Hj,l(cl,1c∗
k,1 + cl,2c∗
k,2 + · · ·) = N
l
Hj,lρl,k (3.17)
is the j, k element of the product N ˆH ˆρ. Therefore, we can write the equation of motion for the
whole density matrix as
dˆρ
dt
=
i
¯h
(ˆρ ˆH − ˆH ˆρ) =
i
¯h
[ˆρ, ˆH] = −
i
¯h
[ ˆH, ˆρ] (3.18)
or in the units of (angular) frequency
dˆρ
dt
= i(ˆρH − Hˆρ) = i[ˆρ, H] = −i[H, ˆρ] (3.19)
Eqs. 3.18 and 3.19 are known as the Liouville-von Neumann eqaution.
3.5. ROTATION IN OPERATOR SPACE 31
3.5 Rotation in operator space
Liouville-von Neumann equation can be solved using techniques of linear algebra. However, a very
simple geometric solution is possible, if the Hamiltonian does not change in time and consists solely of
matrices which commute (e.g., It and Iz, but not Ix and Iz).
Example for H = εIt + ω0Iz and ˆρ = cxIx + cyIy + czIz + ctIt:
Let’s first evaluate the commutators from the Liouville-von Neumann equation:
It is proportional to a unit matrix ⇒ it must commute with all matrices:
[It, Ij] = 0 (j = x, y, z, t). (3.20)
Commutators of Iz are given by the definition of angular momentum operators:
[Iz, Iz] = [Iz, It] = 0 [Iz, Ix] = iIy [Iz, Iy] = −iIx. (3.21)
Let’s write the Liouville-von Neumann equation with the evaluated commutators:
dcx
dt
Ix +
dcy
dt
Iy +
dcz
dt
Iz +
dct
dt
It = −iω0cxIy + iω0cyIx. (3.22)
Written in a matrix representation (noticing that cz and ct do not evolve because the czIz and ctIt
components of the density matrix commute with both matrices constituting the Hamiltonian),
1
2
0 dcx
dt
dcx
dt 0
+
1
2
0 −i
dcy
dt
i
dcy
dt 0
+ 0 + 0 =
1
2
0 −ω0cx
ω0cx 0
−
1
2
0 iω0cy
iω0cy 0
. (3.23)
This corresponds to a set of two differential equations
dcx
dt
= −iω0cy (3.24)
dcy
dt
= +iω0cx (3.25)
with the same structure as Eqs. 1.63 and 1.64. The solution is
cx = c0 cos(ω0t + φ0) (3.26)
cy = c0 sin(ω0t + φ0) (3.27)
with the amplitude c0 and phase φ0 given by the initial conditions.
We see that coefficients cx, cy, cz play the same roles as coordinates rx, ry, rz in Eqs. 1.63–1.65,
respectively, and operators Ix, Iy, Iz play the same role as unit vectors ı, , k, defining directions of the
axes of the Cartesian coordinate system. Therefore, the evolution of ˆρ can be described as a rotation
in an abstract three-dimensional operator space with the dimensions given by Ix, Iy, and Iz.
3.6 General strategy of analyzing NMR experiments
The Liouville-von Neumann equation is the most important tool in the analysis of evolution of the spin
system during the NMR experiment. The general strategy consists of three steps:
1. Define ˆρ at t = 0
32 CHAPTER 3. ENSEMBLES OF SPINS NOT INTERACTING WITH OTHER SPINS
2. Describe evolution of ˆρ using the relevant Hamiltonians – this is usually done in several steps
3. Calculate the expectation value M of the measured quantity according to Eq. 1.17
Obviously, the procedure requires knowledge of
1. relation(s) describing the initial state of the system (ˆρ(0))
2. all Hamiltonians
3. the operator representing the measurable quantity
Here, we start from the end and define first the operator of the measurable quantity. Then we spend
a lot of time defining all necessary Hamiltonians. Finally, we use the knowledge of the Hamiltonians
and basic thermodynamics to describe the initial state.
3.7 Operator of the observed quantity
The quantity observed in the NMR experiment is the total magnetization, i.e., the sum of magnetic moments
of all nuclei. Technically, we observe oscillations in the plane perpendicular to the homogeneous
field of the magnet B0. The associated oscillations of the magnetic fields of nuclei induce electromotive
force in the detector coil. Since a complex signal is usually recorded, the operator of complex
magnetization M+ = Mx + iMy is used (M− = Mx − iMy can be used as well).
ˆM+ = N
n
γn(ˆIx,n − iˆIx,n) = N
n
γn
ˆI+,n, (3.28)
where the index n distinguishes different types of nuclei and N is the number of nuclei of each type
in the sample.
3.8 Static field B0
We already defined the Hamiltonian of the static homogeneous magnetic field B0, following the classical
description of energy of a magnetic moment in a magnetic field. Since B0 defines direction of the z axis,
ˆH0,lab = −γB0
ˆIz. (3.29)
3.9 Radio-frequency field B1
The oscillating magnetic field of radio waves irradiating the sample is formally decomposed into two
rotating magnetic fields (with the same speed given by the frequency of the radio waves ωradio, but with
opposite sense of rotation). The component resonating (approximately) with the precession frequency
of the observed nuclei usually defines the x axis of the rotating coordinate frame used most often in
NMR spectroscopy. In this system, frequency of the resonating component2
is subtracted from the
precession frequency and the difference Ω = ω0 − ωrot = −γB0 − ωrot is the frequency offset defining
the evolution in the rotating frame in the absence of other fields:
In the absence of other fields than B0:
ˆH0,rot = (−γB0 − ωrot)ˆIz = ΩˆIz. (3.30)
2Formally opposite to ωradio
3.10. PHENOMENOLOGY OF CHEMICAL SHIFT 33
During irradiation by waves with the phase defining x
ˆH1,rot = (−γB0 − ωrot)ˆIz − γB1
ˆIx = ΩˆIz + ω1
ˆIy. (3.31)
During irradiation by waves shifted by π/2 from the phase defining x
ˆH1,rot = (−γB0 − ωrot)ˆIz − γB1
ˆIy = ΩˆIz + ω1
ˆIy. (3.32)
If the radio frequency is close to resonance, −γB0 ≈ ωrot, Ω ω1, and the ˆIz component of the
Hamiltonian can be neglected.
3.10 Phenomenology of chemical shift
The energy of the magnetic moment of the observed nucleus is influenced by magnetic fields associated
with motions of nearby electrons. Before we write the Hamiltonian describing this contribution to the
energy of the system, we describe the magnetic fields of moving electrons.
If a moving electron enters a homogeneous magnetic field, it experiences a Lorentz force and moves
in a circle in a plane perpendicular to the field (cyclotron motions). Such an electron represents an
electric current in a circular loop, and is a source of a magnetic field induced by the homogeneous
magnetic field. The homogeneous magnetic field B0 in NMR spectrometers induces a similar motion of
electrons in atoms, which generates microscopic magnetic fields.
The observed nucleus feels the external magnetic field B0 slightly modified by the microscopic fields
of electrons.
If the electron distribution is spherically symmetric, with the observed nucleus in the center (e.g.
electrons in the 1s orbital of the hydrogen atom), the induced field of the electrons decreases the effective
magnetic field felt by the nucleus in the center. Since the induced field of electrons is proportional to
the inducing external field B0, the effective field can be described as
B = B0 + Be = (1 + δ)B0. (3.33)
The constant δ is known as chemical shift and does not depend on the orientation of the molecule
in such a case. The precession frequency of the nucleus is equal to (1 + δ)ω0
Electron distribution is not spherically symmetric in most molecules. As a consequence, the effective
field depends on the orientation of the whole molecule and on mutual orientations of atoms, defining the
shapes of molecular orbitals. Therefore, the effective field fluctuates as a result of rotational diffusion of
the molecule and of internal motions changing mutual positions of atoms. The induced field of electrons
is still proportional to the inducing external field B0, but the proportionality constants are different for
each combination of components of Be and B0 in the coordination frame used. Therefore, we need six3
constants δjk to describe the effect of electrons:
Be,x = δxxB0,x + δxyB0,y + δxzB0,z (3.34)
Be,y = δyxB0,x + δyyB0,y + δyzB0,z (3.35)
Be,z = δzxB0,x + δzyB0,y + δzzB0,z (3.36)
Eqs. 3.34–3.36 can be written in more compact forms


Be,x
Be,y
Be,z

 =


δxx δxy δxz
δyx δyy δyz
δzx δzy δzz

 ·


B0,x
B0,y
B0,z

 (3.37)
3There are nine constants in Eqs. 3.34–3.36, but δxy = δyx, δxz = δzx, and δyz = δzy.
34 CHAPTER 3. ENSEMBLES OF SPINS NOT INTERACTING WITH OTHER SPINS
or
Be = δ · B0, (3.38)
where δ is the chemical shift tensor.
It is always possible to find a coordinate system X, Y, Z known as the principal frame, where δ is
represented by a diagonal matrix. In such a system, we need only three constants (principal values of
the chemical shift tensor): δXX, δY Y , δZZ. However, three more parameters must be specified: three
Euler angles (written as ϕ, ϑ, and ψ in this text) defining orientation of the coordinate system X, Y, Z
in the laboratory coordinate system x, y, z. Note that δXX, δY Y , δZZ are true constants because they
do not change as the molecule tumbles in solution (but they may change due to internal motions or
chemical changes of the molecule). The orientation is completely described by the Euler angles.
The chemical shift tensor in its principal frame can be also written as a sum of three simple matrices,
each multiplied by one characteristic constant:


δXX 0 0
0 δY Y 0
0 0 δZZ

 = δi


1 0 0
0 1 0
0 0 1

 + δa


−1 0 0
0 −1 0
0 0 2

 + δr


1 0 0
0 −1 0
0 0 0

 , (3.39)
where
δi =
1
3
Tr{δ} =
1
3
(δXX + δY Y + δZZ) (3.40)
is the isotropic component of the chemical shift tensor,
δa =
1
3
∆δ =
1
6
(2δZZ − (δXX + δY Y )) (3.41)
is the axial component of the chemical shift tensor (∆δ is the chemical shift anisotropy), and
δr =
1
3
ηδ∆δ
1
2
(δXX − δY Y ) (3.42)
is the rhombic component of the chemical shift tensor (ηδ is the asymmetry of the chemical shift
tensor).
The chemical shift tensor written in its principle frame is relatively simple, but we need its description
in the laboratory coordinate frame. Changing the coordinate systems represents a rotation in a threedimensional
space. Equations describing such a simple operation are relatively complicated. On the
other hand, the equations simplify if B0 defines the z axis of the coordinate frame:
Be = δiB0


1
1
1

+δaB0


3 sin ϑ cos ϑ cos ϕ
3 sin ϑ cos ϑ sin ϕ
3 cos2
ϑ − 1

+δrB0


−(2 cos2
ψ − 1) sin ϑ cos ϑ cos ϕ + 2 sin ψ cos ψ sin ϑ sin ϕ
−(2 cos2
ψ − 1) sin ϑ cos ϑ sin ϕ − 2 sin ψ cos ψ sin ϑ cos ϕ
+(2 cos2
ψ − 1) sin2
ϑ

 .
(3.43)
The first, isotropic contribution does not change upon rotation (it is a scalar). The second, axial
contribution, is insensitive to the rotation about the symmetry axis a, described by ψ. Rotation of the
chemical shift anisotropy tensor from its principal frame to the laboratory frame can be also described
by orientation of a in the laboratory frame:
δa


−1 0 0
0 −1 0
0 0 2

 −→


3a2
x − a2
3axay 3axaz
3axay 3a2
y − a2
3ayaz
3axaz 3ayaz 3a2
z − a2

 , (3.44)
where ax = δa sin ϑ cos ϕ, ay = δa sin ϑ sin ϕ, and az = δa cos ϑ.
3.11. HAMILTONIAN OF CHEMICAL SHIFT 35
3.11 Hamiltonian of chemical shift
Once the magnetic fields of moving electrons are described, definition of the chemical shift Hamiltonian
is straightforward:
ˆHδ = −γ(ˆIxBe,x + ˆIyBe,y + ˆIzBe,z) = −γ( ˆIx
ˆIy
ˆIz )


Be,x
Be,y
Be,z

 =
= −γ( ˆIx
ˆIy
ˆIz )


