# The commutator [L,p]

How do i compute the commutator [L,p]?

You should know from your class that the commutator [x, y] = xy - yx

you can express the L operator in terms of the coordinates x,y,z and the momentum operator p. Apply the commutator to a wavefunction psi and simplify!

Hope that gave you a clue.

Meir Achuz
Homework Helper
Gold Member
use L=rXp in the commkuator.

I find 2ihp, is that correct? do you know the correct answer?

dextercioby
Homework Helper
How do i compute the commutator [L,p]?

First of all, both L and p are vectors, so the commutator should be computed componentwise. Next, you need to find a common dense everywhere domain for the commutator, it's not difficult to see that on the Schwartz space over R^3 both the momentum and the angular momentum operators are essentially self-adjoint and the invariance conditions are met. Therefore,

$$[L_{i},p_{j}]_{-}\psi (\vec{r})=...$$

and , without doing any specific calculations (derivatives i mean), using the fundamental comm. relations (also valid on the Schwartz space) and some simple Levi-Civita pseudotensor manipulations, you can find the answer.

Meir Achuz
Homework Helper
Gold Member
I find 2ihp, is that correct? do you know the correct answer?
No its more complicated than that. Use Cartesian coordinates with
[x,px]=i and (rXp)_i=epsilon_ijk x_ip_j.

r = (x, y, z) and p = (px, py, pz).

I assume you know how to take a cross product. The only other thing is that p = -i\hbarh\del which acts on the wavefunction \Psi, and you can't exchange r and p (ie. rxp is not the same as pxr)

I hope that helps