This content has been downloaded from IOPscience. Please scroll down to see the full text. Download details: IP Address: 147.251.27.31 This content was downloaded on 07/02/2015 at 13:42 Please note that terms and conditions apply. Exact Aharonov-Bohm wavefunction obtained by applying Dirac's magnetic phase factor View the table of contents for this issue, or go to the journal homepage for more 1980 Eur. J. Phys. 1 240 (http://iopscience.iop.org/0143-0807/1/4/011) Home Search Collections Journals About Contact us My IOPscience 240 B Diu Eur. J. Phys. l(1980)240-244. Rinted in Northern Ireland +J dkf(k)Fk(8,cp) f exp(-iE(k)t/h).(62) 0 Exact- -~ Although the second one is not exactly of the form studiedabove,they can bothbe treated by the Aharonov-Bo hmstationary-phase method (P2.1.2). The central-position of the fist packet is thenfound at wavefunction hko z,(t) =--m t x, = y p= 0. (63) obtained by applying Dirac’sAs for thescattered wave packet, the location of its the distance between this maximum and the origin of the coordinate system isgivenby factor maximum dependsonthe direction (e, chosen; magnetlc phase rde, cp; t)=-S;,(@, cp)+; t (64) M V Berry?hk” where S;(@ p) is the derivative with respect to k of the phase of the scatteringamplitude &(e, cp). These formulae are valid only in the asymptotic region (that is, for large It\). Their discussion goes along the same lines as above. Forlarge negative valuesof t, thereisno scattered wavepacket: the waves that build it interfere constructively only for negative values of r andthese are, of course,not permitted. Therefore, all that is found long before the collision is the plane waue packet which is then to be identified with the incident waue packet. For large positive values of t, both packets are effectively present: the first one continues along thepath of the incident packet and the second one diverges in all directions. The scattering cross section can then be deducedfrom the consideration of these wave packets. (Actually, one should also allow for slightly different orientations of the incident wavevector k, in orderto limit the plane wave packetnot only along Oz, but also in the perpendicular directions.) Heretoo, the same result can be obtained much more simply using the ‘probability fluid’ andthe corresponding ‘improper interpretation’ of a stationary state (Cohen-Tannoudji et a1 1977, p 912). So the questions raised and arguments discussed in the present paper can actually be applied to a much wider domain than just the one-dimensional square potential problems. References Cohen-Tannoudji C, Diu B and Laloe F 1977 Quantum Mechanics (Paris: Herman; New York: Wiley) Goldberger M L and Watson K M 1964 Collision Theory (New York: Wiley) Institute for Theoretical Physics, Princetonplein 5 , PO Box 80006, 3508 TA Utrecht, The Netherlands Received 3 December 1980 Abstracl Asolution of Schrodinger’s equation fora particle in a magnetic field B can be obtained from the wavefunction when B = 0 by Dirac’s prescription of multiplication by a phase factor. But this solution is often multiple-valued and hence unsatisfactory. It is shown that in the case of the Aharonov-Bohm effect the Dirac prescription can nevertheless be made to yield the single-valued exact wavefunction, provided it is applied not to the total wave with B =0 but to its separate components in a ‘whirling-wave’ representation. The mth whirling wave at a point r is a contribution that hasarrived at r after travelling m times around the region containing B. R&mC Onpeutobtenirune solution de l’bquation de Schrodinger pour une particule placke dans un champ rnagnttique B,i partir de la solution en champ nul, en appliquant la prescription de Dirac de multiplication par un facteur de phase. Mais cette solution est souvent multiforme et, par suite, non satisfaisante. On montre ici que, dans le cas de l’effet Aharonov-Bohm, on peut ndanmoins tirer de la prescription de Dirac la fonction d’onde exacte, (et uniforme); mais il faut pour cela appliquer cette prescription, non pas a l’inttgralitt de la fonction d‘onde en champ nul, mais, stpartment, i ses diverses composantes dans une reprtsentation o t ~la est parvenue aupoint r aprbs avoir tourne m fois autour de la rCgion oh se manifeste le champ B. m*rne composante en r correspond a la contributionqui t Permanent address: H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 lTL, England. 