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
 A: Comments to the question (v5): 


*

*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.

*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. 

*The formula expression (3) with commutator inside the expectation value is correct, while the formula (2) with the expectation value inside a commutator seems to be an error.   
References: 


*

*Yung-kuo Lim, Problems and Solutions on Quantum Mechanics; Problem 1072(a), p.120.

