Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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is then simply written as:

      (B-14)

where is the occupation number operator in the state |uk〉 defined in (A-28).

B-3. Examples

A first very simple example is the operator , already described in (A-29), and corresponding to the total number of particles:

      (B-15)

      As expected, this operator does not depend on the basis {|ui〉} chosen to count the particles, as we now show. Using the unitary transformations of operators (A-51) and (A-52), and with the full notation for the creation and annihilation operators to avoid any ambiguity, we get:

      (B-16)

      which shows that:

      (B-17)

      For a spinless particle one can also define the operator corresponding to the probability density at point r₀:

      (B-18)

Relation (B-12) then leads to the “particle local density” (or “single density”) operator:

(B-19)

      The same procedure as above shows that this operator is independent of the basis {|ui〉} chosen in the individual states space.

Let us assume now that the chosen basis is formed by the eigenvectors |K_i〉 of a particle’s momentum ħk_i, and that the corresponding annihilation operators are noted a_ki. The operator associated with the total momentum of the system can be written as:

      (B-20)

      As for the kinetic energy of the particles, its associated operator is expressed as:

      (B-21)

B-4. Single 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

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