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A critical re-examination of the Aharonov-Bohm effect
Abstract
The controversy concerning theoretical treatment of the Aharonov-Bohm (AB) effect is critically investigated. A new approach, taking the time-dependent aspect of this effect into consideration, is proposed which seeks to resolve the prevalent ambiguities. This provides interesting insight into the origin of the AB effect. Some clarifications are also made regarding the experimental verification of this effect.
... The Aharonov-Bohm (AB) effect is an authentic quantum phenomenon and was first considered in [19,18,1]. There are many discussions about the justification of the famous Aharonov-Bohm Hamiltonian and interpretations (see, for instance, [2,7,8,9,11,15,20,22,26,28,30] and references therein); the questions are particularly interesting for the more realistic case of solenoids S of radii greater than zero. Sometimes it involves the quantization in multiply connected regions, and the main points to be clarified are the presence of the vector potential A in the operator action (occasionally in regions with no magnetic field), and the (natural) choice of Dirichlet boundary conditions at the solenoid border. ...
Consider bounded solenoids in the space or planar approximate models of infinitely long solenoids. It is proved that the solenoid
impenetrability, by means of a sequence of Hamiltonians with diverging potentials, converges in the norm resolvent sense to
the usual Aharonov–Bohm model with Dirichlet boundary condition. The framework is that of nonrelativistic quantum mechanics.
Mathematics Subject Classification (2000)82D99–47B25–81Q10
... There are controversies over the interpretation of A as a real physical variable, that is, mistrusts of the existence of the Aharonov-Bohm effect as stated above. For instance, that the phase difference could be eliminated by using gauge transformations [6,7]; explanation via the hydrodynamical viewpoint in quantum mechanics [8], whose equations admit a solution where the vector potential appears explicitly, and such solution corresponds to a hamiltonian with the vector potential included; some authors argue that the experimental results could be explained by a border effect and the magnetic field (also due to poor solenoid impermeability) in a region accessible to the electric particles [27,16]-see comments and critiques in [15,19,21]. ...
It is presented, in the framework of nonrelativistic quantum mechanics, a justification of the usual Aharonov-Bohm hamiltonian (with solenoid of radius greater than zero). This is obtained by way of increasing sequences of finitely long solenoids together with a natural impermeability procedure; further, both limits commute. Such rigorous limits are in the strong resolvent sense and in both R 2 and R 3 spaces. PACS: 03.65.Ta; 03.65.Db; 02.30.Sa Given a cylindrical current-carrying solenoid S of infinite length and radius a> 0, centered at the origin and axis in the z direction, there is a constant magnetic field B = (0,0,B) confined in S ◦ , the interior of S, and vanishing in its exterior region S ′. The solenoid is considered impermeable (impenetrable), in the sense that the motion of a spinless particle (of mass m = 1/2 and electric charge q) outside the solenoid has no contact with its interior, particularly with the magnetic field B. If A is the vector potential generating this magnetic field, that is, B = ∇ × A, the usual hamiltonian operator describing the quantum motion of this charged particle is given by
In quantum mechanics, the magnetic field has a physical effect beyond its local presence. For axially symmetric vector potentials, A=A(s,z)ϕ̂, it is the magnetic flux that plays a central role in both interference experiments (as in the Aharonov-Bohm effect) and in bound state spectra for a particle constrained to move in a circle. The flux dependence is evident in various configurations, and we have chosen two applications to highlight the non-local nature of the quantum mechanical magnetic field dependence: A finite solenoid and a Dirac string with a non-zero radius. The finite solenoid is interesting because it represents the actual experimental apparatus for measuring the Aharonov-Bohm effect. The Dirac string provides an electric charge quantization mechanism that depends on where the quantization argument is applied, yielding different fundamental charge units for different locations.
In 1959, Aharonov and Bohm presented a paper entitled “Significance of electromagnetic potentials in quantum theory” [6.1]. Its content can be roughly summarized as follows: In classical electrodynamics, potentials are merely a convenient mathematical tool for calculations concerning electromagnetic fields. The fundamental equations can always be formulated using these fields. However, in quantum mechanics potentials cannot be eliminated from the Schrödinger equation and consequently seem to have physical significance. Aharonov and Bohm went beyond this conjecture and proposed actual electron-interference experiments. These experiments were intended to clarify how potentials affect electrons passing through field-free regions. The phenomenon these researchers described came to be called the Aharonov-Bohm (AB) effect in their honor.
It is argued that much of the recent controversy over the Aharonov-Bohm effect has been fuelled by the widespread consideration of solenoids of infinite length: in this limit, the field and the vector potential have somewhat anomalous properties and it appears superficially that the Aharonov-Bohm phase shift can be changed by a gauge transformation. This incorrect impression is here removed by considering a solenoid of finite length and by carefully distinguishing the longitudinal and transverse parts of the vector potential. It is shown that the phase shift depends only on the transverse part of the vector potential and it cannot be changed by a gauge transformation. The nature of the difficulty that occurs in the limit of a solenoid of infinite length is examined in detail. The finite-length theory given here also provides guidelines for the design of experiments on the Aharonov-Bohm effect.
The electromagnetic fields associated with a time-dependent infinitely long solenoid are analysed. It is proved that if the magnetic field vanishes at the solenoidal outer region then its flux is a linear function of the time. It is also shown that a time-dependent solenoid does not satisfy the field-free requirement of the Aharonov-Bohm effects.
