a free particle of mass $m$, with Hamiltonian
$\hat{H} = \frac {\hat{P}^2} {2m}$,
where $\hat{P} = -i \hbar \frac{\partial} {\partial x}$.
The commutative relation is given by
$[\hat{X}, \hat{H}] = \frac {i\hbar} {m} \hat{P}$ (1)
In the common eigen-state of $\hat{H}$ and $\hat{P}$, $|e, p>$, can wo do the following?
$\lt e, p| [\hat{X}$, $\hat{H}] |e, p\gt$
$ = \lt e, p|\hat{X} (\hat{H}|e, p\gt) - (\lt e, p|\hat{H}) \hat{X}|e, p\gt $
$ = \lt e, p|\hat{X} (e|e, p\gt) - (\lt e, p|e) \hat{X}|e, p\gt $
$ = e( \lt e, p|\hat{X}|e, p\gt - \lt e, p|\hat{X}|e, p\gt ) $
$ = 0 $
Since the $\hat{H}$ is Hermitian, the above derivation doesn't seem to show any flaw. Given the commutative relation, Eq (1), we know the result is wrong. What's wrong with the above derivation?