0143-08071801040240105501.50 0 The Institute of Physics & the EPS The Aharonov-Bohm effect 241 1 Dirac’s prescription Thisarticle is aboutthequantum mechanics of a particle in the presence of a magnetic field B, which will be represented by a vectorpotential A. This must satisfy where the lirst integral is around any closed curve C and thesecond is over any surface bounded by C. Locally, (1)implies B=VAA. (2) Then as iswell known the Hamiltonian operator describing the particle is Hb,P)=Hob, P -qA) (3) where Ho(r,p) is the Hamiltonianwithout the magnetic field but with all other forces unaltered, q is the charge on the particle and, working in position representation, p is the momentum operator -ihV. I shall consider wavefunctions +(r) corresponding to particles with fixed energy E, which must therefore satisfy Schrodinger’s equation H(r, -ihV)+(r) = E+(r). (4) As pointed out by Dirac (1931), a function +,(r) satisfying (4) can be constructed very simply, in terms of the wave in the absence of the field, i.e. in terms of t+h0(r)which satisfies Hob, -ihV)Jl0(r)= E+dr). (5) The construction consists in multiplying Jl0 by a ‘magnetic phase factor’ as follows: where ro is an arbitrary fixed position. The trouble with Dirac’s prescription is that although $tD satisfies the wave equation (4) it is not single-valued andthereforecannot correctly represent the true quantum statein the presence of B. To see why (6)is multivalued, let r be transported round a loop C back to the same point. During this process, the phase of (6)changes by where according to (1) CPc is the flu of B through C. As well as being multivalued, JID would, if it really represented the state,imply that the magnetic field has noeffect on theprobability density I+Iz, which is obviously not the case. Despite these difficulties,Dirac’s prescription has been used in an inexact way to make predictions about the effect of a magnetic field, and these have been experimentally verified. In this inexact procedure, it is imagined that Go(r)consists of two parts, written as corresponding to waves reaching r by different routeslabelled 1 and 2. The effect of B is then included by incorporating the magnetic phase factor (6)into each part separately. This gives path 1 ++F’(r) expc; 1:A dr) path 2 where CP is the flux through the loop from ro to r along path 2 and back to r along path 1.In physical terms,thisresultredicts a change in the interference between 4:‘ and +Lz’ because theirrelative phase has been changed by the factor involving CP. But the exponential prefactor remains, and will still cause JID to be multivalued andhence unsatisfac- tory. My purpose here is to show, using the example of the Aharonov-Bohm effect, how Dirac’s prescription can in fact be used to obtain the exact wavefunction in the presence of a field. The procedure will be to decompose IL0, which of course is single-valued, into an infinite number of components (‘whirling waves’), each of which is multivalued, then to apply (6) to each whirling wave, and finally to resum the magnetically phase-shifted whirling waves to get the exact single-valued wavefunction 9. 2 Aharonov-Bob effect for thin solenoids Aharonovand Bohm (1959) considered a field B confined within a longstraightsolenoiddirected along the z axis andcontaining flux CP. Charged particles with energy E and mass m are incident from the positive x direction (figure 1). They are scattered by the solenoid but cannot penetrate into it. Aharonovand Bohm came tothe surprising conclusion (whichis still controversial-see Casati and Guarneri (1979), Roy (1980) and the remarks at the end of the paper by Berry et al (1980)) that the flux CP can affect particles eventhough the region containing the field is inaccessible to the particles. This comes about because the Hamiltonian (3) involves CP notthrough its field B but through the vector potential A, which, because of (l),cannot vanish outside the solenoid,since its line integral must equal CP. I shall use the simplest potential satisfying this condition, namely A(r) = “ 8 01 2 m 242 M V Beny Path 1 ( m= 01 f"""""""" r. I" Incident C" particles Figure 1 Geometry of the Aharonov-Bohm effect. The solenoid is shown in cross section as a black circle and carries flux of a magnetic field normal to the page. Two paths reaching a point r are shown; the broken path is equivalent to path 2. where r, 6 are plane polar coordinates and 6 is the