Department of physics

Seminar - 4. letnik

Aharonov-Bohm effect

Author: Ambrož Kregar

Mentor: prof. dr. Anton Ramšak

Ljubljana, marec 2011

Abstract

In this seminar I present Aharonov-Bohm effect, a quantum phenomenon in which aparticle is effected by electomagnetic fields even when traveling through a region of spacein which both electric and magnetic field are zero. I will describe theoretical backgroundof the effect, present some experimental verifications and show how this phenomenon canbe practicaly used in modern devices for precise measurement of magnetic field.

Contents1 Introduction 1

2 Maxwell’s equations and gauge symmetry 22.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Charged particle in proximity of solenoid magnet 33.1 Schrödinger equation in electromagnetic field . . . . . . . . . . . . . . . . 33.2 Vector potential of solenoid magnet . . . . . . . . . . . . . . . . . . . . . 43.3 Wavefunction in vector potential . . . . . . . . . . . . . . . . . . . . . . . 43.4 Magnetic Aharonov-Bohm effect . . . . . . . . . . . . . . . . . . . . . . . 53.5 Electric Aharonov-Bohm effect . . . . . . . . . . . . . . . . . . . . . . . . 63.6 Aharonov-Bohm effect and superconductors . . . . . . . . . . . . . . . . 7

4 Experimental evidence of Aharonov-Bohm effect 84.1 Solenoid magnet experiment . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Toroidal magnet experiment . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Practical use of Aharonov-Bohm effect 11

6 Conclusion 12

7 Literature 13

1 IntroductionIn classical mechanics, the motion of particles is described by action of forces. Newton’ssecond law ~F = m~a tells us how the particle will move through space under the influenceof force ~F , which is, in general, always Lorentz force which describes interaction betweencharged particle and electric and magnetic fields [1].

Electric and magnetic fields are uniquely described by Maxwell’s equations, which wewill describe in detail in next section. As we will see, description of electromagnetic phe-nomena can be simplified by introduction of electromagnetic potentials: scalar potentialφ and vector potential ~A. To write electric and magnetic field in form of potentials is ausefull because we need only four components (one scalar field and three components ofvector field) to describe electromagnetic field, which in genaral consists of six components(three components for each vector field ~E and ~B). Until the beginnig of the 20. cen-tury it was widely belived that potentials are only a mathematical construct to simplifycalculations and that they contain no physical significance.

With the development of quantum mechanics in the early 20. century, this view wasput under question, because Schrödinger equation, basic equation of quantum mechan-ics, doesn’t contain fields but potentials. So the question arise, which description ofelectromagnetic phenomena is more fundamental, through electric and magnetic fields orthrough scalar and vector potentials. In 1959, Yakir Aharonov and his doctoral advisorDavid Bohm, proposed an experiment to resolve this issue. The hearth of the experiment

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is the effect in which wavefunction aquire some additional phase when traveling throughspace with no electromagnetic fields, only potentials. This is called Aharonov-Bohm effectand will be the main topic of this seminar.

First we will take a closer look at the electromagnetic potentials and their importantproperty of gauge symmetry. We will examine some important properities of wavefunctionof a charged particle in electromagnetic field, show how they lead to Aharonov-Bohmeffect and describe it’s properties in detail. In second part of seminar we will take a lookat some experimental evidence of the effect and discuss it’s importance and weather itcan be use in some practical way.

2 Maxwell’s equations and gauge symmetry

2.1 Maxwell’s equations

In 1861, Scottish physicist and mathematician James Clerk Maxwell wrote four partialdifferential equations which represents a foundation of classical electrodinamics [1].

∇ · ~E =ρ

ε0(1)

∇ · ~B = 0 (2)

∇× ~E = −∂~B

∂t(3)

∇× ~B = µ0~j + µ0ε0

∂ ~E

∂t(4)

These are so-called Maxwell’s equations. They describe dynamics of electromagnetic fieldand together with Lorentz force on charged particle

~F = e( ~E + ~v × ~B), (5)

they describe most of the classic physics.Equations tells us relations between electric and magnetic fields ~E and ~B and their

sources, electric current density ~j and charge density ρ. If we take a look at the secondMaxwell equation, Equation (2), we see that magnetic field is always divergentless. Thismeans that, using Helmholz theorem [1], we can write magnetic field in a form

~B = ∇× ~A. (6)

Because divergence of a curl of some vector field is always zero, this clearly satisfy Equa-tion (2). ~A is called vector potential.

