Thanks! One place where I struggle with this idea is that people go around saying things like “Given a quantum particle with nonzero electric charge, you can just pick what phase its wavefunction has”. I don’t know how to think of an electron having a wavefunction whose phase I can pick. The wavefunctions that I know assign amplitudes to configurations, not to particles; if I have a wavefunctionover three-electron configurations then I don’t know how to “choose the phase” for each electron, because a three-particle wave-functions doesn’t (in general) factor as a product of three one-particle wave-functions.
I can make guesses about what it means to “choose electron phases” when there’s multiple electrons, but if Vanessa’s right (and in my experience she usually is about this sort of thing), then my initial guess was wrong.
(I appreciate how you gave the equation for how a one-particle wavefunction ψ changes with a change in gauge, but it precisely dodges the case I was unclear about; what I was unclear on is how the wavefunction changes with the gauge when a configuration comprises two or more charged particles. Overall you’ve given quite a lovely demonstration of how the standard story manages to leave me unclear on this key point :-p.)
That said, your comment seems to me like fine exposition about how one might stumble across the electromagnetic 4-potential, and discover that it has gauge freedom. Thanks again!
My recollection is that in nonrelativistic N-particle QM you would have a wavefunction ψ(r1,r2,…,rN) (a complex number for every possible choice of N positions in space for the N particles), and when you change gauge you multiply that that wavefunction by eiq1Λ(r1)/ℏeiq2Λ(r2)/ℏ⋯eiqNΛ(rN)/ℏ. I think this is equivalent to what Vanessa said, and if not she’s probably right :-P
That was my original guess! I think Vanessa suggested something different. IIUC, she suggested
ψ(r1,r2,…,rN)↦N∏k=1(eiqkΛ(rk)/ℏψ(r1,r2,…,rN))
which has N factors of the wavefunction, instead of 1.
(You having the same guess as me does update me towards the hypothesis that Vanessa just forgot some parentheses, and now I’m uncertain again :-p. Having N factors of the wavefunction sure does seem pretty wacky!)
(...or perhaps there’s an even more embarassing misunderstanding, where I’ve misunderstood physicist norms about parenthesis-insertion!)
I think it’s just one copy of the ψ—I don’t think Vanessa intended for ψ to be included in the Π product thing here. I agree that an extra pair of parentheses could have made it clearer. (Hope I’m not putting words in her mouth.)
Thanks! One place where I struggle with this idea is that people go around saying things like “Given a quantum particle with nonzero electric charge, you can just pick what phase its wavefunction has”. I don’t know how to think of an electron having a wavefunction whose phase I can pick. The wavefunctions that I know assign amplitudes to configurations, not to particles; if I have a wavefunctionover three-electron configurations then I don’t know how to “choose the phase” for each electron, because a three-particle wave-functions doesn’t (in general) factor as a product of three one-particle wave-functions.
I can make guesses about what it means to “choose electron phases” when there’s multiple electrons, but if Vanessa’s right (and in my experience she usually is about this sort of thing), then my initial guess was wrong.
(I appreciate how you gave the equation for how a one-particle wavefunction ψ changes with a change in gauge, but it precisely dodges the case I was unclear about; what I was unclear on is how the wavefunction changes with the gauge when a configuration comprises two or more charged particles. Overall you’ve given quite a lovely demonstration of how the standard story manages to leave me unclear on this key point :-p.)
That said, your comment seems to me like fine exposition about how one might stumble across the electromagnetic 4-potential, and discover that it has gauge freedom. Thanks again!
My recollection is that in nonrelativistic N-particle QM you would have a wavefunction ψ(r1,r2,…,rN) (a complex number for every possible choice of N positions in space for the N particles), and when you change gauge you multiply that that wavefunction by eiq1Λ(r1)/ℏeiq2Λ(r2)/ℏ⋯eiqNΛ(rN)/ℏ. I think this is equivalent to what Vanessa said, and if not she’s probably right :-P
That was my original guess! I think Vanessa suggested something different. IIUC, she suggested
ψ(r1,r2,…,rN)↦N∏k=1(eiqkΛ(rk)/ℏψ(r1,r2,…,rN))
which has N factors of the wavefunction, instead of 1.
(You having the same guess as me does update me towards the hypothesis that Vanessa just forgot some parentheses, and now I’m uncertain again :-p. Having N factors of the wavefunction sure does seem pretty wacky!)
(...or perhaps there’s an even more embarassing misunderstanding, where I’ve misunderstood physicist norms about parenthesis-insertion!)
I think it’s just one copy of the ψ—I don’t think Vanessa intended for ψ to be included in the Π product thing here. I agree that an extra pair of parentheses could have made it clearer. (Hope I’m not putting words in her mouth.)