# AHARONOV-BOHM EFFECT: A QUANTUM VARIATION AND A CLASSICAL ANALOGY

Vladan Panković , Darko V. Kapor , Stevica Djurović, Milan Pantić

Department of Physics, Faculty of Sciences, 21000 Novi Sad, Trg Dositeja Obradovića 4., Serbia , This email address is being protected from spambots. You need JavaScript enabled to view it.

PACS number: 03.65.Ta

**Abstract**

In this work we consider a quantum variation of the usual Aharonov-Bohm effect with two solenoids sufficiently close one to the other so that (external) electron cannot propagate between two solenoids but only around both solenoids. Here magnetic field (or classical vector potential of the electromagnetic field) acting at quantum propagating (external) electron represents the quantum mechanical average value or statistical mixture. It is obtained by wave function of single (internal, quantum propagating within some solenoid wire) electron (or homogeneous ensemble of such (internal) electrons) representing a quantum superposition with two practically non-interfering terms. All this implies that phase difference and interference shape translation of the quantum propagating (external) electron represent the quantum mechanical average value or statistical mixture. On the other hand we consider a classical analogy and variation of the usual Aharonov-Bohm effect in which Aharonov-Bohm solenoid is used for the primary coil inside secondary large coil in the remarkable classical Faraday experiment of the electromagnetic induction.

**1. Introduction**

As it is well-known in the usual experimental arrangement corresponding to the Aharonov-Bohm effect [1]-[4] (representing an especial case of the Berry phase phenomenon [4], [5]) behind a diaphragm with two slits there is a long and very thin solenoid (placed perpendicularly, i.e. in the z-axis direction, to the x-0-y plane of the propagation of the (external, without solenoid wire and solenoid) electron passing through the diaphragm). When through the solenoid there is no any classical (constant) electrical current, there is neither any classical (homogeneous) magnetic field within the solenoid nor any classical vector electromagnetic potential without solenoid. In this case on the remote detection plate usual quantum interference shape, corresponding to the (external) electron (precisely statistical ensemble of the electrons) starting from a source, passing through both diaphragm slits and propagating externally around solenoid, will be detected. But, when through the solenoid there is a constant classical electrical current J that induces a homogeneous classical magnetic field B (directed along z-axis) proportional to J within the solenoid and corresponding classical vector electromagnetic potential A (whose lines represent the circumferences in x-0-y plane) without solenoid, a phase difference in the wave function of the quantum propagating (external) electron. This phase difference equals

(1) Δφ= eΦ/ћ = eBS/ћ

where e – represents the electron electric charge, Φ - magnetic flux through the solenoid base equivalent to product of the intensity of the magnetic field B and surface of the solenoid base S, and ћ=h/(2π) – reduced Planck constant. Also, this phase difference causes experimentally measurable translation of the mentioned usual (external) electron interference shape on the detection plate along x-axis for value

(2) Δx = - (L/d) (λ/2π) eΦ/ћ = - (L/d) (e/m) Φ/v = - (L/d) (e/m) BS/v

where L represents the distance between diaphragm and detection plate, d – distance between two diaphragm slits (much larger than solenoid base radius R=(πS)1/2) and λ=h/(mv) – de Broglie wavelength of the quantum propagating (external) electron with mass m and speed v. (It can be observed and pointed out that in (2) after introduction of the explicit form of the de Broglie wavelength Planck constant effectively disappears and (2), or, generally speaking, Aharonov-Bohm effect obtains formally a non-quantum, classical form.) In this way it can be concluded, seemingly paradoxically, that classical magnetic field within solenoid definitely influences at the quantum mechanical propagation of the (external) electron without solenoid without any local connection. This paradox can be explained by supposition that, in fact, there is local influence of the classical vector potential of the electromagnetic field without solenoid at the quantum mechanical propagation of the (external) electron without solenoid. But, it implies that quantum mechanical description of the electromagnetic phenomena by vector (and scalar) potential is more complete than the quantum mechanical description by magnetic (electromagnetic) field [3]. In other words, while in the classical physics description of the electromagnetic phenomena needs immediate use of the magnetic (electromagnetic) field and only intermediate use of the vector (and scalar) potential in the quantum physics situation is completely opposite.