We can do similar thing with the third Maxwell equation (Equation (3)). If we writeelectric field in a form

~E = −∇φ− ∂ ~A

∂t(7)

and use Equation (6) for magnetic field, we always satisfy Equation (3).

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We see that if we write electric and magnetic field as described above, two Maxwellequations are automaticaly satisfied, so number of equations is reduced by half. We useother two Maxwell equations to determine relations between potentials and sources ofelectromagnetic field and we have all electromagnetic phenomena described in terms ofpotentials, which are much easier to work with than fields. If we want to calculate theforce on charged particle, we only use Equations (6) and (7) to determine fields andLorentz force law (Equation(5)) to calculate force.

2.2 Gauge transformations

Electromagnetic potentials have another important property. If we look at Equations (6)and (7) again, we see that potentials don’t stand for themselves but allways in a form ofderivative. This mean that if we transform potentials φ and ~A in the following way:

~A′ = ~A+∇χ (8)

φ′ = φ− ∂χ

∂t, (9)

potentials φ′ and ~A′ correspond to the same electric and magnetic field that φ and ~A.This is easy to show, using the identity ∇×∇f ≡ 0 and the fact that spatial and timederivatives commute. Transformation, written above, is ussualy called ”gauge transfor-mation” and function χ is called ”gauge function”. Because we can satisfy Maxwell’sequations with different potentials we say that the equations are ”gauge invariant”.

Gauge invariance of Maxwell’s equations is the main reason why potentials were ussu-aly considered as purely mathematical construct without any physical significance. Thatview changed with development of quantum mechanics in 20. century and especially withintroduction of Aharonov-Bohm effect.

3 Charged particle in proximity of solenoid magnet

3.1 Schrödinger equation in electromagnetic field

In classical mechanics, we can describe the motion of a particle either using a Lorentzforce law, which contains fields, or by using canonical formalism and hamilton function,which is expressed in terms of potentials. But if we want to describe dynamics of particlein quantum mechanics, we are forced to use canonical formalism because Schrödingerequation explicitly contains Hamilton function. For a charged particle in electromagneticfield, it is of a form [2]

H =1

2m(~p− e ~A(~r))2 + eφ(~r) + V (~r), (10)

where V stands for possible non-electric potential. If we write momentum in form ~p =−ih̄∇ and put Hamiltonian in Schrödinger equation, we get

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2m(ih̄∇+ e ~A(~r))2 + eφ(~r) + V (~r)]Ψ = ih̄

∂Ψ

∂t, (11)

which describes dinamics of charged particle in electromagnetic field.

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3.2 Vector potential of solenoid magnet

Now let us consider a charged particle in vicinity of a long solenoid, carrying magnetic field~B. If solenoid is extremely long, the field inside is uniform and the field outside is zero.We will use polar coordinate sistem with z axis in the middle of solenoid and pointingin direction of magnetic field. To solve Schrödinger equation, we must first determinethe potentials ~A and φ. Because solenoid is uncharged, the electric field ~E = −∇φ = 0,so we choose φ = 0, which clearly satisfy previous equation. Vector potential ~A outsidesolenoid must satisfy two conditions. First, ~B = ∇ × ~A = 0, is simply the definition ofvector potential. The second is a consequence of Stokes theorem and states∮

C

~A · d~r =

∫S

(∇× ~A) · d~S =

∫S

~B · d~S = Φm, (12)

if path of integration C is contracted curve around solenoid and Φm is total magneticflux through solenoid. Usually we choose vector potential to be

~A =Φm

2πrΦ̂, (13)

where r is distance from the z axis and Φ̂ is unit vector in direction Φ of our polarcoordinate sistem. It is easy to show that vector potential, given by Equation (), satisfiesboth conditions. We see that even though magnetic field ~B is confined to the interiorof solenoid, the vector potential ~A outside solenoid is not zero. If we use proper gaugefunction χ (Equation (8)), we can put it to zero almost everywhere outside solenoid, butwe still have to satisfy the condition from Equation (12).

