Solving [H,x] Operator Algebra

[FarticleFysics]

I'm trying to practice some operator algebra..

Solve [H,x] where H is the Hamiltonian operator, x is position operator, and assuming one dimensional potential energy, U(x).

I know the commutator comes out as -ih_bar(p_op)/m


Here is my work so far.

[H,x] = Hx - xH

note: H = [ -ih_bar(p/2m) + U(x) ]

so.. plug it in.. [ -ih_bar(p/2m) + U(x) ] x - x [ -ih_bar(p/2m) + U(x) ]

( x U(x) and -x U(x) cancel)

now.. -(ih_bar/2m)(p x) + (ih_bar/2m)(x p)

-(ih_bar/2m)[ p x - x p ]

x and p are both operators.. so I know they don't cancel.. I'm kind of lost at this point.

 

[Chopin]

Well, $px - xp$ is just another way of writing $[p,x]$, which by definition is $i\hbar$. However, this doesn't give you the answer you're looking for, because your Hamiltonian isn't right. The standard nonrelativistic Hamiltonian is $\frac{\hat{p}^2}{2m} + U(\hat{x})$, which should give you the commutation relations you're looking for.

 

[FarticleFysics]

Oh yeah.. haha thanks.