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.