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American Journal of Physics

When charge meets flux: Classical view of a quantum paradigm

M. L. Horner, Alfred S. Goldhaber

Published Oct 1, 1996• 1 citation
Publisher

Abstract

Quantum mechanics is known to forbid charged, spinless particles from entering an infinitely thin tube of magnetic flux, though particles which do possess spin may even be bound inside. For spin 1/2 the particles may be ‘‘quasibound,’’ so that some penetration from outside to inside is allowed. We use freshman physics and plane geometry to explore such charge-flux systems in classical electrodynamics, finding remarkable similarities to the quantum case, along with characteristic differences. Both aspects illuminate the relationship between classical and quantum physics.

DOI
10.1119/1.18385

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Quantum mechanics is known to forbid charged, spinless particles from entering an infinitely thin tube of magnetic flux, though particles which do possess spin may even be bound inside. For spin 1/2 the particles may be ‘‘quasibound,’’ so that some penetration from outside to inside is allowed. We use freshman physics and plane geometry to explore such charge-flux systems in classical electrodynamics, finding remarkable similarities to the quantum case, along with characteristic differences. Both aspects illuminate the relationship between classical and quantum physics.
In a previous note (1), we argued that both free particle quantum and classical mechanics follow from an energy flow/flux scheme based on: d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)) A(x,t) turns out to be the classical action tL(v) (L=Lagrangian) with x/t=v and the above hold for both the relativistic and nonrelativistic cases. We further argued that ((1a)) and ((1b)) lead to two sets of solutions, one classical and the other quantum. The classical solution assumes dt→0 and dx→0 i.e. flows may be followed at each point in space (and time). In (2) we argued that ((1a)) and ((1b)) are consistent with p=dL/dv (partial) and H=E= pv - L. From these together with p=Ev one may find: E=mo/sqrt(1-vv) (c=1). For the nonrelativistic case, E= mocc + .5mvv. In this classical analysis one may resolve each t and x, thus if one has a function V(x) (potential) representing energy, the nonrelativistic result is: (E-V(x)) = .5mvv. The quantum solution to ((1a)) and ((1b)) is very different. It establishes eigenvalue equations i.e. id/dt (partial) exp(iA) = E exp(iA) and -id/dx (partial) exp(iA) = p exp(iA) for the free particle. The importance of this is that new physical mechanisms seem to come into play. A= -Et+px so exp(-iEt+ipx) suggests time and spatial regions proportional to 1/E and 1/p in which there is some ambiguity i.e. there are two dimensions and there seems to be some kind of probability for the particle to be in one dimension or the other at x (and similar results for t) i.e. cos(px) and sin(px). As a result, one may not have dx→0 and dt→0. Furthermore, it is argued that a two dimensional associated probability scheme must accompany energy flow. At first this may seem strange, but if one considers a photon which has an energy flow Ev, one has electric and magnetic fields perpendicular to each other and perpendicular to the motion which undergo periodic motion linked to cos(wt) and sin(wt). Thus, the quantum particle periodic probability mimics an already known result from Maxwell’s equations. In fact, we have argued that for fermions there is additional angular motion i.e. spin linked with the recognition of 3-space. Thus we argue that quantum mechanics is involved with periodic type probability motions which are needed to resolve dx and dt (linked to exp(-iEt+ipx) and periodic probability motion (spin) and are also needed for fermions to resolve 3-space as two p vectors may have point in different directions (i.e. spin). Quantum mechanics therefore seems to be a very different theory from classical mechanics, but both appear to be based on energy flow/flux. As a result, the average of a quantum bound state should be linked to a classical system we argue in this note as one only deals with energy flow/flux.
The “particle-in-a-box” problem is investigated for a relativistic particle obeying the Klein–Gordon equation. To find the bound states, the standard methods known from elementary non-relativistic quantum mechanics can only be employed for “shallow” wells. For deeper wells, when the confining potentials become supercritical, we show that a method based on a scattering expansion accounts for Klein tunneling (undamped propagation outside the well) and the Klein paradox (charge density increase inside the well). We will see that in the infinite well limit, the wave function outside the well vanishes, and Klein tunneling is suppressed: Quantization is, thus, recovered, similar to the non-relativistic particle in a box. In addition, we show how wave packets can be constructed semi-analytically from the scattering expansion, accounting for the dynamics of Klein tunneling in a physically intuitive way.
Ever since quantum theory was developed during the first quarter of the 20th century, we have lived with a strange division. Objects in our daily lives behave “normally” – they appear to obey classical physics – whereas microscopic objects can behave counter-intuitively and reveal intriguing features of quantum physics. But where exactly is the boundary between the quantum and classical worlds – if, indeed, there is one? If quantum physics is a universal theory, why is it respectable to talk about the quantum behaviour of electrons but not, say, of footballs?
As a consequence of the Aharonov-Bohm effect, there is a quantum-induced attraction between a charged particle and a rigid, impenetrable hoop made from an arbitrarily thin tube containing a superconductor quantum of magnetic flux. This is remarkable because in classical physics there is no force between the two objects, and quantum-mechanical effects (associated with uncertainty-principle energy) generally are repulsive rather than attractive. For an incident spinless charged particle in a P wave (in a configuration with total angular momentum zero) we verify a resonance just above threshold using the Kohn variational principle in its S-matrix form. Even if optimistic choices of parameters describing a model system with these properties were feasible, the temperature required to observe the resonance would be far lower than has yet been attained in the laboratory.
As a consequence of the Aharonov-Bohm effect, there is a quantum-induced attraction between a charged particle and a rigid, impenetrable hoop made from an arbitrarily thin tube containing a superconductor quantum of magnetic flux. This is remarkable because in classical physics there is no force between the two objects, and quantum-mechanical effects (associated with uncertainty principle energy) generally are repulsive rather than attractive. For an incident spinless charged particle in a P wave (in a configuration with total angular momentum zero) we verify a resonance just above threshold using the Kohn variational principle in its S-matrix form. Even if optimistic choices of parameters describing a model system with these properties turned out to be feasible, the temperature required to observe the resonance would be far lower than has yet been attained in the laboratory.