How Does the Commutator [p-hat_x, H-hat] Reveal Quantum Mechanics Fundamentals?

[blueyellow]

Homework Statement

by considering the action of [p-hat (subscript x), H-hat] on a general state, show that

[p-hat (subscript x), H-hat] =-ihbar dV/dx

Homework Equations

H-hat = (((p-hat)^2)/2m) +V(x)

p-hat (subscript x)= -i*h d/dx (partial derivative)

The Attempt at a Solution

tried lookin it up couldn't find anything

 

[Mike Pemulis]

It's right in front of you. Two hints:

1. What is the definition of the commutator of two operators? In other words, how would I write out [A, B]?

2. What is the derivative of a constant?

 

[blueyellow]

so the commutator is AB-BA

so it equals
i*h-bar d/dx(((p-hat)^2)/2m)+V(x)) -(((p-hat)^2)/2m)+V(x)) i*h-bar d/dx
=-ih-bar dV/dx ...
but why does -(((p-hat)^2)/2m)+V(x)) i*h-bar d/dx go to nothing? what do i do please, if the i*h-bar d/dx is put after the hamiltonian?

 

[Mike Pemulis]

what do i do please, if the i*h-bar d/dx is put after the hamiltonian?

In that case, you consider the derivative to be operating on 1 -- in other words d/dx = d/dx(1) = 0. So in the commutator, the only piece that doesn't vanish is the piece that features d/dx operating on a function of x -- which gives you -ih-bar dV/dx.