Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

Quantum Mechanics, Volume 3

particle density operator

Consider the average value of a one-particle operator in an arbitrary N-particle quantum state. It can be expressed, using relation (B-12), as a function of the average values of operator products :

(B-22)

This expression is close to that of the average value of an operator for a physical system composed of a single particle. Remember (Complement E_III, § 4-b) that if a system is described by a single particle density operator , the average value of any operator is written as:

(B-23)

The above two expressions can be made to coincide if, for the system of identical particles, we introduce a “density operator reduced to a single particle” whose matrix elements are defined by:

(B-24)

This reduced operator allows computing average values of all the single particle operators as if the system consisted only of a single particle:

(B-25)

where the trace is taken in the state space of a single particle.

The trace of the reduced density operator thus defined is not equal to unity, but to the average particle number as can be shown using (B-24) and (B-15):

(B-26)

This normalization convention can be useful. For example, the diagonal matrix element of in the position representation is simply the average of the particle local density defined in (B-19):

(B-27)

It is however easy to choose a different normalization for the reduced density operator: its trace can be made equal to 1 by dividing the right-hand side of definition (B-24) by the factor .

C. Two-particle operators

We now extend the previous results to the case of two-particle operators.

C-1. Definition

Consider a physical quantity involving two particles, labeled q and q′. It is associated with an operator acting in the state space of these two particles (the tensor product of the two individual state’s spaces). Starting from this binary operator, the easiest way to obtain a symmetric N-particle operator is to sum all the over all the particles q and q′, where the two subscripts q and q′ range from 1 to N. Note, however, that in this sum all the terms where q = q′ add up to form a one-particle operator of exactly the same type as those studied in § B-1. Consequently, to obtain a real two-particle operator we shall exclude the terms where q = q′ and define:

(C-1)

The factor 1/2 present in this expression is arbitrary but often handy. If for example the operator describes an interaction energy that is the sum of the contributions of all the distinct pairs of particles, and corresponding to the same pair are equal and appear twice in the sum over q and q′: the factor 1/2 avoids counting them twice. Whenever , it is equivalent to write in the form:

(C-2)

As with the one-particle operators, expression (C-1) defines symmetric operators separately in each physical state’s space having a given particle number N. This definition may be extended to the entire Fock space, which is their direct sum over all N. This results in a more general operator , following the same scheme as for (B-2):

(C-3)

C-2. A simple case: factorization

Let us first assume the operator can be factored as:

(C-4)

The operator written in (C-1) then becomes:

(C-5)

The right-hand side of this expression starts with a product of one-particle operators, each of which can be replaced, following (B-11), by its expression as a function of the creation and annihilation operators:

(C-6)

As for the last term on the right-hand side of (C-5), it is already a single particle operator:

(C-7)

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