Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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

(A-4)

where n₁ is the occupation number of the first basis vector |u₁〉 (i.e. the number of particles in |u₁〉), n₂ that of |u₂〉, ..,nk that of |uk〉. In this series of numbers, some (even many) may be zero: a given state has no particular reason to always be occupied. It is therefore often easier to specify only the non-zero occupation numbers, which will be noted ni, nj, .., ni,.. . This series indicates that the first basis state that has at least one particle is |ui〉 and it contains ni particles; the second occupied state is |uj〉 with a population nj, etc. As in (A-4), these occupation numbers add up to N.

       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

For bosons, the basis vectors can be written as in (C-15) of Chapter XIV:

(A-5)

      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).

Let us calculate the norm of the right-hand term. It is composed of N! terms, coming from each of the N! permutations included in SN, but only some of them are orthogonal to each other: all the permutations Pα leading to redistributions of the nj first particles among themselves, of the next nj particles among themselves, etc. yield the same initial ket. On the other hand, if a permutation changes the individual state of one (or more than one) particle, it yields a different ket, actually orthogonal to the initial ket. This means that the different permutations contained in SN can be grouped into families of ni!nj!..nl!.. equivalent permutations, all yielding the same ket; taking into account the factor N! appearing in the definition of SN, the coefficient in front of this ket becomes ni!nj!..nl!../N! and its contribution to the norm of the ket is equal to the square of this number. On the other hand, the number of orthogonal kets is N!/ni!nj!..nl!.. Consequently if c was equal to 1 in formula (A-5), the ket thus defined would have a norm equal to:

      (A-6)

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

(A-7)

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

For the Fock states, it is sometimes handy to use a slightly different but equivalent notation. In (A-7), these states are defined by specifying the occupation numbers of all the states that are actually occupied (ni ≥ 1). Another option would be to indicate all the occupation numbers including those which are zero³ – this is what we have explicitly done in (A-4). We then write the same kets as:

      (A-8)

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

(A-9)

      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:

(A-10)

Taking into account the 1/N factor appearing in definition (A-3) of AN the right-hand side of this equation is a linear superposition, with coefficients

, 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,