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Symmetry of many electron state functions (Read 2881 times)
Gerrit-Jan Linker
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Symmetry of many electron state functions
07.02.11 at 19:35:07
Symmetry properties of many electron state functions
The many body Schrodingerequation can in most cases not be solved in an exact way.
Formal, exact soltuions have symmetry properties:

  • Permutation symmetry
    The Pauli principle demands that ψ changes sign when 2 particles (indices) are interchanged.  
    OAψ(1,2,...,N) = ψ(1,2,....,N)
  • Symmetry of the nuclear configuration
    Ojψ(1,2,...,N) = ψj(1,2,....,N)
    where Oj is an operation that brings the nuclei in equivalent positions.
  • Spin symmetry
    If the N-electron Hamiltonian does not contain spin-orbit interactions, H will commute with the N-electron spin-operators S2 and Sz
    S2ψ(1,2,...,N) = S(S+1)ψj(1,2,....,N)
    Szψ(1,2,...,N) = Msψj(1,2,....,N)
    S=0,1,2 (N even)
    S=1/2, 3/2, 5/2, ... (N odd)
    Ms=S, S-1, S-2, ... , -S (2S+1 fold degeneracy)
    Solutions ψ can then be labeled with S and Ms: ψ(S,Ms).
    It is common to denote the whole multiplet ψ(S,-S), ψ(S,-S+1) ... ψ(S,S-1), ψ(S,S) with 2S+1ψ

See also:
Term Symbol
Spin operators
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« Last Edit: 09.02.11 at 21:26:43 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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