azimuthal unit vector. In the simplest case, the solenoid is idealised as being infinitely thin, so that we are considering scattering by a single flux line. The incident beam has the wavefunction $o(r)=exp(-ikx) =exp(-ikr cos 8) (11) where k = m / h . (12) We seek the wave $ when @ is non-zero. This must satisfy the 'inpenetrability' condition that t/t = 0 on the flux line at r =0. Consider first the Dirac prescription (6). With ro taken at a point on the positive x axis this involves the phase where CY is the magnetic flux parameter, defined by a q@/h. (14) Therefore (6) converts (11)intothe magnetically phase-shifted wave qD(r)=exp(-ikr cos 0 +icre). (15) The multivaluedness is now explicit: $D changes by a factor exp(2aia) during a circuit of the solenoid, and this factor is not unity unless a is an integer, i.e. unless the flux is quantised. Moreover, JID does not vanish at r = 0. 3 Poisson transformationto whirling waves In obtaining the exact solution $ from the solution (11) without the field, the first step is to express Go as an angular momentum decomposition into partial waves by making use of the relation (Gradshteyn and Ryzhik 1965) LC exp(-ikr cos 8) = c (-i)"'J,,,(kr)exp(il8). (16) The modulus signs on twoof the 1 indices do not affect the value of the sum butare nevertheless important for a reason soon to be explained. Next, we transform the summationover 1 by means of the Poisson summation formula (Lighthill 1958). For any summand F(I), this replaces the sum by a series of integrals over F(h) which is any 'interpolation' of F(1) to non-integral values of its variable. The formula is I=-r 3 m .cc c F(O= c J dhF(A)exp(2~imh). (17) l=-= m=-- -cc When applied to (16) it gives LC t/to(r, 8) = c Tm(r,8) (18) m="" where T,(r, e)= dA exp(-$ia(A()JIAl(kr) 1-1 x exp[ih(8 +2am)]. (19) The terms T, are not single-valued. In fact T,,,(r, 8 + 2 ~ ) =Tmtl(r, 8) (20) as follows easily from(19). It is this relation that ensures the single-valuedness of the sum (18) despite the multivaluedness of its terms. Now comes the most important step, which consists of an interpretation of T, based on the fact that these terms contain 8 in the combination 8 + 2am. This is to restrict 8 by -a <8 +T and then interpret T,,,(r, 8) as a wane am'uing at 8 after making m anticlockwise circuits of the origin. I shall call T,,,(r, e) the mth 'whirling-wave' component of &. Each whirling wave is a (multivalued) solution of Schrodinger's equation without the magnetic flux. Therefore, Dirac's prescription (6) can be applied to yield a whirling wave satisfying Schrodinger's equation in the presence of the flux. The phase is given by (13), but instead of 8 we must write 8+2am because that is the total angle turned through. The new whirling waves are there- fore TE(r,e) = Tm(r,8) exp[ia(8+2~m)]. (21) Summing over m gives, on using (19), the wave m The Aharonov-Bohm effect 243 On defining A + a as a new variable this can be reverse-Poisson-transformed to give m J,(r, 0) = c (-i)l"a'J,l-a,(kr) exp(iI0). (23) The wave (23) satisfies Schrodinger's equation and is manifestly single-valued. In fact it is the exact solutionof the problem originally obtained by Aharonovand Bohm (1959), who used a quite different procedure. When the flux is quantised, i.e. when a is an integer,(23)reduces simply to the incident wave (11) multiplied by aphasefactor exp(ia0) which is single-valued. When a is not an integer, (23) contains a wave scattered out towards r = m by the flux line, as well asa complicated phase structure centred on the flux line; since both these aspects of the Aharonov-Bohm wavefunction havebeen recently discussed by Berry et al (1980), I shall not dwell or, them furtherhere, except tomakeone point.This is thatfor nonintegral a,when the flux has aphysical effect on the wave, J, as given by (23) vanishes as r +0, showing thatthe wave is indeed zero where the fluxis non-zero. If the modulus signs had notbeen inserted into the original summation (16), this rzsult would not havebeenobtained,and indeed the integrals (19) forthe individual whirling waves would have diverged. [=-m 4 Generalisation and discossion Now let the solenoid be of finite radius. Thiscan be modelled by a cylindrically symmetric scalar potential fie!d with a'hardcore' at a finite radius, preventing the particles fromenteringthe region where the flux is.The wave J,o in the absence of the flux is now not justthe