  For quantum mechanics of a charged particle in a classical external
electromagnetic field, there is an apparent puzzle that the matrix element of
the canonical momentum and Hamiltonian operators is gauge dependent. A
resolution to this puzzle is recently provided by us in [2]. Based on the
separation of the electromagnetic potential into pure gauge and gauge invariant
parts, we have proposed a new set of momentum and Hamiltonian operators which
satisfy both the requirement of gauge invariance and the relevant commutation
relations. In this paper we report a check for the case of the hydrogen atom
problem: Starting from the Hamiltonian of the coupled electron, proton and
electromagnetic field, under the infinite proton mass approximation, we derive
the gauge invariant hydrogen atom Hamiltonian and verify explicitly that this
Hamiltonian is different from the Dirac Hamiltonian, which is the time
translation generator of the system. The gauge invariant Hamiltonian is the
energy operator, whose eigenvalue is the energy of the hydrogen atom. It is
generally time-dependent. In this case, one can solve the energy eigenvalue
equation at any specific instant of time. It is shown that the energy
eigenvalues are gauge independent, and by suitably choosing the phase factor of
the time-dependent eigenfunction, one can ensure that the time-dependent
eigenfunction satisfies the Dirac equation.
