Evaluating commutator with hamiltonian operator

[spybear]

Evaluate the commutator [H,x], where H is Hamiltonian operator (including terms for kinetic and potential energy). How does it relate to p_x, momentum operator (-ih_bar d/dx)?

 

[Altabeh]

Evaluate the commutator [H,x], where H is Hamiltonian operator (including terms for kinetic and potential energy). How does it relate to p_x, momentum operator (-ih_bar d/dx)?

The Hamiltoinan in a one dimensional space is defined as $$H=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)$$. So the commutator [H,x] is

$$[H,x]=Hx-xH=[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)]x-x[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)]$$. Continue this calculation and then by catching a glimpse of the definition of the operator $$p$$, you can get what the relation is.

AB