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Symmetric and Anti-Symmetric Wavefunctions

  1. Apr 11, 2005 #1
    I am not sure if my title to this thread is appropriate for the question I am about to ask, but it is what we are currently studying in my Quantum Mechanics class so here it goes.

    Two non-interacting particles with mass m, are in 1-d potential which is zero along a length 2a and infinite elsewhere

    a) What are the values of the values of the 4 lowest energies of the system?
    b) What are degeneracies of these energies if two particles are
    i)identical, with spin 1/2
    ii)not identical but both have spin 1/2
    iii)identical with spin 1

    What I am having trouble with is part a, but once I am helped with that can someone point me in the right direction to solve part b which is where I guess the anti-symmetric and symmetric wave functions come into play. I know spin 1/2 corresponds to fermions, while spin 1 correspnds to bosons. But I am still not sure how to this. Thanks and peace.
     
  2. jcsd
  3. Apr 11, 2005 #2

    dextercioby

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    There's only one advice:solve the SE for point a).The energy spectrum should be degenerate.

    Daniel.
     
  4. Apr 11, 2005 #3
    I know, but how do you solve the Schroding equation for 2 particles?
     
  5. Apr 11, 2005 #4

    dextercioby

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    What is the Hamiltonian?

    Daniel.
     
  6. Apr 11, 2005 #5

    Gokul43201

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    Dex, I think you scared Ed away ! :wink:

    Ed, it might help to start with the classical hamiltonian.
     
  7. Apr 11, 2005 #6
    ok, E=(n1^2 + n2^2)E1

    So the 4 lowest energy levels would be when n1=1 n2=1
    1,2
    2,1
    2,2

    So we have 2E1,5E1,5E1,8E1.

    Now can someone help me with part b if part a is correct here? I am unsure how to apply the various spin situations to this to find degeneracies in the energy.
     
  8. Apr 11, 2005 #7

    dextercioby

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    Point b)#1.I say 2.What do you say?

    Daniel.
     
  9. Apr 11, 2005 #8
    I am sorry. I am not clear what you mean. Since in this case, there are two electrons or fermions, it seems to me that the degenerate energy level would be 5E1 since this is the only possible energy level of the 4 that I listed. n1,and n2 cannot have the same value in the case of fermions. The other two parts to b involve bosons though so I am not sure what to do with these.
     
  10. Apr 11, 2005 #9

    Gokul43201

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    For a spin-S particle, what are the allowed values of [itex]m_s[/itex] ?
     
  11. Apr 11, 2005 #10
    1/2,-1/2? I don't think I learned this.
     
  12. Apr 11, 2005 #11

    Gokul43201

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    Correct (for spin 1/2).

    What about for a spin-1, spin-3/2 or spin-2 particle ? And in general, for a spin-S particle ?
     
  13. Apr 11, 2005 #12
    Spin S particle has 2s + 1 possible states?
     
  14. Apr 11, 2005 #13
    Then in answer to part (b)...

    2 non-identical spin S particles.... degneracy=(2S+1)(2S+1)
    2 identical spin S particles.... degneracy=(2S+1)(2S)
     
  15. Apr 11, 2005 #14
    Yeah but how does this pertain to degeneracies in energy values?
     
  16. Apr 11, 2005 #15
    These degeneracies are total possible of states (characterized by quantum numbers say [n,ms] in this case) that gives rise to same E. Only identical particle have to subject to Pauli principal which forbids 2 particles having same set of [n,ms].
     
  17. Apr 11, 2005 #16
    Sorry... it should be [n1,n2,ms] not [n,ms]
     
  18. Apr 11, 2005 #17

    dextercioby

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    ...make that fermions and Pauli principle (sic).

    Daniel.
     
  19. Apr 11, 2005 #18
    Ok,that makes sense to me. Where I wrote down the four lowest energy values, I wrote 5E1 down twice, since this energy could be obtained where n1=1 and n2=2 and vice versa. Thus this energy level is doubly degenerate. So if I were to calculate the total degeneracy of this level in the case of lets say two identical spin 1/2 particles, would the total number of degeneracies increase by 2? What about for two non identical spin 1/2 particles?

    And I didn't think the Pauli exclusion applied to identical spin 1 particles since they are bosons.
     
  20. Apr 11, 2005 #19

    dextercioby

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    Yeah,spin degeneration is 2s+1...For each level.So 2 identical 1/2 particles should have distinct spin eigenvalues.For different 1/2 particles could have the same spins states simultaneously.

    Daniel.
     
  21. Apr 11, 2005 #20
    Hmm... i supposed Pauli exclusion applies only to Fermions (identical particles whose total wave functions have to be anti-symmetric). So if your questions says its Boson, then we can ignore Pauli exclusion..
     
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