incident wave (ll),but includes the wave scattered by the cylinder. Therefore the partial-wave decomposition of Jl0 is m +O(r, e) = C Rllltr)exp(ile) (24) I =-m where Rlll(r)is the solution of the radial equation with angular momentum (I) obtained by separation of the variables r and 8 in the two-dimensional Schrodinger equation obtained from (5) and including a repulsive potential excluding particlesfrom the cylinder. Again the Poisson formula (17) can be employed to transform the s u m over l, with the result that J,o is given by (18) with T,, defined by Tm(r,e) = jmdl\ Rl,l(r)exp[iA(8+ 2mn)] (25) instead of (19). And again the T, (r, 8 ) can be regarded as whirling waves and magnetically phaseshifted as in (21),to give the wave +scattered by a finite cylinder containing flux m: -m This can be reverse-Poisson-transformed to give m +(r, e) = 2 +a,(r) e x p W (27) I=" which again is the exact solution. The outcome of this analysis is that it is possible to obtain single-valued wavefunctions by means of the Dirac prescription (6), provided this is applied to thecorrect representation of the wavefunction in the absence of the field. Inthe Aharonov-Bohm effect this representation consists of a decomposition into whirling waves T,. Mathematically, these arise because the impenetrable cylinder makes the space multiply connected, so that paths encircling the origin differentnumbers of times cannot be deformed intooneanotherand mustbe given magneticphase shifts which take account of the differentnumbers of circuits. It seems likely that the same idea could be employed to solve other problems involving magnetic fields. We are now in a position to understand why the inexact argument presented in Q1is often successful. Depending on the precise scattering properties of the cylinder, it may be the case that in a particular angular region only two of the whirling waves (25) have appreciable and comparable magnitudes. These may be represented,for example, by the two paths shown in figure 1, which correspond to m =0 (path 1) and m = -1 (path 2) (to see thatpath 2 doesindeedcorrespond to m = -1, simply note thatit can be obtainedfrom path 1 by adding a single clockwise circuit of the cylinder as illustrated by the broken path in figure 1). Then in this angular region we may approximate J, by 8 )--. To(r,e)+ T-l(r,e) (28) which is precisely of the previously assumed form (8). Applying the Dirac prescription (21) now gives the analogue of (2), namely J,(r, e) =exp(ia8)[To(r, 6) +exp(-2i~a)T-~(r,e)]. (29) As noted earlier, this is not single-valued. Now we can see why: although the neglected whirling waves ( m# 0 or -1) are small in the angular region considered, they become large when 8 increases by 2 a (because of (20)) and must be included if J, is to be single-valued after a circuit of the cylinder. The whirling waves T,,into which Go is decomposed are unfamiliar, and I conclude with a brief discussion of them. For a cylinder whose radius is large in comparison with the deBroglie wavelength 2a/k of the incident wave, it is valid to employ semiclassical methods to obtain an asymptotic approximationfor Go. Such analysis isnow standard (see e.g. Rubinov 1961, Berry andMount 1972) and yields the result that the Poisson formula leads Eur. J. Phys. 1 (1980)244-248. Printed in Northern Ireland 244 M V Berry to contributing T,,,(r,e) which can be expressed in terms of trajectories arriving at r, 8 after encircling the origin m times. If the cylinder is surrounded by a region of attracting potential, some of these trajectories can be actual classical orbits winding smoothly arovnd several times before emerging. Otherwise,they can be ‘diffracted rays’ (Keller 1958) which skim around the impenetrable surface of the cylinder before emerging tangentially. But I emphasise that the whirling-wave representation is exact and fully quantum mechanical, independent of any semiclassical interpretation. Thisshouldbe clear from the fact that it was introduced in 03 as a representation of aplane wave without any scatterer, so that the whirling waves are a consequence of choosing a line in space (later to be occupied by a flux line) around which rotationsareto be counted, and around which no ‘real’ rays are winding. In this case formula (19) forthe whirling waves can be reduced a little by contour integration: T,,(r,e) =exp(-ikr cos e)&,,, (T+iy) exp(ikr cosh y) (T +iy)’- (e+2 ~ m ) ’ (-T