Derivatives, second-quantization representation

Generally speaking, the representation in terms of occupation numbers is considered to be an independent quantum-mechanical representation, distinct from the coordinate (or momentum) one. In that case, the occupation numbers for one-particle states are dynamic variables, and operators are the quantities that act on functions of these variables. In this section, second-quantization representation is directly related to coordinate representation in order that in what follows we may have a one-to-one correspondence between quantities derived in each of these representations. 

In Chapter 14 we derived, in the second-quantization representation, two different forms of the expressions for that operator - (14.61) and (14.63). To begin with, we consider expression (14.63) in which we, by (15.49), go over to triple tensors. Then, after some transformations and coupling the momenta in quasispin space, we arrive at... 

Having obtained the second quantized representation of fundamental spin operators, one may ask under what conditions can the second quantized Hamiltonian be expressed in terms of spin operators. These considerations serve as efficient tools for deriving model Hamiltonians. 

While early work [16, 19] on the CSE assumed that Nakatsuji s theorem [37], proved in 1976 for the integrodifferential form of the CSE, remains valid for the second-quantized CSE, the author presented the first formal proof in 1998 [20]. Nakatsuji s theorem is the following if we assume that the density matrices are pure A-representable, then the CSE may be satisfied by and if and only if the preimage density matrix D satisfies the Schrodinger equation (SE). The above derivation clearly proves that the SE imphes the CSE. We only need to prove that the CSE implies the SE. The SE equation can be satisfied if and only if... 

To be able to apply this quasiparticle framework we should derive the quasiparticle representation of the Hamiltonian. The first step along this line is to substitute the ordinary second-quantized operators by quasiparticles using the inverse of Eqs. (5) and (6) as... 

In second quantization, the Pauli antisymmetry principle is incorporated through the algebraic properties of the creation and annihilation operators as discussed in Chapter 1. We note that, in density-functional theory (which bypasses the construction of the wave function and concentrates on the electron density), the fulfilment of the A -representability condition on the density represents a less trivial problem. A density is said to be N-representable if it can be derived from an antisymmetric wave function for N particles [1]. 

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