Angular Momentum Commutator

[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)

 

[Lojzek]

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.

 

[pam]

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

 

[malawi_glenn]

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 :

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

 

[eit32]

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

 

[malawi_glenn]

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

 

[reilly]

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

 

[malawi_glenn]

so:

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

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