Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 16

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.