δxx δxy δxz
δyx δyy δyz
δzx δzy δzz




B0,x
B0,y
B0,z

 = −γ
ˆ
I · δ · B (3.45)
The Hamiltonian can be decomposed into
• isotropic contribution, independent of rotation in space:
ˆHδ,i = −γδiB0(ˆIx + ˆIy + ˆIz) (3.46)
• axial component, dependent on ϕ and ϑ:
ˆHδ,a = −γδaB0(3 sin ϑ cos ϑ cos ϕˆIx + 3 sin ϑ cos ϑ sin ϕˆIy + (3 cos2
ϑ − 1)ˆIz)
= −γB0(3axaz
ˆIx + 3ayaz
ˆIy + (3a2
z − a2
)ˆIz) (3.47)
• rhombic component, dependent on ϕ, ϑ, and ψ:
ˆHδ,r = −γδrB0( (−(2 cos2
ψ − 1) sin ϑ cos ϑ cos ϕ + 2 sin ψ cos ψ sin ϑ sin ϕ)ˆIx +
(−(2 cos2
ψ − 1) sin ϑ cos ϑ sin ϕ − 2 sin ψ cos ψ sin ϑ cos ϕ)ˆIy +
((2 cos2
ψ − 1) sin2
ϑ)ˆIz) =
γB0( (cos(2ψ)ax − sin(2ψ)ay)az
ˆIx + (cos(2ψ)ay + sin(2ψ)ax)az
ˆIy + cos(2ψ)(a2
z − a2
)ˆIz)
(3.48)
The complete Hamiltonian of a magnetic moment of a nucleus not interacting with magnetic moments
of other nuclei in the presence of the static field B0 but in the absence of the radio waves is given
by
ˆH = ˆH0,lab + ˆHδ,i + ˆHδ,a + ˆHδ,r. (3.49)
3.12 Secular approximation and averaging
The Hamiltonian ˆH0,lab + ˆHδ,i + ˆHδ,a + ˆHδ,r is complicated, but can be simplified in many cases.
• The components of the induced fields Be,x and Be,y are perpendicular to B0. The contributions
of ˆHδ,i are constant and the contributions of ˆHδ,a and ˆHδ,r fluctuate with the molecular motions
changing values of ϕ, ϑ, and ψ. Since the molecular motions do not resonate (in general) with
the precession frequency −γB0, the components ˆIxBe,x and ˆIyBe,y of the Hamiltonian oscillate
rapidly with a frequency close to −γB0. These oscillations are much faster than the precession
about Be,x and Be,y (because the field Be is much smaller than B0) and effectively average to
36 CHAPTER 3. ENSEMBLES OF SPINS NOT INTERACTING WITH OTHER SPINS
zero on the timescale given by 1/(γB0) (typically nanoseconds). Therefore, the ˆIxBe,x and ˆIyBe,y
terms can be neglected if the effects on longer timescales are studied. Such a simplification is
known as secular approximation.4
The secular approximation simplifies the Hamiltonian to
H = −γB0(1 + δi + (3 cos2
ϑ − 1)δa + cos(2ψ) sin2
ϑδr)ˆIz (3.50)
• If the sample is an isotropic liquid, averaging over all molecules of the sample further simplifies the
Hamiltonian. As no orientation of the molecule is preferred, all values of ψ are equally probable
and independent of ϑ. Therefore, the last term in Eq. 3.50 is averaged to zero. Moreover, average
values of a2
x = cos2
ϕ sin2
ϑ, of a2
y = sin2
ϕ sin2
ϑ, and of a2
z = cos2
ϑ must be the same because
none of the directions x, y, z is preferred:
a2
x = a2
y = a2
z. (3.51)
Finally,
a2
x + a2
y + a2
z = a2
⇒ a2
x + a2
y + a2
z = 3a2
z ⇒ 3a2
z − a2 = (3 cos2 ϑ − 1)δa = 0, (3.52)
and Eq. 3.50 reduces to
H = −γB0(1 + δi)ˆIz. (3.53)
Note that the described simplifications can be used only if they are applicable. Eq. 3.53 is valid only
in isotropic liquids, not in liquid crystals, stretched gels, polycrystalline powders, monocrystals, etc.!
3.13 Thermal equilibrium as the initial state
Knowledge of the Hamiltonian allows us to derive the density matrix at the beginning of the experiment.
Usually, we start from the thermal equilibrium. If the equilibrium is achieved, phases of individual
magnetic moments are random and the magnetic moments precess incoherently. Therefore, the offdiagonal
elements of the equilibrium density matrix (proportional to Ix and Iy) are equal to zero.
Populations of the states can be evaluated using statistical arguments similar to the Boltzmann law
in the classical molecular statistics:
Peq
α =
e−Eα/kBT
e−Eα/kBT + e−Eβ /kBT
(3.54)
Peq
β =
e−Eβ /kBT
e−Eα/kBT + e−Eβ /kBT
, (3.55)
where kB = 1.38064852 × 10−23
m2
kg s−2
K−1
is the Boltzmann constant.
The energies Eα and Eβ are the eigenvalues of the energy operator, the Hamiltonian. Since we use
eigenfunctions of ˆIz as the basis, eigenfunctions of H = −γB0(1 + δi)ˆIz are the diagonal elements of
the matrix representation of ˆH:
4In terms of quantum mechanics, eigenfunctions of ˆIxBe,x and ˆIyBe,y differ from the eigenfunctions of ˆH0,lab (|α and
|β ). Therefore, the matrix representation of ˆIxBe,x and ˆIyBe,y contains off-diagonal elements. Terms proportional to
ˆIz represent so-called secular part of the Hamiltonian, which does not change the |α and |β states (because they are
eigenfunctions of ˆIz). Terms proportional to ˆIx and ˆIy are non-secular because they change the |α and |β states (|α
and |β are not eigenfunctions of ˆIx or ˆIy). However, eigenvalues of ˆIxBe,x and ˆIyBe,y, defining the off-diagonal elements,
are much smaller than the eigenvalues of ˆH0,lab. Secular approximation represents neglecting such small off-diagonal
elements in the matrix representation of the total Hamiltonian and keeping only the diagonal secular terms.
3.14. RELAXATION DUE TO CHEMICAL SHIFT ANISOTROPY 37
ˆH = −γB0(1 + δi)ˆIz = −γB0(1 + δi)
¯h
2
1 0
0 −1
=
−γB0(1 + δi)¯h
2 0
0 +γB0(1 + δi)¯h
2
(3.56)
The thermal energy at 0 ◦
C is more than 12 000 times higher than γB0¯h/2 for the most sensitive
nuclei (protons) at spectrometers with the highest magnetic fields (1 GHz). The effect of chemical shift
is four orders of magnitude lower. Therefore, we can approximate
e
±
γB0(1+δi)¯h
kBT
≈ 1 ±
γB0¯h
2kBT
(3.57)
and calculate the populations as
Peq
α =
e−Eα/kBT
e−Eα/kBT + e−Eβ /kBT
=
1 + γB0¯h
2kBT
1 + γB0¯h
2kBT + 1 − γB0¯h
2kBT
=
1 + γB0¯h
2kBT
2
(3.58)
Peq
β =
e−Eβ /kBT
e−Eα/kBT + e−Eβ /kBT
=
1 − γB0¯h
2kBT
1 + γB0¯h
2kBT + 1 − γB0¯h
2kBT
=
1 − γB0¯h
2kBT
2
. (3.59)
Writing the populations as the diagonal elements, the equilibrium density matrix is
ˆρeq
=
1
2 + γB0¯h
4kBT 0
0 1
2 − γB0¯h
4kBT
=
1
2
1 0
0 1
+
γB0¯h
4kBT
1 0
0 −1
= It + κIz, (3.60)
where
κ =
γB0¯h
2kBT
. (3.61)
Note that we derived the quantum description of a mixed state. Two populations of the density
matrix provide correct results but do not tell us anything about microscopic states of individual magnetic
moments. Two-dimensional density matrix does not imply that all magnetic moments are in one of two
eigenstates.
3.14 Relaxation due to chemical shift anisotropy
The averaged Hamiltonian allowed us to describe the state of the system in thermal equilibrium, but
it does not tell us how is the equilibrium reached. The processes leading to the equilibrium states are
known as relaxation. Description of relaxation represents an example of analysis when the complete
Hamiltonian must be used and when Liouville-von Neumann equation cannot be solved simply as
rotation in an operator space.
Relaxation is return of a system to thermodynamic equilibrium. It takes places e.g. when the sample
is placed into a magnetic field inside the spectrometer or after excitation of the sample by radio wave
pulses.
Spontaneous emission is completely inefficient (due to low energy differences of spin states).
Relaxation in NMR is due to interactions with local fluctuating magnetic fields in the molecule.
One source of fluctuating fields is the anisotropy of chemical shift, described by the axial and rhombic
components of the chemical shift tensor. As the molecule moves, the isotropic component of the chemical
shift tensor does not change because it is spherically symmetric. However, contributions to the local
38 CHAPTER 3. ENSEMBLES OF SPINS NOT INTERACTING WITH OTHER SPINS
fields described by the axial and rhombic components fluctuate even if the constants δa do not change
because the axial part of the chemical shift depends on the orientation of the molecule.
Theoretical description of relaxation is relatively complicated because we cannot neglect the fluctuating
components of the Hamiltonian. Therefore, we first introduce the basic idea by analyzing only
one relaxation effect in a classical manner.
3.14.1 Classical analysis: fluctuations B0 and loss of coherence
Motion of a magnetic moment in a magnetic filed is described classically as
dµ
dt
= ω × µ = −γB × µ, (3.62)
or for individual components:
dµx
dt
= ωyµz − ωzµy (3.63)
dµy
dt
= ωzµx − ωxµz (3.64)
dµz
dt
= ωxµy − ωyµx (3.65)
In this section, we look how fluctuations of Bz affect an ensemble of magnetic moments rotating
coherently about B0 (for the sake of simplicity, let’s assume that we observe only one nucleus in
each molecule). As the precession frequency of magnetic moments is given by the z-component of the
magnetic field (equal to B0 in the absence of radio waves and microscopic fields of the molecule), we can
expect that fluctuations of this component (due to the presence of the microscopic fields of the molecule)
result in fluctuations of the precession frequency. As a consequence, the ensemble of magnetic moments
that originally precessed coherently (with the same frequency) will loose the coherence. This loss of
coherence is manifested as a loss of the macroscopic magnetization in the plane perpendicular to B0.
Let’s now complement the qualitative description with a quantitative analysis. Evolution of each
individual magnetic moment of the ensemble can be described as
dµx
dt
= −ωzµy = γBzµy (3.66)
dµy
dt
= ωzµx = −γBzµx (3.67)
dµz
dt
= 0 (3.68)
Eqs. 3.66–3.68 are very similar to Eqs. 1.63–1.65, so we try the same approach and calculate
dµ+
dt
≡
d(µx + iµy)
dt
= iωz(µx + iµy) = −iγBz(µx + iµy) (3.69)
According to Eq. 3.43,
Bz = B0 + Be,z = B0(1 + δi + δa(3 cos2
ϑ − 1) + δr(2 cos2
ψ − 1) sin2
ϑ). (3.70)
For the sake of simplicity, we assume that the chemical shift tensor is axially symmetric (δr = 0).
Then, ωz can be written as
3.14. RELAXATION DUE TO CHEMICAL SHIFT ANISOTROPY 39
ωz = B0 + Be,z = −γB0(1 + δi) − γB0δa(3 cos2
ϑ − 1) = −(a + bc), (3.71)
where
a = −γB0(1 + δi) (3.72)
b = −2γB0δa (3.73)
c =
3 cos2
ϑ − 1
2
. (3.74)
Note that Eq. 3.69 cannot be solved as easily as we solved 1.63–1.65 because ωz is not constant but
fluctuates in time. But we can assume, that for a very short time ∆t, shorter than the time scale of
molecular motions, the orientation of the molecule does not change and c remains constant. We try
describe evolution of µ+
in such small time steps, when
∆µ+
∆t
≈
dµ+
dt
≈ −i(a + bc)∆µ+
(3.75)
If the initial value of µ+
is µ+
0 and if the values of a, b, c during the first time step are a1, b1, c1,
respectively, µ+
after the first time step is
µ+
1 = µ+
0 + ∆µ+
1 = µ+
0 − i(a1 + b1c1)∆tµ+
0 = [1 − i(a1 + b1c1)∆t]µ+
0 . (3.76)
After the second step,
µ+
2 = µ+
1 + ∆µ+
2 = µ+
1 − i(a2 + b2c2)∆tµ+
1 = [1 − i(a2 + b2c2)∆t][1 − i(a1 + b1c1)∆t]µ+
0 . (3.77)
After k steps,
µ+
k = [1−i(ak +bkck)∆t][1−i(ak−1 +bk−1ck−1)∆t] · · · [1−i(a2)+b2)c2)∆t][1−i(a1 +b1c1)∆t]µ+
0 . (3.78)
3.14.2 Rigid molecules
If the structure of the molecule does not change, the electron distribution is constant and the size and
shape of the chemical shift tensor described by δi and δa does not change in time. Then, a and b
are constant and the only time-dependent parameter is c, fluctuating as the orientation of the molecule
(described by ϑ) changes.5
The parameter a = −γB0(1+δi) represents a constant frequency of coherent
rotation under such circumstances. If we describe the evolution of µ+
in a coordinate frame rotating
with the frequency a, the equation simplifies to
(µ+
k )rot = [1 − ibck∆t][1 − ibck−1∆t] · · · [1 − i + bc2∆t][1 − ibc1∆t]µ+
0 . (3.79)
After multiplying the brackets and sorting the resulting terms according to the power of ∆t,
(µ+
k )rot = [1−ib∆t(ck +ck−1+· · ·+c1)−b2
∆t2
(ck(ck−1+· · · c2+c1)+. . .+c2c1)+ib3
∆t3
(. . .)+· · ·](µ+
0 )rot.
(3.80)
5Obviously, it is not possible to change z-components of the induced field by rotating the molecule and leave the x
and y-components intact. However, we limit our analysis to the z components in order to make the procedure as simple
as possible. Later we will see that the effects of fluctuations can be separated also in a more rigorous treatment.
40 CHAPTER 3. ENSEMBLES OF SPINS NOT INTERACTING WITH OTHER SPINS
We can now return to the question how random fluctuations change µ+
. Let’s express the difference
between µ+
after k and k − 1 steps:
∆(µ+
k )rot = (µ+
k )rot −(µ+
k−1)rot = −[ib∆tck −b2
∆t2
ck(ck−1 +· · ·+c1)−ib3
∆t3
(. . .)+· · ·](µ+
0 )rot. (3.81)
Dividing both sides by ∆t
∆(µ+
k )rot
∆t
= −[ibck + b2
∆tck(ck−1 + · · · + c1) − ib3
∆t2
(. . .) + · · ·](µ+
0 )rot (3.82)
and going back from ∆t to dt (neglecting terms with dt2
, dt3
, . . ., much smaller than dt),
d(µ+
(tk))rot
dt
= −

ibc(tk) + b2
tk
0
c(tk)c(tk − tj)dtj

 (µ+
0 )rot. (3.83)
We see that calculating how fluctuations of Bz affect an individual magnetic moment in time tk
requires knowledge of the orientations of the molecule during the whole evolution (c(tk − tj)). However,
we are not interested in the evolution of a single magnetic moment, but in the evolution of the
total magnetization M+. Total magnetization is given by the sum of all magnetic moments (magnetic
moments in all molecules). Therefore, we must average orientations of all molecules in the sample.
In the case of the axially symmetric chemical shift tensor, the orientations of molecules are given by
orientations of the symmetry axes a of the chemical shift tensors of the observed nuclei in the molecules,
described by the angles ϕ and ϑ. As the angle ϑ(t) is hidden in the function c(t) = (3 cos ϑ2
− 1)/2 in
our equation, the ensemble averaging cen be written as
d(M+
(tk))rot
dt
= −

ibc(tk) + b2
tk
0
c(tk)c(tk − tj)dtj

 (M+
0 )rot. (3.84)
We have already shown that c(tk) = (3 cos ϑ2 − 1)/2 = 0 (Eq. 3.52). It explains why we did not
neglect already the b2
dt term – we would obtain zero on the right-hand side in the rotating coordinate
frame (this level of simplification would neglect the effects of fluctuations and describe just the coherent
motions).
Therefore, the equation describing the loss of coherence (resulting in a loss of transverse magnetization)
is
d(M+
(tk))rot
dt
= −

