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Extra info for Introduction to Many Body Physics. Lecture Notes, University of Geneva, 2008-2013
Example text
For fermions of course the total number of particles is always smaller than the total number of available states because of the Pauli principle. As shown in Fig. 1 we can fully describe the system and reconstruct its wavefunction if we know the number of particles in a given state |αi of the complete basis of single particle states. We can thus fully characterize the wave function of the system by the set of numbers n1 , n2 , . . , nΩ . The total number of particles in the system is of course N = n1 + n2 + · · · + nΩ , and can vary if one varies one of the ni .
NΩ = (1 − nj )(−1) j ci |n1 , . . , ni , . . , nj + 1, . . , nΩ = (1 − nj )(−1) j ni (−1) i |n1 , . . , ni − 1, . . , nj + 1, . . 47) 47 Creation and destruction operators Sect. 3 On the contrary c†j ci |n1 , . . , ni , . . , nj , . . , nΩ = ni (−1) i c†j |n1 , . . , ni − 1, . . , nj , . . , nΩ = (1 − nj )(−1) j ni (−1) i |n1 , . . , ni − 1, . . , nj + 1, . . 48) The term j corresponds to the phase factor with a state with ni − 1 instead of ni . Thus j = j − 1. 48) would be identical and one would have [ci , c†j ] = 0.
47) 47 Creation and destruction operators Sect. 3 On the contrary c†j ci |n1 , . . , ni , . . , nj , . . , nΩ = ni (−1) i c†j |n1 , . . , ni − 1, . . , nj , . . , nΩ = (1 − nj )(−1) j ni (−1) i |n1 , . . , ni − 1, . . , nj + 1, . . 48) The term j corresponds to the phase factor with a state with ni − 1 instead of ni . Thus j = j − 1. 48) would be identical and one would have [ci , c†j ] = 0. 49) which will allow to define the ci operators only in terms of their anticommutators. 20). Wavefunctions and averages can be computed by exactly the same techniques that were given for the bosons.