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Rotational invariance and degeneracy (quantum mechanics)

  1. Mar 24, 2013 #1
    1. The problem statement, all variables and given/known data


    Show that if a Hamiltonian [itex]H[/itex] is invariant under all rotations, then the eigenstates of [itex]H[/itex] are also eigenstates of [itex]L^{2}[/itex] and they have a degeneracy of [itex]2l+1[/itex].


    2. Relevant equations

    The professor told us to recall that

    [itex]J: \vec{L}=(L_x,L_y,L_z) [/itex]

    [itex] L_z|l,m\rangle=m|l,m\rangle [/itex]

    [itex] L_\pm=L_x\pm iL_y [/itex]

    [itex] L_\pm|l,m\rangle= \hbar\sqrt{l(l+1)-m(m\pm1)} |l,m\pm 1\rangle[/itex]


    3. The attempt at a solution

    I have been reading as much materials as I can, but I still have no clue at all on how to solve it at this moment. Can anyone help? Thank you so much!
     
  2. jcsd
  3. Mar 25, 2013 #2
    I solved it!

    Thank goodness! I have solved it now.

    One can do the calculations in either generic Hilbert space for general rotations or in 3D real Hilbert space.
     
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