By Professor Dr. Franz Schwabl (auth.)

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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. ,J . J The exchange term is a nonlocal term which only occurs for m s = m s .. The term in square brackets is equal to the probability amplitude that i ari'd j are at the positions x and x'. For further discussion of the Hartree-Fock equations and their physical implications we refer to Sect. 2 of QM 1. 1 Calculate the statie strueture function for noninteracting fermions SO(q) == ~(4)0 1nq 11- q 14>0), where nq = ~k,,, a~ "ak+q" is the particle density operator in the momentum representation and 14>0} is the ground state.

P,u As was to be expected, the density is homogeneous. The simplest excitation of adegenerate electron gas is obtained by promoting an electron from astate within the Fermi sphere to astate outside this sphere (see Fig. 2). 5) Fig. 2. Excited state of adegenerate electron gas; electronhole pair The absence of an electron in the state Ik1 , 0"1) has an effect similar to that of a positively charged particle (hole). If one defines bku == a~k ,-u and btu == a-k,-u, then the hole annihilation and creation operators band bt likewise satisfy anticommutation relations.

Ni, ... ) = ni( -1)L:;<; I... ,ni a! nj ,ni + 1, ... ) 1, ... ). 4 Fermions 19 It follows from this that aia! I... ,ni, ... ) = (1 - ni)( _1)2 L j

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