Vector Potential applications in Aharonov-Bohm effect

[riezer]

The Aharonov-Bohm effects show how a electro-magnetic field could affect a region of space in which the field had been shielded, although its vector potential did exist there and could interact with the wave function of say the electron.

What practical application(s) (so far) can be derived from this principle?

Also does Aharonov-Bohm effect prove that the wave function is objective in space favoring either Bohmian Mechanics or Objective Collapse or can it still be compatible with Copenhagen in which the wave function is just knowledge of the observer. In the latter case, the vector potential is also knowledge of the observer? As a consequence, electromagnetic field by nature is just knowledge of the observer and not really there in space at all?

 

[Born2bwire]

It's perfectly fine with Copenhagen picture as it demonstrates that the primitive of the electromagnetic field is not the field observables but the potentials. Once you work it from the perspective of the potentials you can drop it into the Schroedinger equation and all that jazz just fine. As for applications, there are many. The basis of the effect is that an electon traversing through a vector potential causes a phase shift in the wavefunction when compared to the wavefunction in the absence of the vector potential. The Aharanov-Bohm effect could be considered as a specific application of this consequence (with the added coolness of having no magnetic field). But there are a lot of other consequences that this causes when you look at things like say a superconductor. The superconductor will expell any applied magnetic field, however, if you form the superconductor into a ring then you can still have the electrons form a path around a contained flux (that is, the electrons flow around the ring which has a magnetic field passing through the hole but the magnetic field is not present inside the superconductor where the trajectory lies).

The resulting phase shifts that arise due to the vector potential can be used for various applications. The simplest is to perform the Aharanov-Bohm experiment (yay) but you can also combine this with Josephson junctions to make a SQUID. A SQUID can measure to a high precision very small magnetic fields.

EDIT: I probably should be clear that the effect of the vector potential only being a phase shift in the wavefunction of course assumes that the B field is zero in that region. Obviously if we had a static B field and solved for the wavefunction of the electron we should get the Landau cyclotron radiation. Obviously a simple phase shift would be insufficient in this situation.

 

[riezer]

It's perfectly fine with Copenhagen picture as it demonstrates that the primitive of the electromagnetic field is not the field observables but the potentials. Once you work it from the perspective of the potentials you can drop it into the Schroedinger equation and all that jazz just fine.

Does the following analogy hold.

Primitive of electromagnetic field are potentials.
Primitive of quantum object are wave functions.
So "wave functions" are equal to "potentials" in the levels.. both knowledge of the observers and both can be in superpositions. ??

Field observables of electromagnetic field are field observables like magnetic, electric field. Observables of quantum objects are position, momentum, etc. observables.
This means field observables like magnetic and electric field are collapsed state of " vector potential" superposition? If not. Which part of my analogy is confused and incorrect?

As for applications, there are many. The basis of the effect is that an electon traversing through a vector potential causes a phase shift in the wavefunction when compared to the wavefunction in the absence of the vector potential. The Aharamov-Bohm effect could be considered as a specific application of this consequence (with the added coolness of having no magnetic field). But there are a lot of other consequences that this causes when you look at things like say a superconductor. The superconductor will expell any applied magnetic field, however, if you form the superconductor into a ring then you can still have the electrons form a path around a contained flux (that is, the electrons flow around the ring which has a magnetic field passing through the hole but the magnetic field is not present inside the superconductor where the trajectory lies).

The resulting phase shifts that arise due to the vector potential can be used for various applications. The simplest is to perform the Aharamov-Bohm experiment (yay) but you can also combine this with Josephson junctions to make a SQUID. A SQUID can measure to a high precision very small magnetic fields.

 

[Born2bwire]

Does the following analogy hold.

Primitive of electromagnetic field are potentials.
Primitive of quantum object are wave functions.
So "wave functions" are equal to "potentials" in the levels.. both knowledge of the observers and both can be in superpositions. ??

Field observables of electromagnetic field are field observables like magnetic, electric field. Observables of quantum objects are position, momentum, etc. observables.
This means field observables like magnetic and electric field are collapsed state of " vector potential" superposition? If not. Which part of my analogy is confused and incorrect?

No I wouldn't say that. The potentials, just like the electromagnetic fields, are operators (or in the case of QED, fields). However, the potentials, unlike the fields, are not observables. What I mean by primitive is in terms of the axiomatic; that the behavior of the fields and potentials differs from classical electrodynamics. True, gauge invariance still holds in quantum mechanics. However, in classical electrodynamics the potential view predicts the same physics as the field view. So if we have a region where no fields are present, then the potentials, even if non-zero, would predict the same null response of charged particles in this region. But in quantum mechanics we cannot assume that this is so. As such, while the fields are still the observables, we need to pay attention to how the potentials affect the particles. So when we work the wave equations, we use the potentials as the operators, not the fields. Most of the time it does not matter. Like I said before, the influence of the vector potential on the wavefunction is akin to a phase shift. The wavefunction relates the probability density of the particle and strictly from this we can see that a phase shift does not influence the probability density since we take the conjugate. But if we allow interaction between multiple particles, then this phase shift can result in interference if electrons can undergo different amounts of shifts (we do this in the Aharanov-Bohm experiment by directing the electrons along paths that experience different vector potentials).

EDIT: I probably should be clear that the effect of the vector potential only being a phase shift in the wavefunction of course assumes that the B field is zero in that region. Obviously if we had a static B field and solved for the wavefunction of the electron we should get the Landau cyclotron radiation. Obviously a simple phase shift would be insufficient in this situation.