Second Quantized Representation of Quantum Mechanical Operators

We are already familiar with the second quantized form of wave functions. In this section the second quantized form of various quantum-mechanical operators is going to be established. Then we shall be in a position to undertake any quantum-mechanical analysis in this representation. 

In principle, there are two equivalent ways for introducing operators in second quantization. The first is a more physical approach one may (in fact, one must) demand that the expectation value of any observable be the same in the second and the first quantized formalisms. Textbooks on second quantization usually choose this way (see, e.g. Szabo Ostlund 1982, and references therein). Here we shall proceed in a more formal manner, which, however, permits us to construct the second quantized operators, not only to introduce them heuristically. 

Consider a wave function and an operator A. If A acts on O, one gets a transformed function which is denoted by F  

Using second quantization, the correspondences of and are known in that representation, cf. (Eqs. 2.52-54). Let and denoted the second quantized counterparts of O and F, respectively. We are interested in the second quantized form of the operator A, which is denoted by A- for the moment. The relation between these quantities can be represented by the following chart  

Here the vertical arrows (j) indicate the mapping by the operator A while the horizontal arrows (- ) denote the second quantized representation. 

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Contents Introduction. - Concept of Creation and Annihilation Operators. -Particle Number Operators. - Second Quantized Representation of Quantum Mechanical Operators. - Evaluation of Matrix Elements. - Advantages of Second Quantization. - Illustrative Examples. - Density Matrices. -Connection to Bra and Ket Formalism. - Using Spatial Orbitals. - Some Model Hamiltonians in Second Quantized Form. - The Brillouin Theorem. -Many-Body Perturbation Theory. -Second Quantization for Nonorthogonal Orbitals. - Second Quantization and Hellmann-Feynman Theorem. - Inter-molecular Interactions. - Quasiparticle Transformations. Miscellaneous Topics Related to Second Quantization -Problem Solutions. - References -Index. 

Second Quantized Representation of Quantum-Mechanical Operators... 

For the sake of simplicity, consider first a special case which is rarely of actual interest the second quantized representation of an operator acting on the coordinates of a single electron. (Actually, this example is relevant only if there is only one electron in the system. Although somewhat artificial, this case is so simple that it enables comprehension of the basic ideas of how to find the second quantized representation of quantum-mechanical operators.)... 

In the preceding section, we constructed second-quantization operators for one- and two-electron operators in such a way that the same matrix elements and hence the same expectation values are obtained in the first and second quantizations. Since the expectation values are the only observables in quantum mechanics, we have arrived at a new representation of many-electron systems with the same physical contents as the standard first-quantization representation. In the present section, we examine this new tool in greater detail by con aring the first- and second-quantization representations of operators. In particular, we show that, for operator products — P, the second-quantization representation of may differ from the product of the second-quantization representations of and unless a complete basis is used. 

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. 

Operators corresponding to physical quantities, in second-quantization representation, are written in a very simple form. In the quantum mechanics of identical particles we normally have to deal with two types of operators symmetric in the coordinates of all particles. The first type includes N-particle operators that are the sum of one-particle operators. An example of such an operator is the Hamiltonian of a system of noninteracting electrons (e.g. the first two terms in (1.15)). The second type are iV-particle operators that are the sum of two-particle operators (e.g. the energy operator for the electrostatic interaction of electrons - the last term in (1.15)). In conventional representations these operators are... 

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. 

Similar to quantum mechanics, which can be formulated in terms of different quantities in addition to the traditional wave function formulation, in quantum chemistry a number of alternative tools are developed for this purpose, which may be useful in the context of the present book. We have already described different approximate models of representing the electronic structure using (many-electronic) wave functions. The coordinate and second quantization representations were employed to get this. However, the entire amount of information contained in the many-electron wave function taken in whatever representation is enormously large. In fact it is mostly excessive for the purpose of describing the properties of any molecular system due to the specific structure of the operators to be averaged to obtain physically relevant information and for the symmetry properties of the wave functions the expectation values have to be calculated over. Thus some reduced descriptions are possible, which will be presented here for reference. 

The most convenient representation of the generators of the Lie algebra of U (25 ) for fermionic quantum mechanics is by second quantized operators acting in Fermi-Fock space, Tfp, which is defined to be the direct sum... 

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