Now what is Aharonov-Bohm really?

In summary, the Aharonov-Bohm effect is a consequence of the different paths of an electron interfering. It causes the energy eigenstates of a charged particle to be shifted by the presence of a magnetic field.
  • #1
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I can only seem to find vague sources explaining Aharonov-Bohm, usually saying things as "the different paths of an electron interfere". I presume this is language borrowed from a Feynman path integral formulation of QM, but I'm not familiar with that yet, so I'd rather see it explained in "ordinary" QM math.

Some sources seem to suggest that the basic consequence of a vector potential is that the phase [itex]S[/itex] of the wavefunction [itex]\psi(\mathbf r,t) = R(\mathbf r,t) e^{iS(\mathbf r,t)}[/itex] gets an extra term, namely the path integral [itex]\int_{\mathbf r_0}^{\mathbf r} \mathbf A(\mathbf r') \cdot \mathrm d \mathbf r'[/itex] (where [itex]\mathbf r_0[/itex] is some arbitrary reference point). Then again, this can't really be true, cause then phase wouldn't be well-defined (since a different path, but also going from r_0 to r, could give a different result). I realize that this last remark is also the key concept in the AB effect, but still, not in the aforementioned way, right? After all, phase shouldn't be ambiguous (except for a 2pi multiple of course). Actually, it's exactly this argument (that different path integrals should give the same phase mod 2pi) that is used to argue flux quantization in a superconducting ring... So why does it not apply more generally?

As you can see, I'm a bit confused. Note that I'm not looking for a vague explanation, I'm looking for something concrete (and well-defined) mathematically. Thank you!
 
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  • #2
One concrete thing you can do easily in the regular Schrodinger wave equation formalism is solve for the energy eigenstates of a charged particle confined to a circular 1D ring through the center of which a solenoid passes carrying some amount of magnetic flux. There is no B-field at the ring's radius, but there is an A-field. Write down the Schrodinger equation including the A-field and solve it. Once you figure out what 1D Schrodinger equation to write down, you should find that the solutions are almost as trivial as free-particle solutions. You should find that the energy eigenstates of the particle are shifted by the presence of the magnetic field, even though the particle cannot enter the B-field. Furthermore (as I recall) the clockwise- and counter-clockwise-moving energy eigenstates cease to be degenerate. This is one manifestation of the Aharonov-Bohm effect.

Not sure if this is helpful, but it is concrete and mathematically well-defined.
 
  • #3
mr. vodka said:
I can only seem to find vague sources explaining Aharonov-Bohm, usually saying things as "the different paths of an electron interfere". I presume this is language borrowed from a Feynman path integral formulation of QM, but I'm not familiar with that yet, so I'd rather see it explained in "ordinary" QM math.
As with most aspects of "ordinary QM", Ballentine is a good place to start.
Try section 11.4.

[...] this can't really be true, cause then phase wouldn't be well-defined (since a different path, but also going from r_0 to r, could give a different result).
This doesn't mean the phase is ill-defined. It just means the phase can be position dependent -- which is also a key concept in gauge theories of interaction.

Take a look at Ballentine and then see if any points remain unclear...
 
  • #4
The setup is similar to a double-slit with photons, where you can add phase shifts at the slits (e.g. with material with a different refractive index). Similar to this, you have to constrain the electron path in order to get a phase which is nearly the same for all electrons in one path.
 
  • #5
I also back up the approach proposed by The Duck.
You will find that the energy states depend on the magnetic flux through the loop which is an observable in contrast to A.
Nevertheless this is astonishing given that the electrons don't enter the region where B is non-vanishing.
 

1. What is the Aharonov-Bohm effect?

The Aharonov-Bohm effect is a quantum phenomenon that describes the influence of a magnetic field on the phase of a quantum particle, even when that particle is located in a region with zero magnetic field. This effect was first proposed by Yakir Aharonov and David Bohm in 1959.

2. How does the Aharonov-Bohm effect work?

The Aharonov-Bohm effect is based on the idea that the phase of a quantum particle is affected not only by the local electromagnetic field, but also by the vector potential of the electromagnetic field. This means that even if there is no magnetic field present, the vector potential can still have an effect on the phase of the particle, leading to observable interference patterns.

3. What are the real-world applications of the Aharonov-Bohm effect?

The Aharonov-Bohm effect has been experimentally verified and has important implications for our understanding of quantum mechanics. It has also been used to explain behavior in superconductors and has potential applications in quantum computing and telecommunications.

4. Can the Aharonov-Bohm effect be observed in everyday life?

The Aharonov-Bohm effect is a purely quantum phenomenon and is not observable in everyday life. It requires highly controlled experimental conditions and extremely small scales to observe the interference patterns caused by the vector potential.

5. What are the current debates and controversies surrounding the Aharonov-Bohm effect?

One major debate surrounding the Aharonov-Bohm effect is whether it is a true physical effect or simply a mathematical artifact. Some scientists argue that the effect can be explained by classical physics, while others maintain that it is a fundamental quantum phenomenon. Additionally, there is ongoing research into how the Aharonov-Bohm effect can be tested and applied in practical settings.

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