By Professor Dr. Franz Schwabl (auth.)
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Extra resources for Advanced Quantum Mechanics
13) independent of the separation. Particles with opposite spin are not affected by the Pauli principle. nq". nq". (x-x)). 8) and the abbreviation x = kF Ix - x'l, we finally obtain 9 . (x - x') = 1 - 6"(smx - xcosx)2 . 0 2. - .. -7r---~ 47r- kF lx - x I I Fig. 4. The pair distribution function 9CT<7(X - x') . 16) plotted in Fig. 4. If a fermion is removed at x, the particle density in the vicinity of this point is strongly reduced. In other words, the probability of finding two fermions with the same spin at separations ;S k F1 is smalI.
For states with equal particle number (from a single subspace), it is identical to the previous scalar product, and for states from different subspaces it always vanishes. Furthermore, we have the completeness relation 1 1 L L ···l n l,n2, ... 3b) nl=O n2=O a! Here, we wish to introduce creation operators once again. 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. We thus define the creation operators by a!
CPj (x) =CiCPi(X) . 7) These are the Hartree-Fock equations. Compared to the Hartree equations, they eontain the additional term J d3x'Ix -L ~2 x'llcpi(X')1 Dm'im'i ) = - ~ Dm,;m' )rt i J J 2 cpi(X) d3x'Ix ~2 x'l IP; (X')CPi(X')CPj (x) d3x' Ix ~2 x'l cP; (X')CPi(X')CPj (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. ,J . J The exchange term is a nonlocal term which only occurs for m s = m s ..