Potential operator in positional space

[qubits]

Is the potential operator (in positional/space basis) of the Hamiltonian always diagonal in that basis? And is the kinetic energy operator always diagonal in complementary momentum space?

[Eye_in_the_Sky]

For a spinless particle in one dimension, the most general Hamiltonian which satisfies Galilean invariance is

Ht = P2/2m + Vt(Q) .

So, in this case, the answer to your question is "yes".
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However, for the case of a spinless particle in three dimensions, the most general Hamiltonian which satisfies Galilean invariance is

Ht = |P - At(Q)|2/2m + Vt(Q) .

The first term on the right hand side is the "kinetic" term ... clearly, it is not diagonal in the momentum representation.