By Professor Dr. Franz Schwabl (auth.)
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Additional info for Advanced Quantum Mechanics
I... ,ni, ... ) = (1 - ni)( _1)2 L j
17) where in the last term we have carried out a translation in the time by -t and write akO" == akO"(O) etc. On the right-hand side there now appears not only GkO"(t), but also a higher-order correlation function. In a systematic treatment we could derive an equation of motion for this, too. ( -t)) = (a~+q O",ak+q 0")( apO"' at,. (-t)) = JO"O", J pk ( a~+q 0"' ap+q 0"' ) ( akO"at,. ( -t) ). 19) From this, we can read off the energy levels E(k) as 10 The other possible factorization (a~+q <7,ap <71) (a~+q <7ak<7( -t) ) requires q = 0, which is excluded in the summation of Eq.
2a) o N247r 12 drre -/lX = -e - . 2c) e 2 47r The leading terms, proportional to N 2 , in the three evaluated energy contributions cancel one another. 5N yields an energy contribution f1 -k per particle of cx: ~ and vanishes in the thermodynamic limit. The limits are taken in the order NV --+ 00 and then jl --+ O. 5) o 0 s and introduced ro, the radius of a sphere of volume equal to the volume per particle. l!.. me ao The potential energy in first-order perturbation theory3 reads: (1) _ e2 471" t t q2 (4)01 ak+q,uak'_q,u,ak'u,aku 14>0) .
Advanced Quantum Mechanics by Professor Dr. Franz Schwabl (auth.)