Expectation values of commutator and anti-commutator (momentum and position)

Question

What are the expectation values of commutator and anti-commutator for momentum and position operators? In the case of commutator: $$\langle[x,p]\rangle=\langle i\hbar\rangle=~?$$ In the case of anti-commutator: $$\langle \{x,p\}\rangle=~?$$

Answer 1

$i\hbar$ is simply a number, or if you must regard it as an operator, a multiple of the identity. So $\langle i\hbar \rangle=i\hbar$, and so is $\langle -i\hbar \rangle$.

By the way, anticommutator of $\hat{x}$ and $\hat{p}$ is not $[\hat{p},\hat{x}]$, but $\{\hat{x},\hat{p}\}=\hat{x}\hat{p}+\hat{p}\hat{x}$.

Answer 2

Expectation values of constants or numbers are just those constants or numbers.

The expectation value of the anti commutator of $\hat x$ and $\hat p$, that is, $\langle\{\hat x,\ \hat p\}\rangle$, for the Harmonic Oscillator, or coherent states of the Harmonic Oscillator, is equal to $0$. $$\langle n|\{\hat x,\ \hat p\}|n \rangle = \langle\hat x \hat p + \hat p \hat x\rangle = \langle \hat x \hat p\rangle + \langle\hat p \hat x\rangle = \langle\frac{i\hbar}2\rangle + \langle\frac{-i\hbar}2\rangle = \frac{i\hbar}2 - \frac{i\hbar}2 = 0$$