In this work we shall consider a variation of the usual Aharonov-Bohm effect with two solenoids sufficiently close one to the other so that (external) electron cannot propagate between two solenoids but only around both solenoids. Here magnetic field (or classical vector potential of the electromagnetic field) acting at quantum propagating (external) electron represents the quantum mechanical average value or statistical mixture. It is obtained by wave function of single (internal, quantum propagating within some solenoid wire) electron (or homogeneous ensemble of such (internal) electrons) representing a quantum superposition with two practically non-interfering terms. All this implies that phase difference and interference shape translation of the quantum propagating (external) electron represent the quantum mechanical average value or statistical mixture. In this way we shall obtain a very interesting generalization of the usual Aharonov-Bohm effect and Berry phase concept. On the other hand we shall consider a classical analogy and variation of the usual Aharonov-Bohm effect in which Aharonov-Bohm solenoid is used for the primary coil inside secondary large coil in the remarkable classical Faraday experiment of the electromagnetic induction. Obtained induced current in the secondary coil represents a classical physical phenomenon whose existence needs immediate use of the vector potential without primary coil locally interacting with electrons in the secondary coil.

**2. Aharonov-Bohm effect for electron phase representing a quantum mechanical average value**

Suppose now that behind diaphragm there are two, identical, long and thin solenoids with the same bases (placed perpendicularly, i.e. in the z-axis direction, to the x-0-y plane of the propagation of the (external) electron passing through the diaphragm). Suppose, also, that solenoids are sufficiently close one to other so that (external) electron practically cannot propagate between solenoids.

Consider situation when classical electric current J1 flows through the first solenoid wire and simultaneously and independently electric current J2 flows through the second solenoid wire. Then within the first solenoid there is classical homogeneous magnetic field B1 proportional to J1 and within the second solenoid there is classical homogeneous magnetic field B2 proportional to J2. It implies that through base of the first solenoid there is classical magnetic flux Φ1 proportional to B1 and through base of the second solenoid there is classical magnetic flux Φ2 proportional to B2. Total classical magnetic flux through surface determined by quantum trajectories of the (external) electron passing through the first and second diaphragm slit, Φ, is simply sum of Φ1 and Φ2, i.e.

(3) Φ = Φ1+Φ2

It implies that total difference of the (external) electron wave function phase and total translation of the (external) electron quantum interference shape on the detection plate along x-axis equal

(4) Δφ = e(Φ1+Φ2)/ћ = Δφ 1+ Δφ 2

(5) Δx = - (L/d) (λ/2π) e(Φ1+Φ2)/ћ = Δx 1+ Δx 2

where Δx k=-(L/d) (e/m)Φk/v represents the translation of the (external) electron quantum interference shape caused by interaction with the k-th solenoid for k=1,2. So, total difference of the (external) electron wave function phase and total translation of the (external) electron quantum interference shape represent simple, classical sum of the corresponding variables through the first and second solenoid.

Especially, for J1=-J2= α/2 it follows B1=-B2=β/2 , Φ1=-Φ2= γ/2, Δφ 1=-Δφ 2=δ/2 and Δx1=- Δx 2=ε/2 so that according to (3)-(5) Φ =0, Δφ =0 and Δx =0, where α, β, γ, δ and ε represent some value of electric current, magnetic field, magnetic flux, phase difference and interference shape translation. Simply speaking, here total phase difference and total translation of the quantum interference shape for (external) electron equal zero.

But consider other, principally different situation. Suppose that there is (internal) single electron that can arrive and stand captured in the first or second solenoid wire only after interaction with an (internal) electron beam splitter (e.g. pair of the Stern-Gerlach magnets or similar). After interaction with beam splitter, wave function of the single (internal) electron represents the quantum superposition of two practically non-interfering terms

(6) Ψ = c1Ψ1+ c2Ψ2 .

First term describes the (internal) electron that arrives and stands captured in the first solenoid wire with probability amplitude c1 while second term describes (internal) electron that arrives and stands captured in the second solenoid wire with probability amplitude c2. Given probability amplitudes or superposition coefficients satisfy normalization condition

(7) | c1|2+ | c2|2 = 1 .

Also, since (internal) electron captured in one solenoid wire cannot turn out in the other solenoid wire Ψ1and Ψ2 represent practically non-interfering wave functions so that practically

(8) Ψ1Ψ*2= Ψ*1Ψ2=0 .

Then, according to general definition and (6), (8), quantum mechanical total electrical current of the single (internal) electron equals

(9) j = iћe/(2m)( Ψ∂Ψ*-Ψ* ∂Ψ) = | c1|2j1+ | c2|2j2 .

