- #1
nonequilibrium
- 1,439
- 2
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!
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!