# What does a line above a commutator, e.g. $\overline{[x, H]}$ mean?

What does this notation mean in relation to quantum mechanics?

$$\overline{[x,H]}\qquad\text{or}\qquad\overline{[p,H]}\tag{1}$$

I know $[x,H]$ is just the commutator e.g $xH-Hx$, and the anti-commutator is $\{x,H\} = xH + Hx$ -- but what does the line above the commutator do?

Here is some context:

$$\frac{d \langle x_0\rangle}{dt} = \frac{1}{i \hbar}[\langle x\rangle \cos{\omega t}, H] -\omega\langle x\rangle \sin{\omega t} - \frac{1}{i \hbar}\bigg[\frac{\langle p\rangle }{m\omega}\sin{\omega t}, H\bigg] -\frac{\langle p\rangle}{m} \cos{\omega t}\qquad \tag{2}$$ $$= \frac{1}{i \hbar}\overline{[x,H]} \cos{\omega t} -\omega\langle x\rangle \sin{\omega t} - \frac{1}{m \omega}\frac{1}{i \hbar}\overline{[p,H]}\sin{\omega t} -\frac{\langle p\rangle}{m} \cos{\omega t} = 0\qquad \tag{3}$$

This is from: Problems and Solutions on Quantum Mechanics By Yung-kuo Lim Page 120 - Problem 1072 ISBN-10: 9810231334

• Never seen this before. Maybe some reverse engineering is possible. What do the brackets in $<x>$ etc. mean. An expectation value? Then, maybe, the overbar also has this meaning. Nov 3, 2015 at 9:56

1. In this quantum case the overline/bar notation $\bar{A}=\langle A\rangle$ is borrowed from statistics and it denotes a quantum expectation value of a quantity $A$. See also Ehrenfest theorem.
2. The problem from Ref. 1 considers a harmonic oscillator with Hamiltonian operator $$\tag{A} H~=~\frac{p^2}{2m} +\frac{m\omega^2}{2}x^2,$$ and defines an initial value position operator $$\tag{B} x_0~:=~x\cos\omega t - \frac{p}{m\omega}\sin\omega t,$$ and asks to show that the expectation value of the operator $$\tag{C} \frac{dx_0}{dt}$$ vanishes. Well, actually, it is straightforward to show with the help of Heisenberg's EOM that the operator (C) itself vanishes, and therefore its expectation value.