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Angular Momentum Commutator

  • Thread starter eit32
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  • #1
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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)
 

Answers and Replies

  • #2
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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.
 
  • #3
pam
455
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If V is a function of phi, H won't commute with L_z.
 
  • #4
malawi_glenn
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it depends on what kind of H you have.


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

[tex] [A,B]\psi(\vec{x}) = A(B\psi(\vec{x}))-B(A\psi(\vec{x})) [/tex]
 
  • #5
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I have been using a test function but everything keeps canceling out and i don't think its suppose to
 
  • #6
malawi_glenn
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just use a general test function, denote it [itex] \psi(r, \phi, \theta) [/itex]

Show how you did, and use Lojzek's hint to use the laplacian in spherical coordinates
 
  • #7
reilly
Science Advisor
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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
 
  • #8
malawi_glenn
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so:

[tex]HL_z \approx H\frac{\partial}{\partial \phi}[/tex]

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

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