3.3 Wavefunction in vector potential

To describe wavefunction of a charged particle, we have to solve Equation (11). In ourcase, it can be simplified by writing the wavefunction in a form

Ψ(~r, t) = eig(~r)Ψ′(~r, t), (14)

where

g(~r) ≡ e

h̄

∫ ~r

0

~A(~r′) · d~r′ (15)

Initial point of integration O is chosen arbitrary, which is consequence of gauge freedomfor electomagnetic potentials. At this point it is crucial that potential ~A is irottational(field ~B is zero), otherwise g(~r) is dependent of path of integration in Equation (15), soit is not a function of ~r.

In terms of Ψ′, the gradient of Ψ is

∇Ψ = eig(~r)(i∇g(~r))Ψ′ + eig(~r)(∇Ψ′). (16)

Because ∇g(~r) = (e/h̄) ~A,

(−ih̄∇− e ~A)Ψ = −ih̄eig(~r)∇Ψ′ (17)

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and further(−ih̄∇− e ~A)2Ψ = −h̄2eig(~r)∇2Ψ′ (18)

Putting this into Equation (11) and canceling the factor eig(~r), we are left with

− h̄2

2m∇2Ψ′ − VΨ′ = ih̄

∂Ψ′

∂t. (19)

We see that wavefunction Ψ′ is a solution of Schrödinger equation in absence of vectorpotential ~A. So if we can solve Equation (19), the solution in presence of vector field isthe same wavefunction, multiplied by phase factor eig(~r).

3.4 Magnetic Aharonov-Bohm effect

Now we make a thought experiment. We take a beam of electrons, split it in to two andsend each beam past solenoid on different side of it (Figure 1).

Figure 1: Schematic picture of magnetic Aharonov-Bohm effect [2]

What we do is in fact very similar to double slit experiment, so we expect that electronbeams will make intreference pattern when they meet on the other side of solenoid. Todescribe interference, we write beams in form of plane waves.

Ψ1 = Aeikx1 ,Ψ2 = Aeikx2 , (20)

where k is wave vector of electron beams and x1 and x2 are lengths each beam travels. Ofcourse this is not exact solution, but it helps us understand whath happens with beams.If solenoid contains no magnetic field, vector potential outside solenoid is set to zero,so the phase shift beetween Ψ1 and Ψ2 and consequetly interference pattern will dependonly on difference between traveled paths: ∆Φ0 = k(x2 − x1).

But when we turn magnetic field on, vector potential ~A is of a form Equation (3.2), sothe wavefunctions Ψ1 and Ψ2 will aquire additional phase factors as shown in Equation

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(14). Consequently, interference pattern will shift for additional phase ∆Φ = g1− g2. Tocalculate the difference, we use Equation (15) and write

∆Φ = g1 − g2 =e

h̄[

∫C1

~A(~r′) · d~r′ −∫C2

~A(~r′) · d~r′] =e

h̄

∮C

~A(~r′) · d~r′ = eΦm

h̄(21)

C1 and C2 stands for paths each beam travels when passing past solenoid. Since they forma closed path around solenoid, total phase difference between beams will be propotional tomagnetic flux inside solenoid. So if we change magnetic field in solenoid, we change phasedifference between beams and interference pattern will shift. This is called Aharonov-Bohm effect.

Let us again point out that when electron beams are traveling past solenoid, they neverpass through regions of space with non-zero magnetic field, so in classical electrodinamics,we would expect no interaction between electron and magnetic field. Nevertheless wesaw that if we describe particles in terms of quantum mechanics, magnetic field in regionisolated from particles produces measurable effect on their motion. So we have to concludethat either interaction between field and particle is not local or that potentials are morefundamental description of physical reality than fields.

The main objection against physical relevance of potentials, their gauge freedom,represents no problem in treatment of our experiment. If we change vector potential to~A′ = ~A + ∇χ and carry out integration in Equation (21), we see that phase difference∆Φ doesn’t change because we integrate gradient of a function over a closed loop. Theresult is identically zero, which means that phase difference is gauge invariant.

3.5 Electric Aharonov-Bohm effect

When we mention Aharonov-Bohm effect, we ussualy refer to phenomenon, describedin previous section, but in fact, there exists two types of the effect. In magnetic effect,described earlier, wavefunctions gain phase difference due to the vector potential, whichussualy describes magnetic field. But in their arcticle, Aharonov and Bohm also describedanother version of same phenomenon, when particles travel through region where electricfield ~E is zero, but scalar potential φ is not.

