To be used in future
 Section 4.2: Hamiltonian Up Chapter 4: Charged Particles and Electromagnetic Fields Section 4.4: Landau Levels 

4.3 Gauge transformations and Gauge invariance

I would suggest you use Ref. [2] as background reading for this section.

4.3.1 Gauge transformation

The potentials and are mathematical constructions, without direct physical reality. Especially, adding fields and to the scalar and vector potentials satisfying does not change the observables fields: Such a gauge transformation can be found--there is an infinite number of solutions to the equations (↓,↓) given above. From the fact that is irrotational (↓), we find that [M]  [M] Using . and from (↓) we now derive and thus we can parametrise a gauge transformation by a single function , the “↓gauge potential”.

4.3.2 Unitary transformations and gauge changes↓

This raises an interesting question. Since a gauge transformation changes the Hamiltonian, through its dependence on and , what is the effect on the physics, c.q., the wave function? As in the case of the symmetry transformations above--and a problem with a set of gauge transformations is often referred to as possessing a gauge symmetry↓--can we find a unitary transformation on the wave function that represents a gauge transformation, just like we have done for rotations in the previous chapter?
The time dependence of the gauge potential makes things more complicated than the cases we had before. Let us therefore first look at the case of time-independent gauge transformations, , where we only make changes to the vector potential. We shall try and find a unitary operator so that we can undo the effect of the gauge change; this is most easily done by calculating the effect of the transformation on the Hamiltonian, which in this case will change due to the change in (the observable physics must still be invariant, however).
From the invariance of the general matrix element with the new wave functions , we see that we must require Since we have the square of an operator on both sides and would like to be unitary, we see that if we can find a unitary solution to we have determined the transformation. [N]  [N] A little thought shows that this is a necessary and sufficient condition. Keeping in mind the type of expressions we have found in the previous chapter, it seems reasonable to try and find an exponential form, where the argument in the exponent, when commuted with , gives rise to the term . Since is proportional to the gradient operator this is easy to solve, and we see that we should take as can easily be checked by the chain rule.
If is time-dependent, we have to look at the full time-dependent Schrödinger equation, most easily using the quantum action, a functional that when varied w.r.t. to or gives the time-dependent Schrödinger equation,  [O]  [O] From now on I shall often use the notation to denote . If we wish this to be invariant, we must have A simple substitution shows that the form (↓) still holds, but now we have a time-dependent gauge potential .

4.3.3 Phase is unobservable

Of course the probability density remains unchanged since the transformation from Eq. (↓) only adds a time- and position-dependent phase to the wave function. We can show that the expectation value of the momentum operator is not invariant--please check--and that the matrix elements of the grand momentum (↓) is again invariant. Can you explain why this behaviour incorporat the right physical properties?
 Section 4.2: Hamiltonian Up Chapter 4: Charged Particles and Electromagnetic Fields Section 4.4: Landau Levels