How Do We Compute the Expectation Value <x̂p̂> in Quantum Mechanics?

[ertansinan]

How can we compute the expectation value, <$\widehat{x}$$\widehat{p}$> where ψ(x) is a normalized wavefunction? (The result is i$\hbar$/2)

 

[tom.stoer]

If you want to do the calculation in x-space you have to use the momentum operator represented by $-i\hbar\partial_x$; then you get the integral

$$\langle\psi|\hat{x}\hat{p}|\psi\rangle = \int_\mathbb{R}dx\,\psi^\ast(x)\,x\,(-i\hbar)\,\partial_x\,\psi(x)$$

Note that your result $i\hbar/2$ cannot be correct in general.