b2
tk
0
c(tk)c(tk − tj)dtj

 (M+
0 )rot, (3.85)
where the time correlation function c(tk)c(tk − tj) plays the key role. Values of c(tk)c(tk − tj) can
be determined easily for two limit cases:
• tj = 0: If tj = 0, c(tk)c(tk − tj) = c(tk)2, i.e., c(tk) and c(tk − tj) are completely correlated. In
spherical coordinates, averaging of any function g(ϑ, ϕ) over all directions (over all values of angles
ϕ and ϑ can be written as
g(ϑ, ϕ) =
1
4π
2π
0
dϕ
π
0
dϑ(sin ϑ)g(ϑ, ϕ) (3.86)
3.14. RELAXATION DUE TO CHEMICAL SHIFT ANISOTROPY 41
Therefore,
c(tk)2 = (3 cos2 ϑ − 1)2 =
1
4π
2π
0
dϕ
π
0
dϑ(sin ϑ)(3 cos2
ϑ − 1)2
=
1
5
(3.87)
• tj → ∞: If the changes of orientation (molecular motions) are random, the correlation between
c(tk) and c(tk − tj) is lost for very long tj and they be averaged separately: c(tk)c(tk − tj) =
c(tk) · c(tk − tj). But we know that average c(t) = 3 cos2 ϑ − 1 = 0. Therefore, c(tk)c(tk − tj) = 0
for t → 0.
If the structure of the molecule does not change (rigid body rotational diffusion), which is the case
we analyze, the analytical form of c(tk)c(tk − tj) can be derived. It is equal to a sum of five exponential
functions for asymmetric rigid body rotational diffusion, to a sum of three exponential functions for
axially symmetric rotational diffusion, or to a single exponential function for spherically symmetric
rotational diffusion.
If the motions are really stochastic, it does not matter when we start to measure time. Therefore,
we can describe the loss of coherence for any tk as
d(M+
)rot
dt
= −

b2
∞
0
c(0)c(t)dt

 (M+
0 )rot, (3.88)
which resembles a first-order chemical kinetics with the rate constant
R0 = b2
∞
0
c(0)c(t)dt. (3.89)
For spherically symmetric rotational diffusion, described by a mono-exponential function characterized
by the rotational correlation time τc,
R0 = b2
∞
0
1
5
et/τc
dt =
b2
5
τc. (3.90)
3.14.3 Internal motions changing orientation of chemical shift tensor
What happens if the structure of the molecule changes? Let’s first assume that the structural changes are
random internal motions which change orientation of the chemical shift tensor relative to the orientation
of the whole molecule, but do not affect its size or shape. Then, Eq. 3.79 can be still used and R0 is still
given by Eq. 3.89, but the correlation function is not mono-exponential even if the rotational diffusion of
the molecule is spherically symmetric. The internal motions contribute to the dynamics together with
the rotational diffusion, and in a way that is very difficult to describe exactly. Yet, useful qualitative
conclusions can be made.
• If the internal motions are much faster than rotational diffusion, correlation between c(tk) and
c(tj) is lost much faster. The faster the correlation decays, the lower is the result of integration.
The internal motions faster than rotational diffusion always decrease the value of R0 (make
relaxation slower). Amplitude and rate of the fast internal motions can be estimated using approximative
approaches.
42 CHAPTER 3. ENSEMBLES OF SPINS NOT INTERACTING WITH OTHER SPINS
• If the internal motions are much slower than rotational diffusion, the rate of decay of the correlation
function is given by the faster contribution, i.e., by the rotational diffusion. The internal
motions slower than rotational diffusion do not change the value of R0. Amplitude and rate of
the fast internal motions cannot be measured if the motions do not change size or shape of the
diffusion tensor.
3.14.4 Chemical/conformational exchange
If the structural changes alter size and/or shape of the chemical shift tensor,6
parameters aj and bj
in Eq. 3.78 vary and cannot be treated as constants. E.g., the parameter aj is not absorbed into
the frequency of the rotating coordinate frame and terms a(tk)a(tk − tj) contribute to R0 even if
a(tk)a(tk − tj) decays much slower than c(tk)c(tk − tj).
• Internal motions or chemical processes changing size and/or shape of the chemical shift tensor
may have a dramatic effect on relaxation even if their frequency is much slower than the rotational
diffusion of the molecule. If the molecule is present in two inter-converting states (e.g. in two
conformations or in a protonated and deprotonated state), the strongest effect is observed if
the differences between the chemical shift tensors of the states are large and if the frequency of
switching between the states is similar to the difference in γB0δa of the states. Such processes are
known as chemical or conformational exchange and increase the value of R0.
3.14.5 Quantum description
The Liouville-von Neumann equation describing the relaxing system of magnetic moments interacting
with moving electrons in a so-called interaction frame (corresponding to the rotating coordinate frame
in the classical description) has the form
d∆ˆρ
dt
= −
i
¯h
[ ˆHδ,a + ˆHδ,r, ∆ˆρ], (3.91)
where ˆHδ,a and ˆHδ,r are defined by Eqs. 3.47 and 3.48, respectively, and ∆ˆρ is a difference (expressed
in the interaction frame) between density matrix at the given time and density matrix in the
thermodynamic equilibrium. Writing ∆ˆρ in the same bases as used for the Hamiltoninan,
∆ˆρ = dt
ˆIt + dz
ˆIz + d+
ˆI+eiω0t
+ d−
ˆI. (3.92)
If the chemical shift is axially symmetric and its size or shape do not change,
d(dz
ˆIz + d+
ˆI+eiω0t
+ d−
ˆI−e−iω0t
)
dt
= −
ib
¯h
cz ˆIz +
3
8
c+ ˆI+eiω0t
+
3
8
c− ˆI−e−iω0t
, dz
ˆIz + d+
ˆI+eiω0t
+ d−
ˆI−e−iω0t
,
(3.93)
where ˆI±e±iω0t
are operators ˆI± = ˆIx ± ˆIy in the interaction frame, ω0 = −γB0(1 + δa), and
cz
=
1
2
(3 cos2
ϑ − 1) (3.94)
c+
=
3
2
sin ϑ cos ϑe−iϕ
(3.95)
c−
=
3
2
sin ϑ cos ϑe+iϕ
(3.96)
6Examples of such changes are internal motions changing torsion angles and therefore distribution of electrons, or
chemical changes (e.g. dissociation of protons) with similar effects.
3.14. RELAXATION DUE TO CHEMICAL SHIFT ANISOTROPY 43
Analogically to the classical analysis, the evolution can be written as
d∆ˆρ
dt
= −
1
¯h2
∞
0
[ ˆHδ,a(0), [ ˆHδ,a(t), ∆ˆρ]]dt. (3.97)
The right-hand side can be simplified dramatically by the secular approximation: all terms with
e±iω0t
are averaged to zero. Only terms with (cz)2 and c+c− are non zero (both equal to 1/5 at tj = 0).7
These are the terms with [ˆIz, [ˆIz, ∆ˆρ]], [ˆI+, [ˆI−, ∆ˆρ]], and [ˆI−, [ˆI+, ∆ˆρ]]. Moreover, averaging over all
molecules makes all three correlation functions identical in isotropic liquids: cz(0)cz(t) = c+(0)c−(t) =
c−(0)c+(t) = c(0)c(t).
In order to proceed, the double commutators must be expressed. We start with
[ˆIz, ˆI±] = [ˆIz, ˆIx] ± i[ˆIz, ˆIy] = ±¯h(ˆIx ± iˆIy) = ±¯hˆI± (3.98)
and
[ˆI+, ˆI−] = [ˆIx, ˆIx] − i[ˆIx, ˆIy] + i[ˆIy, ˆIx] + [ˆIy, ˆIy] = 2¯hˆIz. (3.99)
Our goal is to calculate relaxation rates for the expectation values of components parallel (Mz) and
perpendicular (M+ or M−) to B0.
3.14.6 Relaxation of Mz
Let’s start with Mz. According to Eq. 1.17,
M+ = Tr{ ˆM+∆ˆρ} (3.100)
where ∆ Mz is the difference from the expectation value of Mz in equilibrium. The operator of Mz
for one magnetic moment observed is (Eq. 3.28)
ˆMz = Nγ ˆIz, (3.101)
where N is the number of molecules detected by the spectrometer. Since the basis matrices are
orthogonal, products of ˆIz with the components of the density matrix different from ˆIz are equal to
zero and the left-hand side of Eq. 3.97 reduces to
ddz
dt
ˆIz (3.102)
when calculating relaxation rate of Mz . In the right-hand side, we need to calculate three double
commutators:
[ˆIz, [ˆIz, ˆIz]] = 0 [ˆI+, [ˆI−, ˆIz]] = 2¯h2 ˆIz [ˆI−, [ˆI+, ˆIz]] = 2¯h2 ˆIz (3.103)
After substituting into Eq. 3.97,
ddz
dt
Tr{ˆIz
ˆIz} = −

3
4
b2
∞
0
c+(0)c−(t)eiω0t
dt +
3
4
b2
∞
0
c−(0)c+(t)e−iω0t
dt

 dzTr{ˆIz
ˆIz} (3.104)
d∆ Mz
dt
= −

3
4
b2
∞
0
c+(0)c−(t)eiω0t
dt +
3
4
b2
∞
0
c−(0)c+(t)e−iω0t
dt

 ∆ Mz (3.105)
7We have factored out 3/8 in order to make c+c− = (cz)2.
44 CHAPTER 3. ENSEMBLES OF SPINS NOT INTERACTING WITH OTHER SPINS
The relaxation rate R1 for Mz, known as longitudinal relaxation rate in the literature, is the real
part of the expression in the parentheses
R1 =
3
4
b2



∞
0
c+(0)c−(t)eiω0t
dt +
∞
0
c−(0)c+(t)e−iω0t
dt



(3.106)
For stochastic motions,
∞
0
c+(0)c−(t)eiω0t
dt =
1
2


∞
0
c+(0)c−(t)eiω0t
dt +
0
−∞
c+(0)c−(t)eiω0t
dt

 =
1
2
∞
−∞
c+(0)c−(t)eiω0t
dt.
(3.107)
∞
0
c−(0)c+(t)e−iω0t
dt =
1
2


∞
0
c−(0)c+(t)e−iω0t
dt +
0
−∞
c−(0)c+(t)e−iω0t
dt

 =
1
2
∞
−∞
c−(0)c+(t)e−iω0t
dt,
(3.108)
if the fluctuations are random, they are also stationary: the current orientation of the molecule is
correlated with the orientation in the past in the same manner as it is correlated with the orientation
in the future.
The right-hand side integrals are identical with the mathematical definition of the Fourier transform
of the correlation functions. Real parts of such Fourier transforms are known as spectral density functions
J(ω).
The relaxation rate R1 can be therefore written as
R1 =
3
4
b2 1
2
J(ω0) +
1
2
J(−ω0) ≈
3
4
b2
J(ω0) (3.109)
What is the physical interpretation of the obtained equation? Relaxation of Mz is given by the
correlation functions c+(0)c−(t) and c−(0)c+(t), describing fluctuations of the components of the chemical
shift tensor perpendicular to B0 (ax and ay). Such fluctuating fields resemble the radio waves
with B1 ⊥ B0. If the frequency of such fluctuations matches the precession frequency ω0, the resonance
condition is fulfilled and (random) transitions between the |α and |β states can take place. If
the magnetic moments are described by the quantum theory but their surroundings are treated classically,
J(ω0) = J(−ω0) which corresponds to equal probability of transitions |α → |β and |β → |α .
If the surroundings are described by quantum theory, J(ω0) = e−¯hω0/kBT
J(−ω0), and the transition
|β → |α is slightly more probable. This drives the system back to the equilibrium distribution of
magnetic moments.
3.14.7 Relaxation of M+
Let’s continue with M+. According to Eq. 1.17,
∆ M+ ≡ M+ = Tr{ ˆM+∆ˆρ} (3.110)
The expectation value of M+ in equilibrium is zero, this is why we do not need to calculate the
difference for M+ and why we did not calculate the difference in the classical analysis.
The operator of M+ for one magnetic moment observed is
ˆM+ = Nγ ˆI+ = Nγ(ˆIx + iˆIy). (3.111)
3.14. RELAXATION DUE TO CHEMICAL SHIFT ANISOTROPY 45
Due to the orthogonality of basis matrices, the left-hand side of Eq. 3.97 reduces to
dd+
dt
ˆI+eiω0t
(3.112)
when calculating relaxation rate of ∆ M+ ≡ M+ . In the right-hand side, we need to calculate
three double commutators:
[ˆIz, [ˆIz, ˆI+]] = ¯h2 ˆI+ [ˆI+, [ˆI−, ˆI+]] = 2¯h2 ˆI+ [ˆI−, [ˆI+, ˆI+]] = 0. (3.113)
After substituting into Eq. 3.97,
dd+
dt
Tr{ˆI+
ˆI+} = −

b2
∞
0
cz(0)cz(t)dt +
3
4
b2
∞
0
c+(0)c−(t)eiω0t
dt

 d+Tr{ˆI+
ˆI+} (3.114)
d M+
dt
= −

b2
∞
0
cz(0)cz(t)dt +
3
4
b2
∞
0
c+(0)c−(t)eiω0t
dt

 M+ (3.115)
The relaxation rate R2 for M+, known as transverse relaxation rate in the literature, is the real part
of the expression in the parentheses.
R1 = b2
∞
0
cz(0)cz(t)dt +



