By Professor Dr. Franz Schwabl (auth.)

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14). 25) and, for fermions, particular attention must be paid to the order of the two annihilation operators in the two-particle operator. From this point on, the development of the theory can be presented simultaneously for bosons and fermions. 1 Transformations Between Different Basis Systems Consider two basis systems {li)} and {lA)}. What is the relationship between the operators ai and a)... The state IA) can be expanded in the basis {li)}: IA) = L li) (iIA) . 1) a! The operator creates particles in the state li).

18) Here, g(x) is the prob ability density that a pair of particles has the separation x; in other words, the probability density that a particle is located at x when with certainty there is a particIe at the position O. 20) a,ß Due to translational invariance, this is independent of x' and we may integrate over x', whence (with -b J d3 x' = 1) it follows that 1 G(x) = V L (J(x - Xa + xß)) . a,ß This leads to the relationship 1 ( G(x) = V ~ J(x) + N(NV -1) g(x) ) = nJ(x) + 2 N(N -1) V2 g(x) . , a quantum-mechanical expectation value in a particular state or a thermal expectation value.

2a) the allowed values of the wave vector kare restricted to k n x ny n z = 27f ( L' L' L y x ) z ' n x = 0, ±1, ... , n y = 0, ±1, ... , n z = 0, ±1, ... 3) In order to represent the Hamiltonian in second-quantized form, we need the matrix elements of the operators that it contains. : q Vqe iq . x . 5b) 26 1. Second Quantization For the matrix element of the two-particle potential, one then finds J (p', k'l V(x - x') Ip, k) 1 = V2 . d3 xd 3 x'e- 1P ,xe-I'k ' ,X V(x - x')e l'k ·X e1P 'x 1",", = V3 L Vq 1 J J d3 x 1 1 • .