Compute Commutator [L,p]: Tutorial

[sapplesapple]

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

 

[Mentz114]

Read the forum guidelines.

 

[kcirick]

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]

use L=rXp in the commkuator.

 

[sapplesapple]

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

 

[dextercioby]

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]

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.

 

[kcirick]

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