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

By Professor Dr. Franz Schwabl (auth.)

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Here, we have assumed that the states are different. If the states are identical, the derivation has to be supplemented in a similar way to that for the singleparticle operators. A somewhat shorter derivation, and one which also covers the case of fermions, proceeds as folIows: The commutator and anticommutator for bosons and fermions, respectively, are combined in the form [ak,aj]± = 8kj . 26) I bosons = { . 24). 1 States, Fock Space, Creation and Annihilation Operators For fermions, one needs to consider the states S_ li 1 , i 2 , ...

Ni, ... ) = (1 - ni)( _1)2 L j

Here, we have introduced the factor n/, since, for n/ = 0, the Kronecker delta bn;+l,n;' = 0 always gives zero. The factor n/ also ensures that the right-hand side cannot become equal to the state I... ,n/ - 1, ... ) = I... ,-1, ... ). To summarize, the effects of the creation and annihilation operators are I... ,ni, ... ) = (1 - ni)( -1)L:;<; n; I... ai I... ,ni, ... ) = ni( -1)L:;<; I... ,ni a! nj ,ni + 1, ... ) 1, ... ). 4 Fermions 19 It follows from this that aia! I... ,ni, ... ) = (1 - ni)( _1)2 L j

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