Motivated by a problem in chapter 2 of Sakurai's book Modern Quantum Mechanics, I'm interested in confirming something about the simple harmonic oscillator in quantum mechanics, I have found that the quantity $\langle m| \hat{p} \hat{x} |n \rangle$, where $|n \rangle$ is the n-th energy eigenstate of the Hamiltonian $H = \frac{P^2}{2m} + \frac{m \omega^2 x^2}{2}$, is given by $$\langle m| \hat{p}\hat{x}|n \rangle = \bigg[\frac{-i\hbar}{2}\bigg(\sqrt{m+1}\langle m+1| - \sqrt{m}\langle m-1|)(\sqrt{n}|n-1 \rangle + \sqrt{n+1}|n+1 \rangle\bigg)\bigg] \\= -\frac{i \hbar}{2}\bigg[\sqrt{(m+1)n} \langle m+1|n-1 \rangle + \sqrt{(m+1)(n+1)}\langle m+1| n+1 \rangle - \sqrt{mn} \langle m-1| n-1 \rangle - \sqrt{m(n+1)}\langle m-1|n+1 \rangle\bigg]$$
Hence I get $\langle 0| \hat{p}\hat{x}|0 \rangle = \frac{-i \hbar}{2}$, I believe the correct result is $\langle 0| \hat{p}\hat{x}|0 \rangle = 0$, does anyone know where I went wrong?