The theory of Relativistic Schrödinger Equations offers a local-realistic interpretation for the Bohm-Aharonov effect. The wave function psi(x) is built up by a (scalar) localization field l(x) and a kinetic (vector) field kµ(x), which extends far beyond the support of l(x) and thus transfers electromagnetic effects from far away to the location of the wave packet.
Electromagnetic fields of the toroidal solenoid with a time-dependent current are studied. Their properties and the possible practical applications of the results obtained are discussed.
The charged particle scattering amplitude on the magnetic field of a toroidal solenoid is obtained in the first Born and high-energy approximations. It is shown that multiconnectedness of the accessible space, non-triviality vector potentials in it and the single-valuedness of the wavefunctions used are not enough for the existence of the Aharonov-Bohm effect. Criteria are given for this. Surrounding the toroidal solenoid with barriers of different geometrical forms, the author observes that the magnetic field contribution to the scattering amplitude depends crucially both on the barrier height and its form. The latter could hardly be explained by particle penetration into the H not=0 region. The author proposes an addendum to Tonomura's experiments which probably could clear up any doubts about the existence of the AB effect. The gedanken experiment is discussed which shows that the effect of inaccessible fields could be observed even in a simply connected space.
A quantum mechanical plane rotor with time-dependent magnetic flux on its axis is considered in a representation in which the wavefunction is multivalued. If the representation is chosen such that the new kinetic angular momentum is equal to the old canonical angular momentum, the Schrodinger equation is simplified. An energy eigenfunction expansion in the new representation is used to solve the Schrodinger equation exactly.
We discuss here the prediction, based on a formalism by the author, on the observable effects of a curl-free magnetic vector
potential on the macroscale as against the microscale of the Aharonov-Bohm effect. A new quantum concept — the ‘transition
amplitude wave’ — postulated in the formalism has already been shown to exhibit matter wave manifestations in the form of
one-dimensional interference effects on the macroscale. It was predicted by the formalism that the same entity would lead
to the detection of a curl-free magnetic vector potential on the macroscale. We describe here the manner of generation of
this quantum entity in an inelastic scattering episode and work out an algorithm to observe this radically new phenomenon,
the detection of a curl-free magnetic vector potential on the macroscale. We determine the various characteristic features
of such an observation which can then be looked for experimentally so as to verify the predicted effect, establishing thereby
the physical reality of the new quantum entity, and to fully validate the formalism predicting it. It is also shown that this
‘transition amplitude wave’ can be regarded as a novel kind of ‘quasiparticle’ excited in the charged particle trajectory
as a consequence of the scattering episode.
KeywordsCurl-free vector potential-macroscale quantum effects-transition amplitude wave
Using the path-integral method we analyse the effect of a time-varying (stepped) enclosed magnetic flux on the Aharonov-Bohm
(AB) phase shift in the two-slit interference pattern of electron wavepackets. We find the magnetic phase shift to vary continuously
between zero and the full AB value depending on the timing of the flux increment. This behaviour is amenable to experimental
verification. The special nature of the global features peculiar to the static AB effect emerges clearly from this study which
supports a shift in emphasis away from the enclosed flux itself towards its vector potential, acting locally along the interfering
Feynman (and, in a certain sense, the classical) trajectories.
A recent attempt to deny the crucial role of inaccessible fields in experiments performed to test the predictions of Aharonov and Bohm is critically discussed. It is shown that these experiments may well be considered to show the reality of the Aharonov-Bohm effect even if it is correct to say that the results are completely determined by the accessible field.
It is shown that the existence of Aharonov–Bohm scattering depends upon the criteria used for establishing the stationary states. If one applies Pauli’s criterion, there is no scattering. It is shown further that applying the usual criteria that the wave functions be continuous and single valued, as was done by Aharonov and Bohm, leads to stationary state wave functions which, with two exceptions, are eigenstates of the acceleration operator corresponding to eigenvalue zero. The acceleration operator is undefined for the remaining two states. Thus, only the eigenfunctions satisfying the Pauli criterion lead to well‐defined, sensible physics.
Because the Aharonov-Bohm effect has been challenged on mathematical grounds we give a simple physical proof of its existence, which also gives an insight into why such a strange effect must exist within quantum theory. We also comment on some of the recent literature on the subject.
A recent paper attributes the Aharonov-Bohm effect to the fringing field leaking out of a finite solenoid rather than to the magnetic flux in the excluded region. This argument is seen to be absurd for a very long solenoid where the effect is attributed to the tail of the electron wave function very far from the point of measurement, and is completely refuted by toroidal geometry with no fringing field.
The Aharonov-Bohm effect has recently been questioned on theoretical and experimental grounds. Such discussions have suffered from ambiguities which resulted from their focusing on scattering states. It is noted here that the bound-state problem is much simpler. To avoid the Aharonov-Bohm effect in theory requires us to abandon the most fundamental ideas of quantum mechanics. The quantization of flux in superconducting rings and Josephson junctions is a powerful experimental confirmation.
The Aharonov-Bohm effect is a necessary and easily understood feature of conventional quantum mechanics. Attempts to remove it from the theory must involve a drastic change in our understanding of the quantization and conservation of angular momentum, or of the role of the classical equations of motion in quantum mechanics. The key point is that a charged particle is the source of an electric field which will penetrate a magnetic field from which the particle is excluded. The crossed fields contain angular momentum whose existence alters the motion of the particle because the total angular momentum is quantized.