3
4
b2
∞
0
c+(0)c−(t)eiω0t
dt



. (3.116)
Note that the first integral in 3.116 is a real number, equal to R0 derived by the classical analysis.
Using the same arguments as for Mz,
R2 = b2 1
2
J(0) +
3
4
1
2
J(ω0) ≈ R0 +
1
2
R1. (3.117)
What is the physical interpretation of the obtained equation? Two terms in Eq. 3.117 describe two
processes contributing to the relaxation of M+. The first one is the loss of coherence with the rate
R0, given by the correlation function cz(0)cz(t) and describing fluctuations of the components of the
chemical shift tensor parallel with B0 (az). This contribution was analyzed above using the classical
approach. The second contribution is transitions between the |α and |β states due to fluctuations of the
components of the chemical shift tensor perpendicular to B0 (ax and ay), returning the magnetization
vector M to its direction in the thermodynamic equilibrium. As M is oriented along the z axis in the
equilibrium, the transitions renew the equilibrium value of Mz, as described above, but also make the
Mx and My components to disappear. Note however, that only one correlation function (c+(0)c−(t))
contributes to the relaxation of M+, while both c+(0)c−(t) and c−(0)c+(t) contributes to the relaxation
of Mz and only R1/2 contributes to R2. If we defined R2 as a relaxation rate of M−, c−(0)c+(t) would
contribute8
:
R2 = b2 1
2
J(0) +
3
4
1
2
J(−ω0) ≈ R0 +
1
2
R1. (3.118)
8Fluctuations with frequency +ω0 affect M+ and fluctuations with frequency −ω0 affect M−, but both affect Mz.
Alternatively, we could define R2 as a relaxation rate of Mx or My. Fluctuations of the Be,y component affect Mx but
not My, while fluctuations of the Be,x component affect My but not Mx. On the other hand, both fluctuations of Be,x
and Be,y affect Mz. Working with M+, M− or Mx, My, the relaxation of Mz due to Be,x and Be,y is always twice faster.
46 CHAPTER 3. ENSEMBLES OF SPINS NOT INTERACTING WITH OTHER SPINS
3.15 The one-pulse experiment
At this moment, we have all we need to describe a real NMR experiment for sample consisting of
isolated magnetic moments (not interacting with each other). The basic NMR experiment consists of
two parts. In the first part, the radio-wave transmitter is switched on for a short time, needed to rotate
the magnetization to the plane perpendicular to the magnetic filed B0 (a radio-wave pulse). In the
second time, the radio-wave transmitter is switched off but the receiver is switched on in order to detect
rotation of the magnetization vector about the direction of B0. We will analyze evolution of the density
matrix during these two periods and calculate the magnetization contributing to the detected signal.
3.15.1 Excitation by radio wave pulses
At the beginning of the experiment, the density matrix describes thermal equilibrium (Eq. 3.60):
ˆρ(0) = It + κIz. (3.119)
The Hamiltonian governing evolution of the system during the first part of the experiments consists
of coherent and fluctuating terms. The fluctuating contributions result in relaxation, described by
relaxation rates R1 and R2. The coherent contributions include
H = εt · 2It − γB0(1 + δi)Iz − γB1(1 + δi) cos(ωradiot)Ix − γB1(1 + δi) sin(ωradiot)Iy, (3.120)
where ¯hεt is the total energy of the system outside the magnetic field, and the choice of the directions
x and y is given by the cos(ωradiot) and sin(ωradiot) terms
The Hamiltonian simplifies in a coordinate system rotating with ωrot = ωradio
H = εt · 2It −γB0(1 + δi)
Ω
Iz −γB1(1 + δi)
ω1
Ix, (3.121)
but it still contains non-commuting terms (Ix vs. Iz). Let’s check what can be neglected to keep
only commuting terms, which allows us to solve the Liouville-von Neumann equation using the simple
geometric approach.
• The value of εt is unknown and huge, but It commutes with all matrices (it is proportional to the
unit matrix). As a consequence, this term can be ignored because it does not have any effect on
evolution of ˆρ.
• The value of ω1 defines how much magnetization is rotated to the x, y plane. The maximum effect
is obtained for ω1τp = π/2, where τp is the length of the radio-wave pulse. Typical values of τp for
proton are approximately 10 µs, corresponding to frequency of rotation of 25 kHz (90◦
rotation in
10 µs corresponds to 40 µs corresponds for a full circle, 1/40 µs = 25 kHz).
• Typical values of R1 are 10−1
s−1
to 100
s−1
and typical values of R2 are 10−1
s−1
to 102
s−1
for protons in organic molecules and biomacromolecules. Therefore, effects of relaxations can be
safely neglected during τp.
• When observing a single type of proton (or other nucleus), Ω can be set to zero by the choice of
ωradio. However, variation of Ω is what we observe in real samples, containing protons (or other
nuclei) with various δi. The typical range of proton δi is 10 ppm, corresponding to 5 kHz at a
500 MHz spectrometer.9
The carrier frequency ωradio is often set to the precession frequency of
9Chosen as a compromise here: spectra of small molecules are usually recored at 300 MHz–500 MHz, while spectra of
biomacromolecules are recorded at 500 MHz–1 GHz.
3.15. THE ONE-PULSE EXPERIMENT 47
the solvent. In the case of water, it is roughly in the middle of the spectrum (4.7 ppm at pH 7).
So, we need to cover ±2.5 kHz. We see that |Ω| < |ω1|, but the ratio is only 10 % at the edge of
the spectrum.
In summary, we see that we can safely ignore It and fluctuating contributions, but we must be
careful when neglecting ΩIz. The latter approximation allows us to use the geometric solution of the
Liouville-von Neumann equation, but is definitely not perfect for larger Ω resulting in offset effects.
Using the simplified Hamiltonian H = ω1Ix, evolution of ˆρ during τp can be described as a rotation
about the ”Ix axis”:
ˆρ(0) = It + κIz −→ ˆρ(τp) = It + κ(Iz cos(ω1τp) − Iy sin(ω1τp)). (3.122)
For a 90◦
pulse,
ˆρ(τp) = It − κIy. (3.123)
3.15.2 Evolution of chemical shift after excitation
After switching off the transmitter, ω1Ix disappears from the Hamiltonian, which now contains only
commuting terms. On the other hand, signal is typically acquired for a relatively long time (0.1 s to
10 s) to achieve a good frequency resolution. Therefore, the relaxation effects cannot be neglected.
The coherent evolution can be described as a rotation about the ”Iz axis” with the angular frequency
Ω
ˆρ(t) = It + κ(−Iy cos(Ωt) + Ix sin(Ωt)) = It + κ Ix cos Ωt +
π
2
+ Iy sin Ωt +
π
2
. (3.124)
We see that the system rotates in the operator spacewith angular frequency Ω and the original
phase of π/2. However, this is true only if we the evolution starts exactly at t = 0. In practice, this
is impossible to achieve for various technical reasons (instrumental delays and phase shifts, evolution
starts already during τp, etc.). Therefore, the rotation has an unknown phase shift φ (including the π/2
shift among other contributions), which is removed by an empirical correction during signal processing.
We will ignore the phase shift and write the phase-corrected spectral density
ˆρ(t) = It + κ(Ix cos(Ωt) + Iy sin(Ωt)) (3.125)
The measured quantity M+ can be expressed as (Eq. 1.17)
M+ = Tr{ ˆM+ ˆρ(t)} = NγB0
¯h
2
Tr{I+(It + κ(Ix cos(Ωt) + Iy sin(Ωt))}. (3.126)
The relevant traces are
Tr{I+It} = Tr
0 1
0 0
1
2 0
0 1
2
= Tr
0 1
2
0 0
= 0 (3.127)
Tr{I+Ix} = Tr
0 1
0 0
0 − i
2
i
2 0
= Tr
i
2 0
0 0
=
i
2
(3.128)
Tr{I+Iy} = Tr
0 1
0 0
0 1
2
1
2 0
= Tr
1
2 0
0 0
=
1
2
(3.129)
Including relaxation and expressing κ
48 CHAPTER 3. ENSEMBLES OF SPINS NOT INTERACTING WITH OTHER SPINS
M+ =
Nγ2
¯h2
B0
4kBT
e−R2t
(cos(Ωt) + i sin(Ωt)) =
Nγ2
¯h2
B0
4kBT
e−R2t
ieiΩt
. (3.130)
In general, the analysis of an ideal one-pulse experiment leads to the following conclusions:
• if the analysis of an NMR experiment shows that the density matrix evolves during analysis as
ˆρ(t) ∝ (Ix cos(Ωt + φ) + Ix sin(Ωt) + φ) + terms orthogonal to I+, (3.131)
the magnetization rotates during signal acquisition as
M+ = |M+|e−R2t
eiΩt
(3.132)
(with some unimportant phase shift which is empirically corrected),
• Fourier transform of the signal gives the complex signal
Nγ2
¯h2
B0
4kBT
R2
R2
2 + (ω − Ω)2
− i
ω − Ω
R2
2 + (ω − Ω)2
, (3.133)
• the cosine modulation of Ix can be taken as the real component of the signal and the sine modulation
of Iy can be taken as the imaginary component of the signal.
Chapter 4
Ensembles of spins interacting
through space
4.1 Product operators
Mutual interactions – interactions with fields generated by other nuclei. Description of such fields
involves spin eigenfunctions.
If two spin magnetic moments interact mutually, they cannot be described using the same basis.
Eigenfunctions are influenced by the interactions. State of the first spin depends on the state of the
second spin. For two spin-1/2 nuclei, 2 × 2 = 4 states.
Density matrix for four states – a 4 × 4 matrix. Basis used for such density matrices must consist
of 42
= 16 matrices.
Density matrix for N states – a N × N matrix. Basis used for such density matrices must consist
of 4N
matrices.
The basis can be derived by the direct product of basis matrices of spins without mutual interactions.
For two spins,
2 · It(1) ⊗ It(2) = It(12) (4.1)
2 · It(1) ⊗ Ix(2) = I1x(12) (4.2)
2 · It(1) ⊗ Iy(2) = I1y(12) (4.3)
2 · It(1) ⊗ Iz(2) = I1z(12) (4.4)
2 · Ix(1) ⊗ It(2) = I2x(12) (4.5)
2 · Iy(1) ⊗ It(2) = I2y(12) (4.6)
2 · Iz(1) ⊗ It(2) = I2z(12) (4.7)
2 · Ix(1) ⊗ Ix(2) = 2I1xI2x(12) (4.8)
2 · Ix(1) ⊗ Iy(2) = 2I1xI2y(12) (4.9)
2 · Ix(1) ⊗ Iz(2) = 2I1xI2z(12) (4.10)
2 · Iy(1) ⊗ Ix(2) = 2I1yI2x(12) (4.11)
2 · Iy(1) ⊗ Iy(2) = 2I1yI2y(12) (4.12)
2 · Iy(1) ⊗ Iz(2) = 2I1yI2z(12) (4.13)
2 · Iz(1) ⊗ Ix(2) = 2I1zI2x(12) (4.14)
49
50 CHAPTER 4. ENSEMBLES OF SPINS INTERACTING THROUGH SPACE
2 · Iz(1) ⊗ Iy(2) = 2I1zI2y(12) (4.15)
2 · Iz(1) ⊗ Iz(2) = 2I1zI2z(12), (4.16)
where the numbers in parentheses specify which nuclei constitute the spin system described by the
given matrix (these numbers are not written in practice). The matrices on the right-hand side are
known as product operators. Note that It, equal to1 1
2
ˆ1, is not written in the product operators for
the sake of simplicity. Note also that e.g. Ix(1) and Ix(2) are the same 2 matrices, but I1x(12) and
I2x(12) are different 4 matrices. Basis matrices for more nuclei are derived in the same manner, e.g.
2I1zI2x(12) ⊗ Iy(3) = 4I1zI2xI3y(123).
4.2 Liouville-von Neumann equation
The Liouville - von Neumann equation can be written in the same form as for spins without mutual
interactions (Eq. 3.19):
dˆρ
dt
= i(ˆρH − Hˆρ) = i[ˆρ, H] = −i[H, ˆρ], (4.17)
but the density matrix and Hamiltonian are now N × N matrices described in the appropriate
basis. The same simple geometric solution as for spins without mutual interactions is possible if the
Hamiltonian does not vary in time and consists of commuting matrices only. However, the operator
space is now N2
dimensional (16-dimensional for two spin-1/2 nuclei). Therefore, the appropriate threedimensional
subspace must be selected for each rotation. The subspaces are defined by the commutator
relations, which can be defined for spin systems consisting of any number of spin-1/2 nuclei using the
following equations.
[In,x, In,y] = iIn,z [In,y, In,z] = iIn,x [In,z, In,x] = iIn,y (4.18)
[In,j, 2In,kIn ,l] = 2[In,j, In,k]In ,l (4.19)
[2In,jIn ,l, 2In,kIn ,m] = 2[In,j, In,k]δlm, (4.20)
where n and n specify the nucleus, j, k, l ∈ {x, y, z}, and δlm = 1 for l = m and δlm = 0 for l = m.
4.3 Through-space dipole-dipole interaction (dipolar coupling)
If spin magnetic moments of two spin-1/2 nuclei interact with each other, the magnetic moment of
nucleus 1 is influenced by the magnetic field B2 of the magnetic moment of nucleus 2. B2 is given by
the classical electrodynamics as
B2 = × A2, (4.21)
where
≡
∂
∂x
,
∂
∂y
,
∂
∂z
(4.22)
and the vector potential of the magnetic moment 2 is
1ˆ1 is the unit matrix.
4.4. HAMILTONIAN OF DIPOLAR COUPLING 51
A2 =
µ0
4π
µ2 × r
r3
, (4.23)
where r is a vector defining the mutual position of nuclei 1 and 2 (inter-nuclear vector).
Calculation of B2 thus includes two vector products
B2 =
µ0
4π
× (µ2 × r)
r3
. (4.24)
As a consequence, each component of B2 depends on all components of µ2:
B2,x =
µ0
4πr5
((3r2
x − r2
)µ2,x + 3rxryµ2,y + 3rxrzµ2,z) (4.25)
B2,y =
µ0
4πr5
(3rxryµ2,x + (3r2
y − r2
)µ2,y + 3ryrzµ2,z) (4.26)
B2,z =
µ0
4πr5
(3rxrzµ2,x + 3ryrzµ2,y + (3r2
z − r2
)µ2,z), (4.27)
which can by described by a matrix equation


B2,x
B2,y
B2,z

 =
µ0
4πr5


3r2
x − r2
3rxry 3rxrz
3rxry 3r2
y − r2
3ryrz
3rxrz 3ryrz 3r2
z − r2

 ·


µ2,x
µ2,y
µ2,z

 . (4.28)
The matrix in Eq. 4.28 represents a tensor describing the geometric relations of the dipolar coupling
and has the same form as the matrix in Eq. 3.44, describing the anisotropic contribution to the chemical
shift tensor: the vector defining the symmetry axis of the chemical shift tensor a is just replaced with
the inter-nuclear vector r in Eq. 4.28. Like the anisotropic part of the chemical shift tensor, the matrix
in Eq. 4.28 simplifies to
µ0
4πr3