In fact, this version of effect is easier to explain. As we know, wavefunction withenergy E evolves with time as

Ψ(~r, t) = Ψ(~r, t = 0)eiEth̄ . (22)

The evolution is similar to Equation (14) if we put E = eφ and write

g =e

h̄

∫ t′=t

t′=0

φdt (23)

We see that Equations (15) and (23) has the same form, only that in first we integratevector potential over space and in second scalar potential over time, which is directconsequence of special relativity, where vector and scalar potential form the same physicalentity Aµ = (φ

c, ~A). In electric Aharonov-Bohm effect, electron beams should travel

through regions of space with different scalar potential to aquire phase difference.Electric Aharonov-Bomh effect could be realized in a form, presented on Figure 2.

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Figure 2: Schematic picture of electric Aharonov-Bohm effect [3]

Each beam travels through different conductive cylinder at potential φ1 or φ2. Itis important that cylinder is sufficently long that field inside is zero and potential isconstant. If potentials of cylinders are different, beams aquire phase difference of

∆Φ =et

h̄∆φ, (24)

where ∆φ is potential difference between cylinders and t is time electron needs to passthrough cylinder. The result of experiment would be similar as in magnetic Aharonov-Bohm effect, a fringe shift would appear in interference pattern. Main problem in thistype of experiment is that it is difficult to carry out and results are harder to interpretatebecause we can’t achieve situation where electrons wouldn’t have to pass through electricfield, which is inevetably present at beginning and end of cylinders. In case of magneticeffect, field can really be localised, so we usally use that type of experiment to measurethe effect.

3.6 Aharonov-Bohm effect and superconductors

In addition to zero resistivity, superconductors posses some other important properties.One of them is Meissner effect [4], which means that magnetic field can’t penetrate intosuperconductor. If magnetic field is to strong, some types of superconductors form specialstructures, called flux lines, which enables field to penetrate through superconductor, butonly in form of thin lines. Interesting property of such lines is that the magnetic flux ineach of them is quantized in units of

Φm =h

2e0, (25)

where e0 is charge of electron. This phenomenon can be explained using formalism,developed previously in this section.

Wavefunction of electron Ψ in superconductor is defined in a plane which is penetratedby a flux line. Magnetic field is localised to flux line and is zero in rest of superconductor,which is the situation we described in Section 3.1. If we move along the path aroundflux line with same starting and ending, we see that value of wavefunction is changedfrom Ψ(~r0) to eigΨ(~r0). We want wavefunction Ψ to be single-valued, which means that

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eig = 1. If we use Equation (15) to calculate phase difference g, we get condition

eΦm

h̄= 2πm, (26)

Φm =h

em, (27)

with m being integer. We see that we get the correct result of we put e = 2e0, whichmeans that particles in superconductor has charge twice larger than electron. It can beexplained by Cooper pairs [4], which are composed of 2 electrons. Quantisation of fluxin superconductor is important for proper interpretation of experimental results whenmeasuring Aharonov-Bohm effect at low temperatures.

4 Experimental evidence of Aharonov-Bohm effectAharonov-Bohm effect was first described in 1959 in an article [3], written by YakirAharonov and his doctoral advisor David Bohm and recieved various responses. Manyphysicist claimed that the effect in fact cannot be measured and that it is, like potentials,only a mathematical construct, so experimental conformation was needed. In fact, whendeveloping their idea, Aharonov and Bohm consulted experimental physicist Robert G.Chambers and in their article, they discribed the experiment which had to be carriedout to prove their theory. Only a year later, in 1960, Chambers performed the proposedexperiment and proved that effect does exist. In following years, effect was confirmed bymore and more precise experiments so today only a few people still doubt it’s existance.

In this section we will discribe Chambers’s experiment and discuss his results. We willalso take a look at a more advanced experiment from year 1986, which was performed indeferent geometry, using superconducting toroidal magnet.

4.1 Solenoid magnet experiment

Geometry, used in experiment, performed by Robert G. Chambers in 1960 [5], is prac-tically identical to geometry described in Section 3. The first problem in designing theexperiment was how to sepatate electron beams to sufficient distance. Spatial coherenceof electrons is determined by the size of electron source. If source is infinitely small, sep-aration of beams can be arbitrary, but in real experiment, separation is limited by finitesize of source. For succesful experiment, we need sufficient spatial coherence to separatebeams enough to put magnet between them. Experimental geometry is shown on Figure3.

