By Professor Dr. Franz Schwabl (auth.)

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Assume a square lattice with lattice constant a and wave functions that are Gaussians, rp(x) = L\. 3/ 2\,3/ 4 exp{ _x 2/2,12}. (h) In the limit ,1 « a, the "on-site" interaction U = Vii is the dominant contribution. Determine for this limiting case the form of the Hamiltonian in second quantization. The model thereby obtained is known as the Hubbard model. ai La! (il T Ij) aj + ~ L ara} (ijl V Ikl) alak . 9 (a) Show, by verifying the relation n(x) Ie/» = 8(x - x') Ie/» , that the state (10) = vacuum state) describes a particle with the position x'.

The only contribution for which every annihilation operator is compensated by a creation operator is proportional to Öuu,Ök"k+qaL+quaLu,ak+qu' aku, thus: 2 1 471" = -~ '" 2V ~ E(l) k,q,a = - -2 nk+qu n ku " q e2 '~ " " , ' 471" 2V ~ q2 8(k p = - 471"e2 V - Iq + kl)8(k p - k) k,q u jd3 k8(k p _ k) jd 3 k l 1 Ik-k/l 28 (k p - k' ). 5. 44 2. Spin-l/2 Fermions Fig. 5. Integration region for E(1) consisting of the region of overlap of two Fermi spheres with relative displacement q; see Eq. 916 N . 916 - ---:;:;- + .

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! 5a) 18 1. 5a) and the commutator of two operators A and Bare defined by {A,B} == [A,B]+ == AB+BA [A,B] == [A,B]_ == 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.