# Angular Momentum Commutator

• eit32

#### eit32

Do the Hamiltonian (H) and the z-component of angular momentum (L_z) commute?
[H, L_z]=0

H = [(-(hbar)^2/2m) dell^2] + V(r, theta, phi)
where dell is the gradient, and V is the potential

L_z = -i(hbar)(d/d phi)
where d is actually a partial derivative

I know how to find a commutator : [H,L_Z] = H*L_z - L_z*H
but I'm having trouble working through it in spherical coordinates (which my problem calls for)

Take a math handbook and look for Laplace operator in spherical coordinates. Then use the rule that derivative with recpect to fi commutes with derivatives with respect to other variables.

If V is a function of phi, H won't commute with L_z.

it depends on what kind of H you have.

When one have operators that contais derivatives, one usally let's the commutatur act on a test function, i.e :

$$[A,B]\psi(\vec{x}) = A(B\psi(\vec{x}))-B(A\psi(\vec{x}))$$

I have been using a test function but everything keeps canceling out and i don't think its suppose to

just use a general test function, denote it $\psi(r, \phi, \theta)$

Show how you did, and use Lojzek's hint to use the laplacian in spherical coordinates

1. On what single spherical coordinate variable does Lz depend?

2.Does the KE depend on this variable?

3. Can a potential depend on this variable?

There is no need to use a test function; you are dealing with operators in the same fashion as with the commutator of p and q.

Regards,
Reilly Atkinson

so:

$$HL_z \approx H\frac{\partial}{\partial \phi}$$

what is the derivative operator acting on if you don't have a test function?