I am reading an article about Bloch-Floquet state. My questions is in Part II.B and Appendix A of this paper, I will describe them below.

The original Schordinger equation we consider is:

$$i\hbar\frac{\partial}{\partial t} \tilde{\Psi}(r,t)=\tilde{H}(t)\tilde{\Psi}(r,t)$$

where:

$$\tilde{H}(t)=\frac{1}{2m_e}(\frac{\hbar}{i}\nabla+\frac{e\vec{A}(t)}{c})^2+V_c(r) $$

with $A(t)$ periodic in time and $V_c(r)$ periodic in space.

According to Floquet theorem, this time-periodic Hamiltonian has the wave function in the form:

$$\tilde{\Psi}(r,t)=e^{-i\tilde{\epsilon}(k)t/\hbar}e^{ik\cdot r}\tilde{\phi}_{\tilde{\epsilon},k}(r,t)$$ with $\tilde{\phi}_{\tilde{\epsilon},k}(r,t)$ periodic in space and time, we call $\tilde{\epsilon}(k)$ the Bloch-Floquet quasienergy.

The author did a following transform to avoid dealing with the square term of $A(t)$:

$$\tilde{\Psi}(r,t)=\exp(-\frac{ie^2}{2m_e\hbar c^2}\int^t dt'A^2(t'))\Psi(r,t)$$

Substitute this to the original Schordinger equation we arrive at:

$$i\hbar\frac{\partial}{\partial t} \Psi(r,t)=H(t)\Psi(r,t)$$

where:

$$H(t)=H_0+\frac{e}{m_ec}\vec{A}(t)\cdot \frac{\hbar}{i}\nabla$$ $H_0$ is the field free Hamiltonian. Finally, the author claimed that the physics quantities are invariant under such a transformation and give an example of the invariance of the current density(I can see the identity).

My question is:

• Is this really true for all observables, how is this related to gauge invariance?

• After the transformation, the author is using this new Schordinger equation to calculate the Bloch-Floquet quasienergy, is these qusienergies same as the ones obtained using the original Schordinger equation?

• If I calculate this quantity $\int d\vec{r}\Psi^*H\Psi$, what's the physical meaning of it? Also what's the difference and relation between this quantity and the floquet energy?

Please help if you have answer(s), also any comments are welcomed.