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My text (Shankar) considers a Hamiltonian operator in a one-dimensional space

H = P^2 / 2m + V(X)

where P and X are the momentum and position operators respectively.

It then asserts that [X,H] = [X,P^2/2m]

That is, it has discarded the potential term of the Hamiltonian without comment or explanation.

How is that justified? I would have thought that if, as indicated, V is a function of X, the hamiltonian operator should be expressed in terms of that function.

For example, if V is a gravitational potential V(X) = -k/X, I would expect the above commutator to be

[X,H] = [X,P^2/2m-k/X] = [X,P] - k[X,1/X]

Why does Shankar discard the second term?

If one didn't discard it, what would [X,1/X] mean? Is there any way to handle the reciprocal of an operator?

Thanks for any help.