What exactly isIn the Schrodinger equation, the statement of electromagnetic gauge invariance? I see a lot of closely related ideas is that float together, and I'm trying to determine if they are actually part and parcel ofobservables don't depend on the electromagnetic gauge invariance. I'm thinking now in That is, if we let:
$$\partial_t \psi(\mathbf x,t) = \hat H \psi(\mathbf x,t) $$ $$\mathbf A' \equiv \mathbf A +\nabla \lambda$$ $$V' \equiv V - \frac{\partial \lambda}{\partial t}$$
then
$$\partial_t \psi(\mathbf x,t) = \hat H' \psi(\mathbf x,t) $$
will also give the context ofsame expectation values, where $\hat H'$ that depends on the electromagnetic gauge invariance ofmodified potentials.
Often, however, I see authors drawing more attention to the fact that you can rewrite the Schrodinger equation in the same form as the original, but obviously this is just an example. Specificallyprovided the wavefunction also transforms by a local phase factor:
- Are 'gauge invariance' and 'form invariance on gauge transformation' the same statement, or subtly different?
- The transformed position and mechanical momentum are equivalent to their untransformed counterparts in the Schrodinger Equation. What would the implication be if this were not true for a similar field equation - say, that violated E.M. gauge invariance but preserved something like it?
- Finally, I observe mathematically the connection between local phase and gauge freedom in the potentials... but what are the implications in this instance?
$$\psi'(\mathbf x,t) \equiv \exp(-i\lambda)\psi(\mathbf x,t).$$
I realize itWhy exactly is policythis 'form invariance' significant, and how is it related to submit questions separatelythe fact that observables (e.g. electric and magnetic fields, but theseexpectation values) are so closely related I'd likenot changed by the choice of gauge? I feel it is sometimes implied that the ability to hear answerswrite a transformed equation in context with each otherthe same form as the original implies that it is gauge invariant. Naively, one might assumed that you've merely 'defined away' observable differences by transforming your wavefunction.
If this is question is still poorly defined and semantic I'll just close it out.
