Advanced Quantum Mechanics by Professor Dr. Franz Schwabl (auth.)

By Professor Dr. Franz Schwabl (auth.)

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These must be defined such that the result of applying them twice is zero. Furthermore, the order in which they are applied must play a role. 5a) 18 1. 5a) and the commutator of two operators A and B are defined by {A,B} [A,B] == == == AB+BA == AB - BA. 6) Given these preliminaries, we can now address the precise formulation. If one wants to characterize the states by means of occupation numbers, one has to choose a particular ordering of the states. This is arbitrary but, once chosen, must be adhered to.

12) e where is the position operator of a single particle and where we have made use of the fact that the matrix element within the integral is equal to 8(3) (x-e)8(3) (e -e'). In general, for a k-particle operator Vk: / d3 6 ... d3 ekd3 6' ... d3 ek' 'ljJt (el) ... 'ljJt (ek)

7) These are the Hartree-Fock equations. Compared to the Hartree equations, they contain the additional term J d3x'ix ~2 x'll'Pi(X')1 2'Pi(X) J = - ~Oms,msJ Jd3X'lx~2x'I'P;(X')'Pi(X')'Pj(X). 8) 52 2. Spin-l/2 Fermions The second term of the interaction on the left-hand side is known as the exchange integral, since it derives from the antisymmetry of the fermion state. The interaction term can also be written in the form J d3 x'ix ~2 x'i L cpj(x') [cpj(X')CPi(X) - CPj(x)CPi(x')8ms,ms,] . J The exchange term is a nonlocal term which only occurs for m s , = m s ,.

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