Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji


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XIV a basis of the state space for N identical particles. Its vectors are characterized by the occupation numbers ni, with:

       Comment:

      In this chapter we constantly use subscripts of different types, which should not be confused. The subscripts i, j, k, l, ..denote different basis vectors {|ui〉} of the state space 1 of a single particle; they span values given by the dimension of this state space, which often goes to infinity. They should not be confused with the subscripts used to number the particles, which can take N different values, and are labeled q, q′, etc. Finally the subscript α distinguishes the different permutations of the N particles, and can therefore take N! different values.

      A-1-a. Fock states for identical bosons

      where c is a normalization constant; on the right-hand side, ni particles occupy the state |ui〉, nj the state |uj〉, etc… (because of symmetrization, their order does not matter).

      (A-6)

      We shall therefore choose for c the inverse of the square root of that number, leading to the normalized ket:

      These states are called the “Fock states”, for which the occupation numbers are well defined.

      (A-8)

      Another possibility is to specify a list of N occupied states, where ui is repeated ni times, uj repeated nj times, etc. :

      As we shall see later, this latter notation is sometimes useful in computations involving both bosons and fermions.

      A-1-b. Fock states for identical fermions

      In the case of fermions, the operator AN acting on a ket where two (or more) numbered particles are in the same individual state yields a zero result: there are no such states in the physical space A(N). Hence we concentrate on the case where all the occupation numbers are either 1 or 0. We denote |ui〉, |uj〉,..,|ul〉,.. all the states having an occupation number equal to 1. The equivalent for fermions of formula (A-7) is written:

, of N! kets which are all orthogonal to each other (as we have chosen an orthonormal basis for the individual states {|uk〉}); hence its norm is equal to 1. Consequently, Fock states for fermions are defined by (A-10). Contrary to bosons, the main concern is no longer how many particles occupy a state, but whether a state is occupied or not. Another difference with the boson case is that, for fermions, the order of the states matters. If for instance the first two states ui and uj are exchanged, we get the opposite ket:

      (A-11)

      but it obviously does not change the physical meaning of the ket.

      A-1-c. Fock space

      The Fock states are the building blocks used to construct this whole chapter. We have until now considered separately the spaces S, A(N) associated with different values of the particle number N. We shall now regroup them into a single space, called the “Fock space”, using the direct sum4 formalism. For bosons:

      (A-12)

      and,


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