Beam of electrons, used in experiment, was produced by electron microscope. Electronwavelenght was smaller than 1nm, which is much smaller than the size of solenoid, sodiffraction can be neglected. Beam is split into two by a electrostatic biprism (e and f onfigure) and interfering on observation plane o. Biprism consists of aluminized quartz fibref and two earthed metal plates e. Effective angle of biprism can be altered by applyingpositive potential on fibre f . In our calculations in Section 3, we assumed that solenoid isinfinitely long so that magnetic field outside is identically zero. In experimental situation,this cannot be achieved, so Chambers performed two experiments to distinguish the effects

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Figure 3: Schematic diagram of inferometer, used by Chambers [5]

of magnetic field, leaking from solenoid, from Aharonov-Bohm effect. In first experiment,electrons travel through magnetic field, extended over region a′ (Figure 3). As shown inFigure 4, field, extended over region a′, leaves fringes on interference pattern unchanged,it only displaces entire patteren over the screen, which can be explained classicaly byLorentz force.

Figure 4: Interferogram due to a) biprism alone and b) magnetic field in region a′ [5]

In the second experiment, magnetic field is localised to region a on Figure 3, which isa situation in which we expect Aharonov-Bohm effect. Instead of solenoid, which is hardto made so tiny, iron whisker 1 µm wide and 0.5 mm long was used. The flux in suchwisker decreases with along it’s lenght with the slope approximately (hc/e)/1µm, whichmeans that in case of Aharonov-Bohm effect, interference pattern will shift for about 1fringe per µm. Whisker is positioned in shadow of cylinder f , so no electron pass throughit. Results of this type of experiment is shown on Figure 5.

We see that fringes in fact shift in vertical direction, which is clear evidence ofAharonov-Bohm effect.

Even though Chambers’s experimental results confirmed predictions of Aharonov and

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Figure 5: Interferogram due to magnetic field in region a, produced by iron whisker [5]

Bohm, some still argued that the effect was not caused by electromagnetic potentials.Because the flux in iron whisker is decreasing, it is obvious that magnetic field leaksout in the region where electrons travel, so many belived that the observed effect couldbe explained by interaction of electrons with magnetic field. Further experiments wereneeded to undisputably prove Aharonov-Bohm effect.

4.2 Toroidal magnet experiment

In this section, we will describe an experiment, performed by a group of japanese physicistfrom Hitachi laboratory in year 1986 [6]. They used a bit different geometry of experimentand supermagnetic properties of materials to get more precise results of Aharonov-Bohmeffect.

Instead of solenoid magnet, proposed in original article in 1959 and used in firstexperiments, toroidal magnet was used. In case of solenoid, magnetic field around it isnever precisly zero because magnetic lines of force have to be closed, so they have to linkboth poles of the magnet. Magnetic field outside of solenoid is decrising with it’s length,but we would need infinitely long magnet to get field exactly zero. Toroidal magnet hasnorth and south pole attached, lines of force are circular so in theory we can achievemagnetic field outside of it to be identically zero.

To furher reduce possibility of field leaking into surrounding area, toroidal magnetwas covered with Nb, which is superconducting at temperatures under Tc = 9K. Becauseof Meissner effect, magnetic field is unable to penetrate through superconductor andit’s value outside is zero. The entire structure is additionaly covered by a layer of Cu,shielding superconductor from electrons.

The scheme of experiment is presented on Figure 6. Part of electron beam is travelingthrough a hole in toroidal magnet, so together with reference beam it forms a closedcircuit around magnetic flux, which causes phase shift. The rest of electrons interfere withreference beam without any phase shift, so we can use that pattern as a reference. Toroidal

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Figure 6: Conceptual diagram of toroidal magnet experiment [6]

magnet was made of ferromagnetic Permalloy. Due to the phase transition, magnetisationof Permalloy is increasing when we lower the temperature and magnetic flux throughmagnet is rising. In experiment, phase shift was observed while cooling magnet fromroom temperature to 5 K. Figure 7 shows us interference patterns for different magnetsat temperatures 15 K and 5 K.