−1 0 0
0 −1 0
0 0 2

 (4.29)
in a coordinate system with axis z r. Rotation to the laboratory frame is described by angles ϕ
and ϑ defining orientation of r in the laboratory frame
δa


−1 0 0
0 −1 0
0 0 2

 −→


3r2
x − r2
3rxry 3rxrz
3rxry 3r2
y − r2
3ryrz
3rxrz 3ryrz 3r2
z − r2

 , (4.30)
where rx = r sin ϑ cos ϕ, ry = r sin ϑ sin ϕ, and rz = r cos ϑ.
4.4 Hamiltonian of dipolar coupling
Describing the magnetic moments by the operators ˆµ1,jγ1
ˆI1,j and ˆµ2,jγ1
ˆI2,j, where j is x, y, and z, the
Hamiltonian of dipolar coupling ˆHD can be written as
ˆHD = −γ1(ˆI1,xB2,x + ˆI1,yB2,y + ˆI1,zB2,z) = −γ1( ˆI1,x
ˆI1,y
ˆI1,z )


B2,x
B2,y
B2,z

 =
52 CHAPTER 4. ENSEMBLES OF SPINS INTERACTING THROUGH SPACE
= −
µ0γ1γ2
4πr5
( ˆI1,x
ˆI1,y
ˆI1,z )


