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

By Professor Dr. Franz Schwabl (auth.)

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Sample text

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 {Ii)} and {IA)}. What is the relationship between the operators ai and a>..? The state IA) can be expanded in the basis {Ii)}: IA) = L Ii) (iIA) . 1) The operator a! creates particles in the state Ii).

Nm -1, ... ajakam I··· , ni,··· , nj, ... , nk,··· , nm, ... ) . 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 follows: The commutator and anticommutator for bosons and fermions, respectively, are combined in the form [ak, aj]'f = Okj. 26) £ bosons or fermions. 24). 1 States, Fock Space, Creation and Annihilation Operators For fermions, one needs to consider the states S_ lil, i2, ...

1. 7) where we have used polar coordinates and introduced the abbreviation ry = cos e. The integration over ry yields e ipr i;,~-ipr with r = Ix - xii. Thus, we have G(7(x - x') = 1 -2- 27r r lkF 0 dppsinpr = 1 ~(sinkFr - kFr cos kFr) 27r r 3n sin kFr - kFr cos kFr (kFr)3 2 The single-particle correlation function oscillates with a characteristic period of l/kF under an envelope which falls off to zero (see Fig. 3). , for large x they are at n21r. (x - x') "2 Fig. 3. (x - x') as a function of kFT 36 2.