Here jk= iћe/(2m)( Ψk∂Ψk*/∂η -Ψk* ∂Ψ1/∂η) represents the quantum electric current of the single (internal) electron in the k-th solenoid wire for k=1,2 where η represents the (internal) electron coordinate. In this way total quantum mechanical electrical current of the single (internal) electron (9) represents the quantum mechanical average value or statistical mixture of the quantum mechanical electrical currents of the single electron within the first and second solenoid.

Further consider a homogeneous statistical ensemble or simply beam of n (internal) electrons all described by wave function (6). Then total quantum electrical current of this ensemble J is given by expression

(10) J = nj = | c1|2nj1+ | c2|2nj2 = | c1|2J1+ | c2|2J2

where Jk= njk represents the ensemble electrical current in the k-th solenoid wire for k=1, 2. In other words total quantum electrical current of the ensemble represents the quantum mechanical average value or statistical mixture of the currents trough the first and second solenoid wire.

It simply implies

(11) B= | c1|2B1+ | c2|2B2

(12) Φ = | c1|2Φ 1+ | c2|2Φ 2 .

Here B, B1, B2 represent the ensemble total quantum magnetic field, ensemble quantum magnetic field in the first and ensemble quantum magnetic field in the second solenoid wire proportional to J, J1 and J2. (It can be added that, according to usual electro-dynamical formalism, quantum magnetic fields B1 and B2 (directed along z-axis) within two solenoids correspond to quantum vector potentials of the electromagnetic fields A1 and A2 (whose lines represent the circumferences in x-0-y plane with centers corresponding to the bases of corresponding solenoids) without solenoids.) Also, Φ, Φ1, Φ2 represent the ensemble total quantum magnetic flux, ensemble quantum magnetic flux through the base of the first and ensemble quantum magnetic flux through the base of the second solenoid corresponding to B, B1, B2 . In other words (internal) electron ensemble total quantum magnetic field and magnetic flux represent quantum mechanical average values or statistical mixtures of corresponding variables trough the first and second solenoid.

All this implies that total translation of the wave function phase and total interference shape on the detection plate along x-axis of single (external) electron propagating around both solenoids equal

(13) Δφ = e(| c1|2 Φ1+ | c2|2 Φ2)/ћ = | c1|2 Δφ 1+ | c2|2 Δφ 2

(14) Δx = - (L/d) (λ/2π) e(| c1|2 Φ1+ | c2|2 Φ2)/ћ = |c1|2 Δx 1+ | c2|2 Δx 2

where Δx k=-(L/d) (e/m)Φk/v represents the translation of the (external) electron quantum interference shape caused by interaction with the k-th solenoid for k=1,2. So, total translation of the wave function phase and total interference shape on the detection plate along x-axis of single (external) electron represent quantum mechanical average values or statistical mixtures of corresponding variables trough the first and second solenoid.

It represents a result principally different from corresponding classical case (3), (4). Especially, for | c1|2=| c2|2=1/2, for J1=-J2= α, B1=-B2=β , Φ1=-Φ2= γ, Δφ 1=-Δφ 2=δ and Δx1=- Δx 2=ε, where α, β, γ, δ and ε represent some value of electric current, magnetic field, magnetic flux, phase difference and interference shape translation, (13), (14) imply

(15) Δφ = | c1|2 Δφ 1+ | c2|2 Δφ 2 = 1/2 δ + 1/2 (-δ)

(16) Δx = |c1|2 Δx 1+ | c2|2 Δx 2 = 1/2 ε + 1/2 (-ε) .

It means that with the 1/2 probability phase difference δ and interference shape translation ε will be detected, either that with the same probability 1/2 phase difference -δ and interference shape translation -ε will be detected. It is principally different from corresponding especial classical case, previously considered, where both, total phase difference and total interference shape translation, are zero.

In this way we obtain a very interesting generalization of the usual Aharonov-Bohm effect and Berry phase concept since we obtain here a quantum mechanical average value or statistical mixture of the corresponding Berry phases.

**3. Aharonov-Bohm paradox analogy in the classical Faraday electromagnetic induction experiment**

In remarkable Faraday iron ring or torus apparatus for electromagnetic induction primary, left coil warped around a part of the left hand of iron ring or torus by a switch mechanism can be connected or disconnected with a voltage source, while secondary, right coil warped around a part of the right hand of the iron ring or torus is permanently connected with an ammeter. Then by a short connection or disconnection of the primary coil with voltage source there is a change of the magnetic field in this coil that immediately continues and appears as the magnetic field and corresponding magnetic flux change in the secondary coil. For this reason, according to the famous Faraday law of the electromagnetic induction, in the secondary coil induced electric current or voltage appears.

