Quantum Harmonic Oscillator

  • #1
208
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Homework Statement



Consider the Hamiltonian

[tex]H=\frac{p^2}{2M}+\frac{1}{2}\omega^2r^2-\omega_z L_z[/tex]

Determine its eigenstates and energies.

2. The attempt at a solution

I want to check my comprehension; by eigenstate they mean

[tex]\psi(r)[/tex]
from the good old

[tex]H\psi(r)=E\psi(r)[/tex]
and then the energies would then be solutions for E?
 

Answers and Replies

  • #2
208
7
To sort of answer my own question the eigenstates would more properly probably be
[tex]|\psi>[/tex]
in
[tex]H|\psi>=E|\psi>[/tex]
 
  • #3
vela
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It's a three-dimensional problem, so the wave function will be a function of r, θ, and φ.
 
  • #4
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Ah, right. (and just as I was getting comfortable in 2D)
 
  • #5
dextercioby
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Convert L_z and p to spherical coordinates and separate variables just like in the H-atom case, or the isotropic 3D oscillator.
 

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