3r2
x − r2
3rxry 3rxrz
3rxry 3r2
y − r2
3ryrz
3rxrz 3ryrz 3r2
z − r2




ˆI2,x
ˆI2,y
ˆI2,z

 =
ˆ
I1 · D ·
ˆ
I2, (4.31)
where D is the tensor of direct dipole-dipole interactions (dipolar coupling).
The Hamiltonian can be written in spherical coordinates as
ˆHD = −
µ0γ1γ2
4πr3
+ (3 sin2
ϑ cos2
ϕ − 1)ˆI1x
ˆI2x + (3 sin2
ϑ sin2
ϕ − 1)ˆI1y
ˆI2y + (3 cos2
ϑ − 1)ˆI1z
ˆI2z+
+ 3 sin2
ϑ sin ϕ cos ϕˆI1x
ˆI2y + 3 sin ϑ cos ϑ cos ϕˆI1x
ˆI2z + 3 sin ϑ cos ϑ sin ϕˆI1y
ˆI2z
+ 3 sin2
ϑ sin ϕ cos ϕˆI1y
ˆI2x + 3 sin ϑ cos ϑ cos ϕˆI1z
ˆI2x + 3 sin ϑ cos ϑ sin ϕˆI1z
ˆI2y .(4.32)
4.5 Secular approximation and averaging
The Hamiltonian of dipolar coupling can be simplified in many cases.
• Magnetic moments with the same γ and chemical shift precess about the z axis with the same
precession frequency. In addition to the precession, the magnetic moments moves with random
molecular motions, described by re-orientation of r. In a coordinate system rotating with the
common precession frequency, r quickly rotates about the z axis in addition to the random molecular
motions. On a time scale slower than nanoseconds, the rapid oscillations of rx, ry, and rz
are neglected (secular approximation). The values of r2
x and r2
y do not oscillate about zero, but
about a value r2
x = r2
y , which is equal to2
(r2
− r2
z )/2 because r2
x + r2
y + r2
z = r2
= r2
.
Therefore, the secular approximations (i.e., neglecting the oscillations and keeping the average
values) simplifies the Hamiltonian to
ˆHD = −
µ0γ1γ2
4πr5
3 r2
z − r2 ˆI1,z
ˆI2,z −
1
2
ˆI1,x
ˆI2,x −
1
2
ˆI1,y
ˆI2,y (4.33)
= −
µ0γ1γ2
4πr3
3 cos2
ϑ − 1
2
2ˆI1,z
ˆI2,z − ˆI1,x
ˆI2,x − ˆI1,y
ˆI2,y . (4.34)
• Magnetic moments with different γ and/or chemical shift precess with different precession frequencies.
Therefore, the x and y components of µ2 rapidly oscillate in a frame rotating with
the precession frequency of µ1 and vice versa. When neglecting the oscillating terms (secular
approximation), the Hamiltonian reduces to
ˆHD = −
µ0γ1γ2
4πr5
3 r2
z − r2 ˆI1,z
ˆI2,z = −
µ0γ1γ2
4πr3
3 cos2
ϑ − 1
2
2ˆI1,z
ˆI2,z. (4.35)
• Averaging over all molecules in isotropic liquids has the same effect as described for the anisotropic
part of the chemical shielding tensor because both tensors have the same form:
r2
x = r2
y = r2
z. (4.36)
Finally,
r2
x + r2
y + r2
z = r2
⇒ r2
x + r2
y + r2
z = 3r2
z = r2
⇒ 3r2
z − r2 = r(3 cos2 ϑ − 1) = 0. (4.37)
2Note that r2
x = r2
y = r2
z in general.
4.6. RELAXATION DUE TO THE DIPOLE-DIPOLE INTERACTIONS 53
Unlike the chemical shift Hamiltonian, the Hamiltonian of the dipolar coupling does not have any
isotropic part. As a consequence, the dipole-dipole interactions are not observable in isotropic
liquids. On the other hand, their effect is huge in solid state NMR and they can be also be
measured e.g. in liquid crystals or mechanically stretched gels.
4.6 Relaxation due to the dipole-dipole interactions
Rotation of the molecule (and internal motions) change the orientation of the inter-nuclear vector and
cause fluctuations of the field of magnetic moment µ2 sensed by the magnetic moment µ1. It leads to
the loss of coherence in the same manner as described for the anisotropic part of the chemical shift (cf.
Eqs 3.44 and 4.28. However, the relaxation effects of the dipole-dipole interactions are more complex,
reflecting the higher complexity of the Hamiltonian of the dipolar coupling.
In order to describe the dipole-dipole relaxation on the quantum level, it is useful to work in spherical
coordinates and to convert the product operators to a different basis. Single quantum operators are
transformed using the relation ˆI± = ˆIx
± iˆIy):
ˆI1x
ˆI2z =
1
2
(+ˆI1+
ˆI2z + ˆI1−
ˆI2z) (4.38)
ˆI1y
ˆI2z =
i
2
(−ˆI1+
ˆI2z + ˆI1−
ˆI2z) (4.39)
ˆI1z
ˆI2x =
1
2
(+ˆI1z
ˆI2+ + ˆI1z
ˆI2−) (4.40)
ˆI1z
ˆI2y =
i
2
(−ˆI1z
ˆI2+ + ˆI1z
ˆI2−). (4.41)
Since
cos ϕ + i sin ϕ = eiϕ
(4.42)
cos ϕ − i sin ϕ = e−iϕ
, (4.43)
3 sin ϑ cos ϑ(ˆI1x
ˆI2z cos ϕ + ˆI1y
ˆI2z sin ϕ + ˆI1z
ˆI2x cos ϕ + ˆI1z
ˆI2y sin ϕ)
=
3
2
sin ϑ cos ϑ(ˆI1+
ˆI2ze−iϕ
+ ˆI1−
ˆI2zeiϕ
+ ˆI1z
ˆI2+e−iϕ
+ ˆI1z
ˆI2−eiϕ
) (4.44)
The double-quantum/zero-quantum operators are transformed in a similar fashion
ˆI1x
ˆI2y =
i
4
(+ˆI1+
ˆI2− − ˆI1−
ˆI2+ − ˆI1+
ˆI2+ + ˆI1−
ˆI2−)
ˆI1y
ˆI2x =
i
4
(−ˆI1+
ˆI2− + ˆI1−
ˆI2+ − ˆI1+
ˆI2+ + ˆI1−
ˆI2−)
ˆI1x
ˆI2x =
1
4
(+ˆI1+
ˆI2− + ˆI1−
ˆI2+ + ˆI1+
ˆI2+ + ˆI1−
ˆI2−)
ˆI1y
ˆI2y =
1
4
(+ˆI1+
ˆI2− + ˆI1−
ˆI2+ − ˆI1+
ˆI2+ − ˆI1−
ˆI2−)
and
3 sin2
ϑ(ˆI1x
ˆI2x cos2
ϕ + ˆI1y
ˆI2y sin2
ϕ + ˆI1x
ˆI2y sin ϕ cos ϕ + ˆI1y
ˆI2x sin ϕ cos ϕ) − (ˆI1x
ˆI2x + ˆI1y
ˆI2y)
54 CHAPTER 4. ENSEMBLES OF SPINS INTERACTING THROUGH SPACE
=
3
4
sin2
ϑ ( ˆI1+
ˆI2−(cos2
ϕ + sin2
ϕ + i sin ϕ cos ϕ − i sin ϕ cos ϕ)
+ˆI1−
ˆI2+(cos2
ϕ + sin2
ϕ − i sin ϕ cos ϕ + i sin ϕ cos ϕ)
+ˆI1+
ˆI2+(cos2
ϕ − sin2
ϕ − i sin ϕ cos ϕ − i sin ϕ cos ϕ)
+ˆI1−
ˆI2−(cos2
ϕ − sin2
ϕ + i sin ϕ cos ϕ + i sin ϕ cos ϕ) )
−
1
4
(2ˆI1+
ˆI2− + 2ˆI1−
ˆI2+)
=
1
4
ˆI1+
ˆI2−(3 sin2
ϑ − 2) +
1
4
ˆI1−
ˆI2+(3 sin2
ϑ − 2)
+
1
4
ˆI1+
ˆI2+ sin2
ϑe−i2ϕ
+
1
4
ˆI1−
ˆI2− sin2
ϑei2ϕ
= −
1
4
ˆI1+
ˆI2−(3 cos2
ϑ − 1) −
1
4
ˆI1−
ˆI2+(3 cos2
ϑ − 1)
+
1
4
ˆI1+
ˆI2+ sin2
ϑe−i2ϕ
+
1
4
ˆI1−
ˆI2− sin2
ϑei2ϕ
. (4.45)
Using Eqs. 4.44 and 4.45 and moving to the interaction frame (ˆIn± → ˆIn±e±iωnt
), Eq. 4.32 is
converted to
ˆHI
D = −
µ0γ1γ2
4πr3
ˆI1z
ˆI2z(3 cos2
ϑ − 1)
−
1
4
ˆI1+
ˆI2−(3 cos2
ϑ − 1)ei(ω1−ω2)t
−
1
4
ˆI1−
ˆI2+(3 cos2
ϑ − 1)e−i(ω1−ω2)t
+
3
2
ˆI1+
ˆI2z sin ϑ cos ϑe−iϕ
ei(ω1)t
+
3
2
ˆI1−
ˆI2z sin ϑ cos ϑeiϕ
e−i(ω1)t
+
3
2
ˆI1z
ˆI2+ sin ϑ cos ϑe−iϕ
ei(ω2)t
+
3
2
ˆI1z
ˆI2− sin ϑ cos ϑeiϕ
e−i(ω2)t
+
3
4
ˆI1+
ˆI2+ sin2
ϑe−i2ϕ
ei(ω1+ω2)t
+
3
4
ˆI1−
ˆI2− sin2
ϑei2ϕ
e−i(ω1+ω2)t
= −
µ0γ1γ2
4πr3
2ˆI1z
ˆI2zczz
−
1
2
c+− ˆI1+
ˆI2− −
1
2
c−+ ˆI1−
ˆI2+
+
3
2
c+z ˆI1+
ˆI2z + c−z ˆI1−
ˆI2z + cz+ ˆI1z
ˆI2+ + cz− ˆI1z
ˆI2− + c++ ˆI1+
ˆI2+ + c−− ˆI1−
ˆI2− (4.46)
Similarly to Eq. 3.97, the dipole-dipole relaxation is described by
d∆ˆρ
dt
= −
1
¯h2
∞
0
[ ˆHD(0), [ ˆHD(t), ∆ˆρ]]dt. (4.47)
The right-hand side can be simplified dramatically by the secular approximation as in Eq. 3.97:
all terms with e±iωnt
are averaged to zero. Only terms with (czz)2, cz+cz−, c+zc−z, c+−c−+, and
c++c−− are non zero (all equal to 1/5 at tj = 0).3
This reduces the number of double commutators
to be expressed from 81 to 9 for each density matrix component. The double commutators needed to
describe relaxation rates of the contributions of the first nucleus to the magnetization M1z and M1+
are, respectively,
ˆI1z
ˆI2z, [ˆI1z
ˆI2z, ˆI1z] = 0 (4.48)
3Averaging over all molecules makes all correlation functions identical in isotropic liquids.
4.6. RELAXATION DUE TO THE DIPOLE-DIPOLE INTERACTIONS 55
ˆI1−
ˆI2+, [ˆI1+
ˆI2−, ˆI1z] = ¯h2
(ˆI1z − ˆI2z) (4.49)
ˆI1+
ˆI2−, [ˆI1−
ˆI2+, ˆI1z] = ¯h2
(ˆI1z − ˆI2z) (4.50)
ˆI1+
ˆI2z, [ˆI1−
ˆI2z, ˆI1z] =
1
2
¯h2 ˆI1z (4.51)
ˆI1−
ˆI2z, [ˆI1+
ˆI2z, ˆI1z] =
1
2
¯h2 ˆI1z (4.52)
ˆI1z
ˆI2+, [ˆI1z
ˆI2−, ˆI1z] = 0 (4.53)
ˆI1z
ˆI2−, [ˆI1z
ˆI2+, ˆI1z] = 0 (4.54)
ˆI1+
ˆI2+, [ˆI1−
ˆI2−, ˆI1z] = ¯h2
(ˆI1z + ˆI2z) (4.55)
ˆI1−
ˆI2−, [ˆI1+
ˆI2+, ˆI1z] = ¯h2
(ˆI1z + ˆI2z) (4.56)
ˆI1z
ˆI2z, [ˆI1z
ˆI2z, ˆI1+] =
1
4
¯h2 ˆI1+ (4.57)
ˆI1+
ˆI2−, [ˆI1−
ˆI2+, ˆI1+] = ¯h2 ˆI1+ (4.58)
ˆI1−
ˆI2+, [ˆI1+
ˆI2−, ˆI1+] = 0 (4.59)
ˆI1+
ˆI2z, [ˆI1−
ˆI2z, ˆI1+] =
1
2
¯h2 ˆI1+ (4.60)
ˆI1−
ˆI2z, [ˆI1+
ˆI2z, ˆI1+] = 0 (4.61)
ˆI1z
ˆI2+, [ˆI1z
ˆI2−, ˆI1+] =
1
2
¯h2 ˆI1+ (4.62)
ˆI1z
ˆI2−, [ˆI1z
ˆI2+, ˆI1+] =
1
2
¯h2 ˆI1+ (4.63)
ˆI1+
ˆI2+, [ˆI1−
ˆI2−, ˆI1+] = 0 (4.64)
ˆI1−
ˆI2−, [ˆI1+
ˆI2+, ˆI1+] =
1
2
¯h2 ˆI1+. (4.65)
The relaxation rates can be then derived as described for the relaxation due to the chemical shift.
The following equations are obtained:
d∆ M1z
dt
= −
1
8
b2
(2J(ω1 − ω2) + 6J(ω1) + 12J(ω1 + ω2))∆ M1z
+
1
8
b2
(2J(ω1 − ω2) − 12J(ω1 + ω2))∆ M2z
= −Ra1∆ M1z + Rx∆ M2z (4.66)
d∆ M2z
dt
= −
1
8
b2
(2J(ω1 − ω2) + 6J(ω2) + 12J(ω1 + ω2))∆ M2z
+
1
8
b2
(2J(ω1 − ω2) − 12J(ω1 + ω2))∆ M1z
= −Ra2∆ M2z + Rx∆ M1z (4.67)
d M1+
dt
= −
1
8
b2
(4J(0) + 6J(ω2) + J(ω1 − ω2) + 3J(ω1) + 6J(ω1 + ω2)) M1+ (4.68)
56 CHAPTER 4. ENSEMBLES OF SPINS INTERACTING THROUGH SPACE
= −R2 M1+ = − R0 +
1
2
R1 M1+ (4.69)
where
b = −
µ0γ1γ2¯h
4πr3
. (4.70)
The relaxation rate R1 of the dipole-dipole relaxation is the rate of relaxation of the z-component
of the total magnetization Mz = M1z + M2z . R1 is derived by solving the set of Eqs. 4.66 and
4.67. The solution is simple if J(ω1) = J(ω2) = J(ω) ⇒ Ra1 = Ra2 = Ra (this is correct e.g. if both
nuclei have the same γ, if the molecule rotates as a sphere, and if internal motions are negligible or
identical for both nuclei).4
d∆ Mz
dt
= −
1
8
b2
(6J(ω) + 24J(2ω1))∆ Mz = −(Ra − Rx)∆ Mz (4.71)
There are several remarkable differences between relaxation due to the chemical shift anisotropy and
dipole-dipole interactions:
• The rate constants describing the return to the equilibrium polarization is more complex than
for the chemical shift anisotropy relaxation. In addition to the 3b2
J(ω1)/4 term, describing the
|α ↔ |β transition5
of nucleus 1, the auto-relaxation rate Ra1 contains terms depending on
the sum and difference of the precession frequency of µ1 and µ2. These terms correspond to the
zero-quantum (|αβ ↔ |βα ) and double-quantum (|αα ↔ |ββ ) transitions, respectively.
• Return to the equilibrium polarization of nucleus 1 depends also on the actual polarization of
nucleus 2. This effect, resembling chemical kinetics of a reversible reaction, is known as crossrelaxation,
or nuclear Overhauser effect (NOE), and described by the cross-relaxation constant
Rx. The value of Rx is proportional to r−6
and thus provides information about inter-atomic
distances. NOE is a useful tool in analysis of small molecules and the most important source of
structural information for large biological molecules.
• The relaxation constant R0, describing the loss of coherence, contains an additional term, depending
on the frequency of the other nucleus, 3b2
J(ω2)/4. This term has the following physical
significance. The field generated by the second magnetic moment depends on its state. The state
is changing due to |α ↔ |β transitions the with the rate given by 3b2
J(ω2)/4. Such changes have
the similar effect as the chemical or conformational exchange, modifying the size of the chemical
shift tensor. Therefore, 3b2
J(ω2)/4 adds to R0 like the exchange contribution.
4.7 2D spectroscopy based on dipole-dipole interactions
Three 90◦
pulses and two delays before data acquisition:
a(π/2)xb − t1 −c (π/2)xd − τm −e (π/2)xf − t2(acquire)
Homonuclear experiments - all nuclei the same γ.
4The general solution gives R1 = 1
2
Ra1 + Ra2 − (Ra1 − Ra2)2 + 4R2
x .
5The |αα ↔ |βα and |αβ ↔ |ββ transitions in a two-spin system
4.7. 2D SPECTROSCOPY BASED ON DIPOLE-DIPOLE INTERACTIONS 57
4.7.1 Two-dimensional spectroscopy
In order to describe principles of 2D spectroscopy, we first analyze the experiment for two non-interacting
magnetic moments, e.g. of two protons with different chemical shift δi too far from each other.
We describe the density matrix just before and after pulses, as labeled by letters ”a” to ”f”.
• ˆρ(a) = It + κ(I1z + I2z)
thermal equilibrium, the matrices are different than for the noninteracting spin, but the constant
is the same.
• ˆρ(b) = It + κ(−I1y − I2y)
90◦
pulse, see the one-pulse experiment
• ˆρ(c) = It+κ −e−R2t1
cos(Ω1t1)I1y + e−R2t1
sin(Ω1t1)I1x − e−R2t1
cos(Ω2t1)I2x + e−R2t1
sin(Ω2t1)I2y
ˆρ(c) = It + κ (−c11I1y + s11I1x − c21I2y + s21I2x)
evolution after excitation, the same as in the one-pulse experiment. No effect of the dipolar coupling
(averaged to zero in isotropic liquids). Relaxation included as the exponential factors with
the same R2 (differ in general).
• ˆρ(d) = It + κ (−c11I1z + s11I1x − c21I2z + s21I2x)
90◦
x-pulse does not affect x magnetization, rotates −y magnetization further to −z - B0 but
inverted polarization, similar to a → b.
• ˆρ(e) =? Delay τm is usually longer than 0.1 s. New, should be analyzed (here for a large molecule
such as a small protein): In proteins, Mx, My relax with R2 > 10 s−1
and Mz with R1 ≈ 1 s−1
.
Let’s assume τm = 0.2 s and R2 = 20 s−1
. After 0.2 s, e−R2τm
= e−20×0.2
= e−4
≈ 0.02. We see
that Mx, My relaxes almost completely ⇒ I1x, I1y, I2x, I2y can be neglected. On the other hand,
e−R1τm
= e−1×0.2
= e−0.2
≈ 0.82. We see that Mz does not relax too much ⇒ we continue analysis
with I1z, I2z. The I1z, I2z terms do not evolve because they commute with H = Ω1I1z + Ω2I2z.
Therefore,
ˆρ(e) = It + κ −e−R1τm
c11I1z − e−R1τm
c21I2z = It − A1I1z − A2I2z
• ˆρ(e) = It + A1I1y + A2I2y
see the first pulse
• ˆρ(t2) =
It +A1(e−R2t2
cos(Ω1t2)I1y −e−R2t2
sin(Ω1t2)I1x)+A2(e−R2t2
cos(Ω2t2)I2y −e−R2t2
sin(Ω2t2)I2x)
evolution during data acquisition, correction of the phase gives ˆρ in the same form as in Eq. 3.131:
ˆρ(t2) =
It +A1(e−R2t2
cos(Ω1t2)I1x +e−R2t2
sin(Ω1t2)I1y)+A2(e−R2t2
cos(Ω2t2)I2x +e−R2t2
sin(Ω2t2)I2y)
Therefore, Fourier transform of the signal provides spectrum in the form (see Eq. 3.133)
Nγ2
¯h2
B0
4kBT
A1R2
R2
2 + (ω − Ω1)2
+
A2R2
R2
2 + (ω − Ω2)2
− i
A1(ω − Ω1)
R2
2 + (ω − Ω1)2
+
A2(ω − Ω2)
R2
2 + (ω − Ω2)2
.
(4.72)
In the one-dimensional experiment, A1 and A2 just scale the peak height. However, they depend on
the length of the delay t1. If the measurement is repeated many times and t1 is increased by an increment
∆t each time, the obtained series of 1D spectra is amplitude modulated by c11 = e−R2t2
cos(Ω1t1) and
c21 = e−R2t2
cos(Ω2t1). Since the data are stored in a computer in a digital form, they can be treated
as a two-dimensional array (table), depending on the real time t2 in one direction and on the length of
the incremented delay t1 in the other directions. These directions are referred to as direct dimension
and indirect dimension. Fourier transform can be performed in each dimension.
58 CHAPTER 4. ENSEMBLES OF SPINS INTERACTING THROUGH SPACE
Transmitter on
Transmitter off
Receiver on
Receiver off
t1 τm t2
t1 f1 f1
t1
f1
Figure 4.1: Principle of two-dimensional spectroscopy (experiment NOESY). The acquired signal is shown in red, the
signal after Fourier transform in the direct dimension is shown in magenta, and the signal after Fourier transform in both
dimensions is shown in blue.
Since we acquire signal as a series of complex numbers, it is useful to introduce the complex numbers
in the indirect dimension as well. It is possible e.g. by repeating the measurement twice for each value
of t1, once with the x-phase (the same phase as the first pulse) of the second pulse, as described above,
and then with the y-phase (phase-shifted from the first pulse by 90◦
). In the latter case, the I1y and
I2y components are not affected and relax during τm, while the I1x and I2x are rotated to −I1z and
−I2z, respectively, and converted to the measurable signal by the third pulse. Because the I1x and I2x
coherences are modulated by s11 and s21, A1 and A2 oscillate as a sine function, not cosine function, in
the even spectra. So, we obtain cosine modulation in odd spectra and sine modulation in even spectra.
The cosine- and sine- signals are then treated as the real and imaginary component of the complex signal
in the indirect dimension. Complex Fourier transform in both dimensions provides a two-dimensional
spectrum.
4.7.2 Nuclear Overhauser efect spectroscopy (NOESY)
The two-dimensional spectra described in the preceding section are not very useful because they do not
bring any new information. The same frequencies are measured in the direct and indirect dimension
and all peaks are found along the diagonal of the spectrum. What makes the experiment really useful
is the interaction between magnetic moments during τm.
As described by Eq. 4.66, relaxation of nucleus 1 is influenced by the state of nucleus 2 (and vice
versa):
−
d∆ M1z
dt
= Ra1∆ M1z + Rx∆ M2z (4.73)
4.7. 2D SPECTROSCOPY BASED ON DIPOLE-DIPOLE INTERACTIONS 59
−
d∆ M2z
dt
= Ra2∆ M2z + Rx∆ M1z . (4.74)
The analysis greatly simplifies if the auto-relaxation rates are identical for both magnetic moments.
Then,
∆ M1z = ((1 − ζ)∆ M1z (0) + ζ∆ M2z (0)) e−(Ra+Rx)t
, (4.75)
where ζ = (e2Rxt
− 1)/2. Therefore,
ˆρ(e) = It−A1I1z−A2I2z = It−κ ((1 − ζ)c11 + ζc21) e−(Ra+Rx)τm
I1z−κ ((1 − ζ)c21 + ζc11) e−(Ra+Rx)τm
I2z
Now, the amplitudes A1 and A2 depend on both frequencies Ω1 and Ω2 (contain both c11 and c21. Therefore,
the spectrum contains both diagonal peaks (with the frequencies of the given magnetic moment
in both dimensions) and off-diagonal cross-peaks (with the frequencies of the given magnetic moment
in the direct dimension and the frequency of its interaction partner in the indirect dimension). The
overall loss of signal (”leakage”) due to the R1 relaxation is given by e−(Ra−Rx)τm
and intensities of the
cross-peaks are given by the factor
ζe−(Ra+Rx)τm
=
1
2
eRxτm
− e−Rxτm
≈ Rxτm =
µ0
8π
γ4
¯h2
r6
(J(0) − 6J(2ω))τm, (4.76)
where the difference of the precession frequencies due to different chemical shifts was neglected (ω1 = ω2
because γ1 = γ2). Hence, the cross-peak intensity is proportional to r−6
in the linear approximation.
60 CHAPTER 4. ENSEMBLES OF SPINS INTERACTING THROUGH SPACE
Chapter 5
Ensembles of spins interacting
through bonds
Magnetic moments of nuclei connected by covalent bonds interact also indirectly, via interactions with
magnetic moments of the electrons of the bonds. The simplest example is a pair of nuclei (e.g., 1
H and
13
C) connected by a σ bond. In such system, the states |αβ and |αβ allow all interacting particles
to be in the opposite state (H↑
-e↓
-e↑
-C↓
and H↓
-e↑
-e↓
-C↑
, respectively) and are energetically more
favorable than the |αα and |ββ states, which require too interacting particles to be in the same state
(H↑
-e↓
-e↑
-C↑
or H↑
-e↑
-e↓
-C↑
and H↓
-e↓
-e↑
-C↓
or H↓
-e↑
-e↓
-C↓
, respectively). The relations are more
complex in the case of interactions through multiple bonds.
Again, each component of the field felt by magnetic moment 1 (e.g. of 1
H) depends on all components
of the magnetic moment 2 (e.g. of 13
C). Therefore, the interaction is described by tensors (like chemical
shift or dipolar coupling):
ˆHJ = −γ(ˆIx1B2,x + ˆIy1B2,y + ˆIzB2,z1) = −γ( ˆIx1
ˆIy1
ˆIz1 )


B2,x
B2,y
B2,z

 =
= 2π( ˆIx
ˆIy
ˆIz )


Jxx Jxy Jxz
Jyx Jyy Jyz
Jzx Jzy Jzz




ˆIx1
ˆIy1
ˆIz1

 = 2π
ˆ
I1 · J ·
ˆ
I2 (5.1)
Anisotropic part of the J-tensor is usually small (and difficult to distinguish from the dipolar coupling)
and is neglected in practice. Therefore, only the isotropic (scalar) part of the tensor is considered
and the interaction is called scalar coupling:
2π


