Problem A$.$ Magnetic Translation Operators. We consider an electron in a three dimensional periodic potential $U(r)=U(r+R)($with $R$ a vector of the direct lattice$)$ in presence of an uniform magnetic field $B.$ The one$-$electron Hamiltonian is given by

hat$(H{)}_1(r)=\frac{1}{2m}(p-eA(r){)}^2+U(r)$

where $A(r)$ is the vector potential such that $B(r)=$grad$A(r).$ We use the symmetric gauge $A(r)=\frac{1}{2}Br$ as is customary in condensed matter physics. In this problem we consider $=c=1.$

$($I$-$A$-1)$ Show that the translation operator hat$(T{)}_R($defined in class$)$ does not commute with hat$(H{)}_1.$

$($I$-$A$-2)$ Show that the magnetic translation operator hat$(T{)}_R^M$ defined as

hat$(T{)}_R^M(r)=(r+R)e^{-\text{i}\text{e}\text{r}*\frac{BR}{2}}$

commutes with hat$(H{)}_1.$ To do so$,$ recall that Schrodinger's equation is invariant under gauge transformations

$A(r)A(r)+$grad$(r)$

$(r)(r)e^{ie(r)}$

where $(r)$ is a scalar potential which you need to find for Eq$.(2)$ to hold.

$($I$-$A$-3)$ Using the above results, show that two magnetic translation operators hat$(T{)}_R^M$ and hat$(T{)}_{R^{\text{'}}}^M$ do not commute.

$($I$-$A$-4)$ Briefly discuss what are the implications of these results for electrons in a magnetic field $($you can search on the web for inspiration$).$