Variation of mentioned electromagnetic induction experiment in which there is no iron ring or torus at all and where both coil in air have mutually parallel cylindrical shapes, so that a change of the magnetic field in the primary coil immediately continues and appears as the magnetic field and corresponding magnetic flux change in the secondary coil, are well known too.

Consider, however, next variation of electromagnetic induction experiment in which primary cylindrical coil, i.e. solenoid, holds small radius and large height while secondary coil holds large radius and small height. Moreover, suppose that both coils are placed coaxially so that secondary coil is placed on the half of the height of the primary coil. By realization of this experiment (that anyone can do very easy) electromagnetic induction definitely appears as well as in all previous experiments. Nevertheless, some interesting details must be considered additionally.

According to introduced suppositions and well-known facts magnetic field of the primary coil (far away form its ends or nearly half of its height) is non-zero and homogeneous exclusively within coil while outside this coil it is zero (in difference from non-zero electromagnetic vector potential). Similar refers to the immediate changes of the magnetic field of the primary coil that, in distinction to the previously discussed experiments of the electromagnetic induction, cannot do any immediate influence at the secondary coil. Obviously it represents a situation deeply conceptually analogous to situation with magnetic field of the solenoid and electron phase in Aharonov-Bohm effect. Here, as well as in Aharonov-Bohm effect, local character of the classical electromagnetic interaction between primary and secondary coil needs necessarily introduction of the vector potential as a real physical field.

Finally, usual, quantum Aharonov-Bohm effect can be considered by such geometry of the experimental arrangement when interference curve has such shape that (external) electron with large probability can be detected in the domain between left and right first local minimums nearly zero maximum of the interference curve. Then, by change of the electrical current, i.e. magnetic field within solenoid, during a small time interval there is discussed phase difference and experimentally measurable translation of the mentioned usual (external) electron interference shape, e.g. mentioned domain of the (external) electron. It with large quantum probability, or, simply speaking, almost exactly, can be quasi-classically effectively treated as the corresponding translation of the electron. In especial case that detection plate represents simply detection wire warped in the coil with ammeter this translation of the electron during small time interval represents corresponding small but measurable in principle electrical current. All this, further, can be simply connected with previous discussed example of the Faraday induction experiment.

**4. Conclusion**

In conclusion we can shortly repeat and pointed out the following. In the usual Aharonov-Bohm effect, representing an especial case of the Berry phase phenomenon, classical magnetic field within long and thin solenoid (or classical vector potential of the electromagnetic field without this solenoid) behind two slits diaphragm causes phase difference and interference shape translation of the quantum propagating electron. In this work we consider a variation of the usual Aharonov-Bohm effect with two solenoids sufficiently close one to the other so that (external) electron cannot propagate between two solenoids but only around both solenoids. Here magnetic field (or classical vector potential of the electromagnetic field) acting at quantum propagating (external) electron represents the quantum mechanical average value or statistical mixture. It is obtained by wave function of single (internal, quantum propagating within some solenoid wire) electron (or homogeneous ensemble of such (internal) electrons) representing a quantum superposition with two practically non-interfering terms. All this implies that phase difference and interference shape translation of the quantum propagating (external) electron represent the quantum mechanical average value or statistical mixture. In this way we obtain a very interesting generalization of the usual Aharonov-Bohm effect and Berry phase concept. On the other hand we consider a classical analogy and variation of the usual Aharonov-Bohm effect in which Aharonov-Bohm solenoid is used for the primary coil inside secondary large coil in the remarkable classical Faraday experiment of the electromagnetic induction.

**Acknowledgements**

Authors are very grateful to Branko Marčeta, Milan Mrđen and Vojislav – Voja Božić "Sremac" for inspiriting discussions, support and technical and other help.

**References**

[1] Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485

[2] M. Peshkin, A. Tonomura, The Aharonov-Bohm Effect (Springer-Verlag, New York-Berlin, 1989)

[3] R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, Vol. 3 (Addison-Wesley Inc., Reading, Mass. 1963)

[4] D. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall Inc., Englewood Cliffs, New Jersey, 1995)

[5] M. V. Berry, Proc. Roy. Soc. A (London) 392 (1984) 45