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Symmetry of many electron state functions (Read 1863 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)
     
    with:  
    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:
Hamiltonian
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1297080156/0#0
Term Symbol
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1238655847
Spin operators
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1297282533/0#0
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« Last Edit: 09.02.11 at 21:26:43 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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