Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 37

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 .