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Energy in a crystal

  1. Dec 19, 2016 #1
    • Thread moved from the technical forums, so no Homework Help Template is shown.
    Hello, I am stuck at the beginning of an exercise because I have some trouble to understand how are the energy level in this problem :

    In a crystal we have Ni2+ ions that we consider independent and they are submitted to an axial symmetry potential. Each ion acts as a free spin S=1. We have the hamiltonian :

    H0=C(Sz2-S2/3)
    C is a constant >0

    - I need to know the energy level of each ion and the eigenstates.

    After that, we add a magnetic field B oriented on the z axis (in interaction with the ion) which is given by the hamiltonian :
    H1=2uBBSz

    - Here again I have to find the levels of energy for H = H0+H1
     
  2. jcsd
  3. Dec 20, 2016 #2

    DrClaude

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    Staff: Mentor

    So what are the eigenvalues and eigenstates of that Hamiltonian? (Hint: think in terms of the quantum numbers for operators that commute with H.)
     
  4. Dec 20, 2016 #3
    I haven't studied much the eigenvalues of the spin but I can try. (I am not sure at all) :

    Sz = +1/2 , -1/2
    S=1

    Then H0=C(1/4-1/3) = -C/12

    eigenvalues K=-C/12

    and I should look for something who satisfy :

    H0Ψ=KΨ
     
  5. Dec 20, 2016 #4

    DrClaude

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    The eigenvalue of the ##\hat{S}^2## is ##S (S+1)##.

    Looking at it in terms of a wave function is not necessary. Do you know Dirac notation?
     
  6. Dec 21, 2016 #5
    So the eigenvalue of S2 should be only 2 ? (since it is given that S=1)

    Yes i know Dirac notation
     
  7. Dec 21, 2016 #6

    DrClaude

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    Correct.

    Edit: assuming ħ = 1.

    Then you should be able to write the eigenstate as a ket,
     
  8. Dec 27, 2016 #7
    I have been a little busy, I'm sorry for not answering sooner.

    It should be on this form then :

    | eigenvalue of S2 , eigenvalue of Sz >
    ##\frac{C}{3}## for |2,1>

    ##\frac{C}{3}## for |2,-1>

    ##\frac{-2C}{3}## for|2,0>

    And if we add a magnetic field B oriented on the z axis (in interaction with the ion), the hamiltonian is ##H=C(S_z^2 -\frac{1}{3}S^2) + 2 \mu_B B S_z## and the eigenvalues are :

    ##\frac{C}{3}+ 2 \mu_B B## for|2,1>
    ##\frac{C}{3} - 2 \mu_B B## for |2,-1>
    ##\frac{-2C}{3}## for |2,0>

    Is that right ?
     
    Last edited: Dec 27, 2016
  9. Dec 27, 2016 #8

    DrClaude

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    Looks fine, except
    is not conventional. Normally, one would use ##| S, M \rangle## (in other words, use the value of the spin ##S##, not the eigenvalue of ##\hat{S}^2##).
     
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