Jxx 0 0
0 Jyy 0
0 0 Jzz

 = 2π
Jxx + Jyy + Jzz
3


1 0 0
0 1 0
0 0 1

 = 2πJ


1 0 0
0 1 0
0 0 1

 . (5.2)
The scalar coupling is observed as splitting of peaks by 2πJ in NMR spectra. Proton-proton coupling
is significant (exceeding 10 Hz) up to three bonds and observable for 4 or 5 bonds in special cases (planar
geometry like in aromatic systems). Interactions of other nuclei are weaker, but the one-bond couplings
are always significant (as strong as 700 Hz for 31
P-1
H, 140 Hz to 200 Hz for 13
C-1
H, 90 Hz for 15
N-1
H
in amides, 30 Hz to 60 Hz for 13
C-13
C, 10 Hz to 15 Hz for 13
C-15
N). The value of J is given by the
distribution of electrons in bonds and thus reflect the local geometry of the molecule. Three-bond
scalar couplings can be used to measure torsion angles in molecules.
61
62 CHAPTER 5. ENSEMBLES OF SPINS INTERACTING THROUGH BONDS
5.1 Secular approximation and averaging
If the anisotropic part of the J-tensor is neglected, the J-coupling does not depend on orientation (scalar
coupling) and no ensemble averaging is needed. The secular approximation is applied like in the case
of the dipolar coupling.
• In the case of magnetic moments with the same γ and chemical shift, precessing about the z axis
with the same precession frequency,
ˆHJ = πJ 2ˆI1,z
ˆI2,z + 2ˆI1,x
ˆI2,x + 2ˆI1,y
ˆI2,y . (5.3)
• In the case of magnetic moments with different γ and/or chemical shift, precessing about the z
axis with different precession frequencies,
ˆHJ = 2πJ ˆI1,z
ˆI2,z = πJ 2ˆI1,z
ˆI2,z . (5.4)
5.2 Relaxation due to the J-coupling
In principle, the anisotropic part of the J-tensor would contribute to relaxation like the anisotropic
part of the chemical shift tensor, but it is small and usually neglected. Scalar coupling (isotropic part
of the J-tensor) does not depend on the orientation. Therefore, it can contribute to the relaxation only
through a conformational or chemical exchange. Conformational effects are usually small: one-bond
and two-bond couplings do not depend on torsion angles and three-bond coupling constants are small.
In summary, relaxation due to the J-coupling is rarely observed.
5.3 2D spectroscopy based on scalar coupling
5.3.1 Evolution in the presence of the scalar coupling
In the presence of the scalar coupling, the Hamiltonian describing evolution after a 90◦
pulse is complicated
even in a coordinate system rotating with ωrot = ωradio
H = εt · 2It −γ1B0(1 + δi1)
Ω1
I1z −γ1B0(1 + δi2)
Ω2
I2z + πJ (2I1zI2z + 2I1xI2x + 2I1yI2y) . (5.5)
However, if the precession frequencies differ, the Hamiltonian simplifies to a form where all components
commute. Therefore, the Liouville - von Neumann equation can be used geometrically as rotations
in three-dimensional subspaces of the 16-dimensional operator space. Rotations described by different
components of the Hamiltonian are independent and can be performed consecutively, in any order.
For a density matrix ˆρ(b) = It + κ(−I1y − I2y) after a 90◦
pulse, the evolution due to the chemical
shift (described by Ω1 and Ω2 and scalar coupling (described by πJ) can be analyzed as follows
I1t −→ I1t −→ I1t (5.6)
−I1y −→



−c1I1y −→
−c1cJ I1y
+c1sJ 2I1xI2z
+s1I1x −→
+s1cJ I1x
+s1sJ 2I1yI2z
(5.7)
5.4. SPIN ECHOES 63
−I2y −→



−c2I2y −→
−c2cJ I2y
+c2sJ 2I2xI1z
+s2I2x −→
+s2cJ I2x
+s2sJ 2I2yI1z
(5.8)
where the first arrows represent rotation ”about” I1z or I2z by the angle Ω1t or Ω2t, the second
arrows represent rotation ”about” 2I1zI2z by the angle πJt, and
c1 = cos(Ω1t) s1 = sin(Ω1t) (5.9)
c2 = cos(Ω2t) s2 = sin(Ω2t) (5.10)
cJ = cos(πJt) sJ = sin(πJt) (5.11)
Only I1x, I1y, I2x, I2y contribute to the expected value of M+, giving non-zero trace when multiplied
by ˆI+ (orthogonality).
Including relaxation and applying a phase shift by 90 ◦
, the expected value of M+ evolves as
1
4
e−R2t
e−i(Ω1−πJ)t
+ e−i(Ω1+πJ)t
+ e−i(Ω2−πJ)t
+ e−i(Ω2+πJ)t
(5.12)
which gives two doublets in the spectrum after Fourier transform:
Nγ2
¯h2
B0
4kBT
R2
R2
2 + (ω − Ω1 + πJ)2
+
R2
R2
2 + (ω − Ω2 − πJ)2
+
R2
R2
2 + (ω − Ω1 + πJ)2
+
R2
R2
2 + (ω − Ω2 − πJ)2
−i
Nγ2
¯h2
B0
4kBT
(ω − Ω1 + πJ)
R2
2 + (ω − Ω1 + πJ)2
+
(ω − Ω2 − πJ)
R2
2 + (ω − Ω2 − πJ)2
+
(ω − Ω1 + πJ)
R2
2 + (ω − Ω1 + πJ)2
+
(ω − Ω2 − πJ)
R2
2 + (ω − Ω2 − πJ)2
.
(5.13)
5.4 Spin echoes
Experiments utilizing scalar coupling are based on ”spin alchemy” - artificial manipulations of quantum
states of the studied system.
Spin echoes are basic tools of spin alchemy, providing the possibility to control evolution of the
chemical shift and scalar coupling separately.
Here we analyze three types of spin echoes for a heteronucler system (two nuclei with different γ,
1
H and 13
C in our example). In order to distinguish the heteronuclear systems from homonuclear ones,
we will use symbols Ij and Sj for operators of nucleus 1 and 2, repsectively, if γ1 = γ2. For the sake of
simplicity, relaxation is not included.
Vector analysis: Solid arrow - component of µ1 ⊥ B0 for spin 2 in |α , dashed arrow - component
of µ1 ⊥ B0 for spin 2 in |β , colors - different δi
5.4.1 Free evolution (Figure 5.1A)
Evolution of the system of two nuclei in the presence of scalar coupling was already described in
Section 5.3.1.
• ˆρ(a) = It + κ1Iz + κ2Sz
thermal equilibrium, the constants κ1 and κ2 are different because the nuclei have different γ.
64 CHAPTER 5. ENSEMBLES OF SPINS INTERACTING THROUGH BONDS
A
B
C
D
Figure 5.1: Vector analysis of spin echoes for 1H (nucleus 1) and 13C (nucleus 2) in an isolated –CH– group. In
individual rows, evolution of magnetization vectors in the plane ⊥ B0 is shown for three protons (distinguished by colors)
with slightly different precession frequency due to the different chemical shifts δi. The protons are bonded to 13C. Solid
arrows are components of proton magnetization for 13C in |β , dashed arrow are components of proton magnetization for
13C in |α . The first column shows magnetization vectors at the beginning of the echo (after the initial 90◦ pulse at the
proton frequency), the second column shows magnetization vectors in the middle of the first delay τ, the third and fourth
columns show magnetization immediately before and after the 180◦ pulse(s) in the middle of the echo, respectively, the
fifths column shows magnetization vectors in the middle of the second delay τ, the sixth column shows magnetization
vectors at the end of the echo. Row A corresponds to an experiment when no 180◦ pulse is applied, row B corresponds to
the echo with the 180◦ pulse applied at the proton frequency, row C corresponds to the echo with the 180◦ pulse applied
at the 13C frequency, and row D corresponds to the echo with the 180◦ pulses applied at both frequencies. The x-axis
points down, the y-axis points to the right.
5.4. SPIN ECHOES 65
• ˆρ(b) = It − κ1Iy + κ2Sz
90◦
pulse applied to nucleus 1 only
• ˆρ(e) = It + κ1 (−c1cJ Iy + s1cJ Ix + c1sJ 2IxSz + s1sJ 2IySz) + κ2Sz
free evolution during 2τ (t → 2τ in c1 etc.)
For nuclei with γ > 0, magnetizations of nucleus 1 (proton) evolve faster if nucleus 2 (13
C) is in
|β (the energy difference between |αβ and |ββ is larger than the energy difference between |αα and
|βα ) - solid arrows rotated by a large angle than dashed arrows in Fig. 5.1A.
The 2IxSz, 2IySz coherences do not give non-zero trace when multiplied by I+ (they are not
measurable per se), but cannot be ignored if the pulse sequence continues because they can evolve into
measurable coherences later (note that the scalar coupling Hamiltonian 2πJIzSz converts them to Iy,
Ix, respectively).
5.4.2 Refocusing echo (Figure 5.1B)
90◦
pulse exciting magnetic moment 1 and 180◦
pulse on the excited nucleus in the middle of the echo
a(π/2)1xb − τ −c (π)1xd − τ−e
The middle 180◦
pulse flips all vectors from left to right (rotation about the vertical axis x by 180 ◦
).
The faster vectors start to evolve with a handicap at the beginning of the second delay τ and they reach
the slower vectors at the end of the echo regardless of the actual speed of rotation.
Even without a detailed analysis of product operators, we see that the final state of the system does
not depend on chemical shift or scalar coupling: the evolution of both chemical shift and scalar coupling
is refocused during this echo.
The initial state of protons was described (after the 90◦
pulse) by −Iy in terms of product operators
and by an arrow with the −y orientation. As the vector only changed its sign at the end of the
experiment (arrow with the +y orientation), we can deduce that the final state of protons is +Iy:
ˆρ(e) = It + κ1Iy + κ2Sz
5.4.3 Decoupling echo (Figure 5.1C)
90◦
pulse exciting magnetic moment 1 and 180◦
pulse on the other nucleus in the middle of the echo
a(π/2)1xb − τ −c (π)2xd − τ−e
The middle 180◦
is applied at the 13
C frequency. It does not affect vectors of proton magnetization
but inverts polarization (populations) of 13
C (solid arrows change to dashed ones and vice versa). The
faster vectors become slower, the slower vectors become faster, and they meet at the end of the echo.
Without a detailed analysis of product operators, we see that the final state of the system does
not depend on scalar coupling (the difference between solid and dashed arrows disappeared) but the
evolution due to the chemical shift took place (arrows of different colors rotated by different angles
2Ω1τ). As the effects of scalar coupling are masked, this echo is known as the decoupling echo.
As the vectors at the end of the echo have the same orientations as if the nuclei were not coupled
at all, we can deduce that the final state of protons is identical to the density matrix evolving due to
the chemical shift only:
ˆρ(e) = It + κ1 (c1Iy − s1Ix) − κ2Sz
66 CHAPTER 5. ENSEMBLES OF SPINS INTERACTING THROUGH BONDS
y
H
N
τ τ
Figure 5.2: INEPT pulse sequence applied to 1H and 15N.
5.4.4 Recoupling echo (Figure 5.1D)
90◦
pulse exciting magnetic moment 1 and 180◦
pulses on both nuclei in the middle of the echo
a(π/2)1xb − τ −c (π)1x(π)2xd − τ−e
180◦
pulses are applied at 1
H and 13
C frequencies in the middle of the echo, resulting in combination
of both effects described in Figs. 5.1B and C. The proton pulse flips vectors of proton magnetization
and the 13
C flips polarization (populations) of 13
C (solid arrows change to dashed ones and vice versa).
As a result, the average direction of dashed and solid arrows is refocused at the end of the echo but the
difference due to the coupling is preserved (the handicapped vectors were made slower by the inversion
of polarization of 13
C).
Without a detailed analysis of product operators, we see that the effect of the chemical shift is
removed (the hypothetical arrows showing average direction of vectors of the same color just change
the sign), but the final state of the system depends on scalar coupling (the solid and dashed arrows
disappeared) but the evolution due to the chemical shift took place (arrows of different colores rotated
by different angles 2Ω1τ). As the effects of scalar coupling are masked, this echo is known as the
decoupling echo.
We can deduce that the final state of the system is obtained by rotation ”about” 2IzSz, but not
”about” Iz in the product operator space, and by changing the sign of the resulting coherences as
indicated by the vector analysis:
ˆρ(e) = It + κ1 (cJ Iy − sJ 2IySz) − κ2Sz
5.5 INEPT
INEPT is an NMR experiment based on the recoupling echo. It differs from the simple echo in two
issues:
• The length of the delay τ is set to 1/4J
• The echo is followed by two 90◦
pulses, one at the frequency of the excited nucleus – this one must
be phase-shifted by 90 ◦
from the excitation pulse, and one at the frequency of the other nucleus
(15
N in Fig. 5.2).
5.5. INEPT 67
With τ = 1/4J, 2πτ = π/2, cJ = 0, and sJ = 0. Therefore, the density matrix at the end of the
echo is
ˆρ(e) = It − κ1 (2IySz) − κ2Sz
−→ It + κ1 (2IzSz) − κ2Sz after the first pulse and
−→ It − κ1 (2IzSy) + κ2Sy after the second pulse.
If the experiment continues by acquisition, the density matrix evolves as
It −→ It −→ It (5.14)
−2IzSy −→



−c1 2IzSy −→
−c1cJ 2IzSy
+c1sJ Sx
+s1 2IxSz −→
+s1cJ 2IzSx
+s1sJ Sy
(5.15)
−Sy −→



