Advanced Quantum Mechanics by Professor Dr. Franz Schwabl (auth.) PDF

By Professor Dr. Franz Schwabl (auth.)

ISBN-10: 3662054183

ISBN-13: 9783662054185

ISBN-10: 3662054205

ISBN-13: 9783662054208

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1 Momentum Eigenfunctions and the Hamiltonian The momentum representation is particularly useful in translationally invariant systems. We base our considerations on a rectangular normalization volume of dimensions Lx, Ly and L z . 1) = LxLyLz. with the volume V eik(x+L x ) By assuming periodic boundary conditions = e ikx , etc. 2a) the allowed values of the wave vector k are restricted to k = 27r ( nx ny nz) Lx' Ly' Lz ,nx = 0, ±1, ... ,ny = 0, ±1, ... ,nz = 0, ±1, ... 3) In order to represent the Hamiltonian in second-quantized form, we need the matrix elements of the operators that it contains.

1, ... ). To summarize, the effects of the creation and annihilation operators are I··· ,ni, ... ) = ai I... ,ni, ... ) = aJ (1 - ni)( -l)L;<, n; ni( -l)Lj<, n; I... I... ,ni + 1, ... ) ,ni -1, ... ). I ... ,ni,"') = = (1-ni)(-1)2LJ<,nJ(ni+1)1 .. · ,ni,"') (1 - ni) I.. 13a) a;ail .. · ,ni,"') =ni(-1)2LJ<,nJ(1-ni+1)1··· ,ni,"') == ni I· . ,ni,' .. 13b) , since for ni E {O, I} we have = ni and (_1)2LJ<' nj = l. 13b) one can regard a; ai as the occupation-number operator for the state Ii). 13a,b), one obtains the anticommutator In the anticommutator [ai, a}l+ with i is different: =1= j, the phase factor of the two terms Likewise, [ai, aj 1+ for i =1= j, also has different phase factors in the two summands and, since aiai I...

48 2. 20) The second term leads to a change in E(k), LlE(k) = - J d 3 k' (21f)3 41fe 2 8(k - k') Ik _ k'I 2 kF F 1 2 = _ e Jdk'k,2 Jd 1f o -1 TJ k2 1 + k,2 - 2kk'TJ kF = - ~ J 1fk = _ 2e 2 k F dk' k' I o Ik + k' I og k - k' (~ + 1- x 2 log 11 + x 1f , 4x 2 . 21) ' I F(x) Here again the function F(x) of Eq. 6") appears. The Hartree-Fock energy levels are reduced in comparison to those of the free electron gas. However, the estimated reduction turns out to be greater than that actually observed. 7 shows F(x) and E(k) in comparison to Eo(k) = 1i;:;'2 for rs = 4.

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Advanced Quantum Mechanics by Professor Dr. Franz Schwabl (auth.)


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