Landau gauge momentum conservation question

I’m having an issue understanding why $hat{P}_x$ commutes with the Hamiltonian in the Landau gauge ($A_x=-Bhat{Y}, A_y=0$) in two dimensions. I’ve been finding elsewhere online the hand-wavey argument that this is because the Hamiltonian is independent of $hat{X}$. I want to be able to show this more rigorously, i.e. by applying the commutator to a test function, but I don’t know what kind of assumptions to make about such test functions (obviously they would be wavefunctions considering the Hamiltonian’s application in the Schrodinger equation). I would appreciate it if anyone could offer a hint.