−c2Sy −→
−c2cJ Sy
+c2sJ 2SxIz
+s2Sx −→
+s2cJ Sx
+s2sJ 2SyIz
(5.16)
Both the ”blue” coherence 2IzSy and the ”green” coherence Sy evolve into measurable product
operators, giving non-zero trace when multiplied by S+.
After calculating the traces, including relaxation, and applying a phase shift by 90 ◦
, the expected
value of M2+ evolves as
κ2
4
e−R2t
−e−i(Ω2−πJ)t
+ e−i(Ω2+πJ)t
+
κ1
4
e−R2t
e−i(Ω2−πJ)t
+ e−i(Ω2+πJ)t
(5.17)
The real part of the spectrum obtained by Fourier transform is
Nγ2
2
¯h2
B0
4kBT
−
R2
R2
2 + (ω − Ω2 + πJ)2
+
R2
R2
2 + (ω − Ω2 − πJ)2
+
Nγ1
2
¯h2
B0
4kBT
+
R2
R2
2 + (ω − Ω2 + πJ)2
+
R2
R2
2 + (ω − Ω2 − πJ)2
(5.18)
• The ”blue” coherence 2IzSy gives a signal with opposite phase of the peaks at Ω2 − πJ and
Ω2 + πJ. Accordingly, it is called the anti-phase coherence.
• The ”green” coherence Sy gives a signal with the same phase of the peaks at Ω2 −πJ and Ω2 +πJ.
Accordingly, it is called the in-phase coherence.
• More importantly, the ”blue” coherence 2IzSy gives a signal proportional to γ2
1 while the ”green”
coherence Sy gives a signal proportional to γ2
2 . The amplitude of the ”green” signal corresponds
to the amplitude of a regular 1D 15
N spectrum. The ”blue” signal ”inherited” the amplitude
with γ2
1 from the excited nucleus, proton. In case of 1
H and 15
N, γ1 is approximately ten times
higher than γ2. Therefore, the blue signal is two orders of magnitude stronger. This is why this
experiment is called Insensitive Nuclei Enhanced by Polarization Transfer.
• As described, the ”blue” and ”green” signals are combined, which results in different heights of
the Ω2 −πJ and Ω2 +πJ peaks. The ”blue” and ”green” signals can be separated if we repeat the
measurement twice with the phase of the proton y pulse shifted by 180 ◦
(i.e., with −y). It does
not affect the ”green” signal, but changes the sign of the ”blue” signal. If we subtract the spectra,
we obtained a pure ”blue” signal. This trick - repeating acquisition with different phases - is
known as phase cycling and is used routinely in NMR spectroscopy to remove unwanted signals.
68 CHAPTER 5. ENSEMBLES OF SPINS INTERACTING THROUGH BONDS
5.6 Heteronuclear Single-Quantum Correlation (HSQC)
HSQC is a 2D pulse sequence using scalar coupling to correlate frequencies of two magnetic moments
with different γ (Fig. 5.3).
• After a 90◦
pulse at the proton frequency, polarization is transfered to the other nucleus (usually
15
N or 13
C). The density matrix at the end of the INEPT is ˆρ(e) = It − κ1 (2IzSy) + κ2Sy
• During an echo with a decoupling 180◦
pulse at the proton frequency (red pulse in Fig. 5.3),
anti-phase single quantum coherences evolve according to the chemical shift
ˆρ(e) −→ It + κ1 (cos(Ω2t1)2IzSy − sin(Ω2t1)2IzSx) + κ2 (c11Sy + s11Sy).
We assume that the green coherences are discarded by phase cycling, as describe above, and ignore
them. Also, we ignore the red term which never evolves to a measurable coherence because it
commutes with all Hamiltonians.
• Two 90◦
pulses convert 2IzSy to 2IySz and 2IzSx to 2IySx. The magenta operator is a multiple
quantum coherence (a combination of zero-quantum and double-quantum coherence), which can
be converted to a single quantum coherence only by a 90◦
pulse. Since the pulse sequence does not
contain any more 90◦
pulses and since no multiple-quantum coherence is measurable, we ignore
2IySx.
• The last echo allows the scalar coupling to evolve but refocuses evolution of the scalar coupling. If
the delays τ = 1/4J, the measurable components of the density matrix evolve to −κ1 cos(Ω2t1)Iy
(rotation ”about” 2IzSz by 90 ◦
and change of the sign by the last 180◦
pulse at the proton
frequency).
• During acquisition, both chemical shift and scalar coupling evolve in the experiment described in
Fig. 5.3. Therefore, we obtain a doublet in the proton dimension of the spectrum. The second
dimension is introduced by repeating the measurement with t1 being incremented. Each increment
is measured twice with a different phase of one of the 90◦
pulses applied to nucleus 2, which
provides real (modulated by cos(Ω2t1)) and imaginary (modulated by sin(Ω2t1)) component of a
complex signal, like in the NOESY experiment. After calculating the trace, including relaxation
(with different rates R2 in the direct and indirect dimensions), phase shift by 90 ◦
and Fourier
transforms in both t1 and t2 dimensions, we obtain a 1D spectrum with peaks at Ω2 chemical
shift in the indirect dimension and a doublet at Ω1 ± πJ in the direct (proton) dimension. Note
that the spitting by ±πJ was removed by the red decoupling pulse in the indirect dimension.
5.6.1 Decoupling trains
Splitting of peaks in the direct dimension in spectra recorded by the pulse sequence in Fig. 5.3 is
undesirable. On the other hand, we acquire signal in real time and cannot remove the splitting by
a decoupling echo. In principle, we can divide the acquisition time into short fragments and apply a
180◦
pulse at the frequency of nucleus 2 (13
C or 15
N) in the middle of each such echo. In practice,
imperfections of such a long series of echoes, affecting especially magnetic moments with large Ω2,
are significant. However, more sophisticated series of pulses have much better performance. Typical
examples of decoupling pulse sequences are
• WALTZ - a series of 90◦
, 180◦
, and 270◦
pulses with phase of 0 ◦
(x), or 180 ◦
(−x), repeating in
complex patterns
• DIPSI - a similar series of pulses with non-integer rotation angles
• GARP - computer-optimized sequence of pulses with non-integer rotation angles and phases.
5.7. SYSTEMS WITH MULTIPLE PROTONS - ATTACHED PROTON TEST (APT 69
y
2
τ τ
t1
2
t1
2
τ τ
H
N
t
Figure 5.3: 1H,15N HSQC pulse sequence.
τ
1
t2
t
H
N
τ τ
y
τ
Figure 5.4: Idea of the decoupling in the direct dimension.
5.6.2 Benefits of HSQC
• 13
C or 15
N frequency measured with high sensitivity (higher by (γ1/γ2)5/2
than provided by the
direct detection)
• expansion to the second dimension and reducing the number of peaks in spectrum (only 13
C or
15
N-bonded protons and only protonated 13
C or 15
N nuclei are visible) provides high resolution
• 1
H-13
C and 1
H-15
N correlation is important structural information (which proton is attached to
which 13
C or 15
N)
5.7 Systems with multiple protons - attached proton test (APT
Systems CHn (C, CH, CH2, CH3).
Refocusing echo, but with excitation of 13
C (nucleus 2), followed by 13
C acquisition with proton
decoupling. The 13
C operators are labeled Sx, Sy, Sz, relaxation is ignored for the sake of simplicity.
• ˆρ(a) = It + κ1
n
j=1
(Ijz) + κ2Sz
• ˆρ(b) = It + κ1
n
j=1
(Ijz) − κ2Sy
• refocusing echo: evolution of Ω2 is refocused, scalar coupling evolves for 2τ as cos(2πjτ) and
sin(2πjτ), nucleus 1 (proton) is never excited (no proton 90◦
pulse) ⇒ only Ijz contributions
70 CHAPTER 5. ENSEMBLES OF SPINS INTERACTING THROUGH BONDS
• ˆρ(a) = It+κ1
n
j=1
(Ijz)+κ2



n = 0 : Sy
n = 1 : cSy − s2I1zSx
n = 2 : c2
Sy − sc(2I1zSx + 2I2zSx) − s2
4I1zI2zSy
n = 3 : c3
Sy − sc2
(I1zSx + I2zSx + I3zSx)
−s2
c(4I1zI2zSy + 4I1zI3zSy + 4I2zI3zSy) + s3
8I1zI2zI3zSx
where s = sin(2πJτ) and c = cos(2πJτ).
• Since decoupling is applied during acquisition, only the Sy coherences give a measurable signal.
They evolve under the influence of chemical shift, exactly like in a one-pulse experiment. If τ is
set to τ = 2J, then c = cos π = −1. Therefore, signals of C and CH2 are positive and signals of
CH and CH3 are negative ⇒ useful chemical information.
5.8 Homonucler correlation based on scalar coupling (COSY)
We started the discussion of experiments based on scalar couplings with heteronuclear correlations
because they are easier to analyze. The basic (and very popular) homonuclear experiment is COSY
(COrrelated SpectroscopY). Its pulse sequence is very simple, consisting of only two 90◦
pulses separated
by an incremented delay t1 (which provides the second dimension), but the evolution of the density
matrix is relatively complex. Here, we analyze evolution for a pair of interacting nuclei (protons).
• ˆρ(a) = It + κ(I1z + I2z)
thermal equilibrium, the matrices are different than for the noninteracting spin, but the constant
is the same.
• ˆρ(b) = It + κ(−I1y − I2y)
90◦
pulse, see the one-pulse experiment
• ˆρ(c) = It
+κ(−c11cJ1I1y + s11cJ1I1x + c11sJ12I1xI2z + s11sJ12I1yI2z)
+κ(−c21cJ1I2y + s21cJ1I2x + c21sJ12I1zI2x + s21sJ12I1zI2y),
where ci1 = cos(Ωit1), si1 = sin(Ωit1), cJ1 = cos(πJt1), and sJ1 = sin(πJt1) – evolution of the
chemical shift and coupling.
• The second 90◦
pulse creates the following coherences ˆρ(d) = It
+κ(−c11cJ1I1z+ s11cJ1I1x −c11sJ12I1xI2y− s11sJ12I1zI2y )
+κ(−c21cJ1I2z+ s21cJ1I2x −c21sJ12I1yI2x− s21sJ12I1yI2z ).
The red terms contain polarization operators, not coherences, they do not contribute to the
signal. The green terms contain in-phase single-quantum coherences, only they give non-zero
trace when multiplied with ˆM+ ∝ (I1x + iI1y + I2x + iI2y). The blue terms contain anti-phase
single-quantum coherences, they do not contribute to the signal directly, but they evolve into
in-phase coherences during acquisition due to the scalar coupling. The magenta terms contain
multiple-quantum coherences. They do not contribute to the signal, but can be converted to
single-quantum coherences by 90◦
pulses. Such pulses are not applied in the discussed pulse
sequence, but are used in some versions of the experiment.
• The terms in black frames evolve with the chemical shift of the first nucleus during acquisition:
s11cJ1I1x → s11cJ1c12cJ2I1x + s11cJ1s12cJ2I1y+ unmeasurable anti-phase coherences
−s21sJ12I1yI2z → s21sJ1c12sJ2I1xs21sJ1s12sJ2I1y+ unmeasurable anti-phase coherences ,
5.8. HOMONUCLER CORRELATION BASED ON SCALAR COUPLING (COSY) 71
where ci2 = cos(Ωit2), si2 = sin(Ωit2), cJ2 = cos(πJt2), and sJ2 = sin(πJt2). Using the following
trigonometric relations
cikciJ =
c−
ik + c+
ik
2
siksiJ =
c−
ik − c+
ik
2
ciksiJ =
−s−
ik + s+
ik
2
sikciJ =
s−
ik + s+
ik
2
, (5.19)
where c±
ik = cos((Ωi ± πJ)tk) and s±
ik = sin((Ωi ± πJ)tk), the terms contributing to the signal can
be written as

(s−
11 + s+
11)(c−
12 + c+
12)
[Ω1,Ω1]
+ (s−
21 + s+
21)(c−
12 + c+
12)
[Ω2,Ω1]


 I1x
+


(s−
11 + s+
11)(c−
12 + c+
12)
[Ω1,Ω1]
+ (s−
21 + s+
21)(c−
12 + c+
12)
[Ω2,Ω1]


 I1y
The first and second line show coherences providing the real and imaginary component of the
complex signal acquired in the direct dimension (t2). The imaginary signal in the indirect dimension
is obtained by repeating acquisition for each increment of t1 with a different phase (shifted
by 90 ◦
).
• The green component of the signal evolves with the same chemical shift in both dimensions,
providing diagonal signal (at frequencies [Ω1, Ω1] in the 2D spectrum). The blue (originally antiphase)
component of the signal also evolves with Ω1 in the direct dimension, but with Ω2. It
provides off-diagonal signal, a cross-peak at frequencies [Ω1, Ω1] in the 2D spectrum. Note that
the blue and green components have the phase different by 90 ◦
. Therefore, either diagonal peaks
or cross-peaks have the undesirable dispersion shape (it is not possible to phase both diagonal
peaks or cross-peaks, they always have phases differing by 90 ◦
). Typically, the spectrum is
phased so that the cross-peaks have a nice absorptive shape because they carry a useful chemical
information - they show which protons are connected by 2 or 3 covalent bonds.
• The diagonal peaks are not interesting, but their dispersive shape may obscure cross-peaks close to
the diagonal. The problem with the phase can be solved if one more 90 ◦
pulse is introduced. Such
a pulse converts the magenta multi-quantum coherences to anti-phase single-quantum coherences,
which evolve into the measurable signal. The point is that other coherences can be removed by
phase cycling, which results in a spectrum with a pure phase.1
This version of the experiment is
known as double-quantum filetered COSY (DQF-COSY). Its disadvantage is a lower sensitivity –
we lose a half of the signal.
• Also, note that each peak is split into doublets in both dimensions. More complex multiplets are
obtained if more than two nuclei are coupled. The distance of peaks in the multiplets is given
by the interaction constant J. In the case of nuclei connected by three bonds, J depends on the
torsion angle defined by these three bonds. So, COSY spectra can be used to determine torsion
angles in the molecule.
• The terms in gray frames evolve with the chemical shift of the second nucleus during acquisition
as s21cJ1I1x → s21cJ1c12cJ2I1x + s21cJ1s12cJ2I1y+ unmeasurable anti-phase coherences
−s11sJ12I1yI2z → s11sJ1c12sJ2I1xs11sJ1s12sJ2I1y+ unmeasurable anti-phase coherences and
give a similar type of signal for the other nucleus:
1Phase cycling can distinguish multi-quantum coherences from single-quantum ones, it cannot distinguish anti-phase
single quantum coherences from in-phase single quantum coherences.
72 CHAPTER 5. ENSEMBLES OF SPINS INTERACTING THROUGH BONDS


(s−
21 + s+
21)(c−
22 + c+
22)
[Ω1,Ω1]
+ (s−
11 + s+
11)(c−
22 + c+
22)
[Ω2,Ω1]


 I1x
+


(s−
21 + s+
21)(c−
22 + c+
22)
[Ω1,Ω1]
+ (s−
11 + s+
11)(c−
22 + c+
22)
[Ω2,Ω1]


 I1y