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I How is the CSCO in an harmonic oscillator?

  1. Jul 4, 2017 #1
    Hi everyone, I have a great doubt in this problem:
    Let a mass m with spin 1/2, subject to the following central potencial V(r):
    V(r)=1/2mω2r2
    Find the constants of motion and the CSCO to solve the Hamiltonian?

    This is my doubt, I can't find the CSCO in this potencial. Is a problem in general quantum physics, not use the Dirac's notation brakets and ket.

    Thanks a lot!
    Dar
     
  2. jcsd
  3. Jul 4, 2017 #2

    vanhees71

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    What the heck is CSCO? Define your acronyms!
     
  4. Jul 4, 2017 #3
    Yes I am sorry, CSCO= Complete Set of Conmuting Observables. This I need for resolve the hamiltonian.
     
  5. Jul 4, 2017 #4

    hilbert2

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    It's the same set as with a hydrogen atom, as this is rotation symmetric. Isn't that term just a "centrifugal" force that actually belongs to the kinetic energy but can also be thought of as an effective noninertial part of the potential field?
     
  6. Jul 4, 2017 #5

    vanhees71

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    For the symmetric 3D harmonic oscillator you have of course several possibilities. On one hand you can use Cartesian coordinates and use the three phonon-number operators ##\hat{N}_j=\hat{a}_j^{\dagger} \hat{a}_j## as the complete set. The relation to the Hamiltonian is
    $$\hat{H}=\hbar \omega \left (\sum_{j=1}^3 \hat{N}_j+\frac{3}{2} \right ).$$
    On the other hand, as any central potential you can as well use ##\hat{H}##, ##\hat{\vec{L}}^2##, and ##\hat{L}_z## as the complete set. The corresponding quantum numbers are then as for any central potential ##E##, ##\ell## and ##m##.
     
  7. Jul 6, 2017 #6
    Ok thanks a lot!

    Then in addition, the ##\hat{L}_z## and ##\hat{L}^2## they are motion constants because the hamiltonian is invariant to rotations and translations, and the energy ##E## only depends of the quantum number ##n##just because the hamiltonian conmute with the operator rotator ##\hat{R}## and translator ##\hat{T}##. It is true no?
     
    Last edited by a moderator: Jul 6, 2017
  8. Aug 1, 2017 #7
    Hi, a question about the spin, this add a new constant of movement, ie., in addition to ##L^2##, ##L_z## and ##H##, ##S_z## also is a constant of movement? My doubt is because this problem have in reality four constant of movement if we take into account that spin.
     
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