In first column, we see interference pattern at T = 15K. At this temperature, Nb isnot in superconducting state, so flux in magnet can take any value and phase shift can bearbitrary. Below Nb phase transition, magnetic flux in toroid is quantized: Φm = h

2e0n,

which means that phase shift of electrons is also quantized

∆Φ =e0Φm

h̄=e0h̄

h

2e0n = πn (28)

Such quantization means that interference pattern can be shifted by either half fringeif n is odd or not shifted at all if n is even. Half fringe shift can be observed only atphase amplification ×1, which is reason why we see no difference in pattern between topand bottom picture in Figure 7. In third column, phase amplification is ×1 and we canclearly see that phase shift is 0 in top picture and π in the bottom picture.

Experiment clearly shows that phase shift between electron beams is present evenwhen magnetic field outside magnet is identicaly zero, which is evidence that electronsare influenced by potentials, not fields.

5 Practical use of Aharonov-Bohm effectIn addition to it’s phylosophical value in understanding electromagnetic and quantumphenomena, Aharonov-Bohm effect also has some important practical implications. We

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Figure 7: Interference pattern for 6 different situatuations. In top line, ((a),(b) and (c)),flux is quantized with even n, and in bottom line, ((d),(e) and (f), flux is quantized withodd n at T < Tc. First column shows results at T = 15K, second at T = 5K and phaseamplification ×2 and third at T = 5K and phase amplification ×1. [6]

saw that phase shift between electron beams strongly depends on enclosed magnetic flux.Interference patteren shifts one fringe for every ∆Φm = h/e0 = 4.1 × 10−15Tm2, whichis very small value. In principle, the effect enables us measurement of extremely smalldifferences in magnetic flux. The most simple case of such magnetometer would be twowires with electron current, forming a closed loop, and we would count oscilations ofcurrent through structure when magnetic flux through loop would change.

In fact, this basic idea can be reallized in even better way, using Josephson effect. Inthat case, loop is formed of two superconductors separated by small insulating barrier.The actual reallizaton of idea is to complicated to describe in this seminar, but it is mucheasier to understand with basic knowledge of Aharonov-Bohm effect and behaviour ofwavefunction around magnetic fluxes.

6 ConclusionThe main objective of this seminar was to show that in quantum mechanics, electromag-netic potentials appear to be more fundamental physical entities than fields. Aharonov-Bohm effect is phenomenon which can’t be describe in terms of classical mechanics andis of purely quantum origin. The effect was confirmed by many different experiments andtoday it’s existence is widely accepted. Anyhow, some physicist belive that mechanism ofthe effect is not properly described [7]. They argue that the effect is not purely quantum,described by phase shift, but that it is consequence of forces, acting on electron wavepackets when traveling past magnet. That problem will probably be resolved by furtherexperiments.

Anyway, we can conclude that electomagnetic interaction can take place in regions ofspace where field is zero and that Aharonov-Bohm effect does exist. Yet again we see thatlaws of quantum mechanics often contradict our intuition, which is a reason why NewScientist magazine proclaimed Aharonov-Bohm effect to be one of the "seven wonders ofthe quantum world" [8].

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7 Literature[1] Griffits, D. J., Introduction to electrodynamics, Prentice Hall, New Jersey (1999)

[2] Griffits, D. J., Introduction to quantum mechanics, Prentice Hall, New Jersey (1995)

[3] Y. Aharonov and D. Bohm, Significance of Electromagnetic Potentials in the Quan-tum Theory Phys. Rev., 115, 485, (1959)

[4] Shigeji Fujita and Salvador Godoy, Quantum Statistical Theory of Superconductiv-ity, Kluwer Academic Publisher, (2002)

[5] R.G. Chambers, Shift of an electron interference pattern by enclosed magnetic flux,Phys. Rev. Lett. 5, 3, (1960)

[6] N. Osakabe et. al., Expperimental conformation of Aharonov-Bohm effect using atoroidal magnetic field confined by a superconductor, Phys. Rev. A 34, 815, (1986)

[7] Boyer, T.H., Classical Electromagnetic Interaction of a Point Charge and a Mag-netic Moment: Considerations Related to the Aharonov-Bohm Phase Shift, Foun-dations of Physics 32, p. 1-39, (2002)

[8] Seven wonders of the quantum world, http://www.newscientist.com/special/seven-wonders-of-the-quantum-world, 25. 3. 2011

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