Occupation-number operators

Properties can be computed by finding the expectation value of the property operator with the natural orbitals weighted by the occupation number of each orbital. This is a much faster way to compute properties than trying to use the expectation value of a multiple-determinant wave function. Natural orbitals are not equivalent to HF or Kohn-Sham orbitals, although the same symmetry properties are present. 

Creation and Annihilation Operators.—In the last section there was a hint that the theory could handle problems in which populations do not remain constant. Thus < , < f>s 2 is the probability density in 3A -coordinate space that the occupation numbers are , and the general symmetrical state, Eq. (8-101), is one in which there is a distribution of probabilities over different sets of occupation numbers the sum over sets could easily be extended to include sets corresponding to different total populations N. 

When dealing with systems described by antisymmetrical states, the creation and annihilation operators are defined in such a way that the occupation numbers can never be greater than unity. Thus we have a creation operator af defined by... 

It is to be remarked that these operators can act only on states of the system expressed in occupation number representation, as explicitly appearing in the definitions, Eqs. (8-105), (8-106), (8-112), and (8-114). We can multiply any one of these operators by a scalar factor, so that we can also define the following operators ... 

Both in Eq. (8-149) and Eq. (8-147), we have written the function in the center of the integrand simply for ease of visual memory in fact both /(q) and F(q,q ) commute with all the B-operators and their positions are immaterial. The B-operators operate on vectors > in occupation number space, so that we can evaluate the matrix elements of F in occupation number representation, viz., Eq. (8-145), either from Eq. (8-147) or from Eq. (8-149). 

The matrix of the projection operator in occupation number representation has a typical element... 

Here the projection operator P is multiplied by the distribution probability te , and the result summed over all states >. A typical element of the matrix of this operator in occupation number representation, called the density matrix, is... 

Let R be any linear operator in occupation number space, and consider the product WNR. A typical matrix element of this operator is... 

It is worthwhile to consider the same theorem in terms of coordinate space instead of occupation number space. Thus, we may envision an ensemble of systems whose states are X>, and whose distribution probabilities among these states are w(X). We define the density operator... 

The corresponding operator expression for the equilibrium entropy in occupation number representation is then also seen to be... 

The expectation value of the density operator, and, indeed, all the components of the density matrix, are stationary in time for an ensemble set up in terms of energy eigenstates. IT we use occupation number representation to set up the density matrix, it is at once seen from Eq. (8-187) that it also is independent of time ... 

Now we shall consider the time dependence of the ensemble average of any operator B not explicitly a function of time, Tr FVR. Because the trace is independent of the representation, we choose the one most convenient, which turns out to be the occupation number representation whose eigenvectors are eigenvectors of H. Thus we write... 

The occupation number operator for particles of momentum k can be defined as = k 1o ok, and the total number of particles operator as... 

The commutations (9-416)-( 9-419) guarantee that the state vectors are antisymmetric and that the occupation number operators N (p,s) and N+(q,t) can have only eigenvalues 0 and 1 (which is, of course, what is meant by the statement that particles and antiparticles separately obey Fermi-Dirac etatistios). In fact one readily verifies that... 

Observables, rate of change of, 477 Occupation number operator, 54 for particles of momentum k, 505 One-antiparticle state, 540 One-dimensional antiferromagnetic Kronig-Penney problem, 747 One-negaton states, 659 One-particle processes Green s function for computing amplitudes under vacuum conditions, 619... 

An appealing way to apply the constraint expressed in Eq. (3.14) is to make connection with Natural Orbitals (31), in particular, to express p as a functional of the occupation numbers, n, and Natural General Spin Ckbitals (NGSO s), yr,, of the First Order Reduced Density Operator (FORDO) associated with the N-particle state appearing in the energy expression Eq. (3.8). In order to introduce the variables n and yr, in a well-defined manner, the... 

In (4.3), all the occupation numbers remain unchanged, except nk. Likewise, the creation operator c is defined by... 

A proof of the bounds for the occupation numbers will be given in Section II. F. Expectation values of (particle-number conserving) operators are easily expressed in terms of the density matrices. For example, for the energy... 

Because polynomials of orbital occupation number operators do not depend on the off-diagonal elements of the reduced density matrix, it is important to generalize the Q, R) constraints to off-diagonal elements. This can be done in two ways. First, because the Q, R) conditions must hold in any orbital basis, unitary transformations of the orbital basis set can be used to constrain the off-diagonal elements of the density matrix. (See Eq. (56) and the surrounding discussion.) Second, one can replace the number operators by creation and annihilation operators on different orbitals according to the rule... 

The occupation number of this oscillator is the operator ala0 = JV its expectation value at time t is... 

In the usual formalism of quantum mechanics, the first quantization formalism, observables are represented by operators and the wave functions are normal functions. In the method of second quantization, the wave functions are also expressed in terms of operators. The formalism starts with the introduction of an abstract vector space, the Fock space. The basis vectors of the Fock space are occupation number vectors, with each vector defined by a set of occupation numbers (0 or 1 for fermions). An occupation number vector represents a Slater determinant with each occupation number giving the occupation of given spin orbital. Creation and annihilation operators that respectively adds and removes electrons are then introduced. Representations of usual operators are expressed in terms of the very same operators. 

The phase factor r(n) is introduced in order to endow the antisymmetry of many-electron wave functions in the Fock space, as we soon will see. The definition that ai operating on an occupation number vector gives zero if spin... 

The action of two operators aj and aj on an occupation number vector with spin orbital i and j, i[Pg.40]

The action of the operators a on occupation number kets can be determined by taking the scalar product of the conjugates of the vectors of Eqs. (1.5) and an arbitrary vector I k>. Using Eq. (1.5a) we obtain... 

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