Quantum second quantization

J0rgensen P and Simons J 1981 Second Quantization-Based Methods in Quantum Chemistry (New York Aoademio) J0rgensen P and Simons J (eds) 1986 Geometrical Derivatives of Energy Surfaces and Molecular Properties (Boston, MA Reidel)... 

J0rgensen P and Simons J 1981 Second Quantization Based Methods in Quantum Chemistry (New York Academic) oh 4... 

We have thus far only considered the relativistic quantum mechanical description of a single spin 0, mass m particle. We next turn to the problem of describing a system of n such noninteracting spin 0, mass m, particles. The most concise description of a system of such identical particles is in terms of an operator formalism known as second quantization. It is described in Chapter 8, The Mathematical Formalism of Quantum Statistics, and Hie reader is referred to that chapter for detailed exposition of the formalism. We here shall assume familiarity with it. 

Second quantization and configuration space description of spin 0 particles Schweber, S. S., An Introduction to Relativistic Quantum Field Theory, Harper and Row, New York, 1961. 

For anything bigger than the hydrogen atom, however, solving directly in terms of the coordinates and momenta becomes extremely difficult. Far more common is to express the wave function in terms of basis functions, introducing the idea of second quantization [45], A simple way to think of second quantization is that it describes the quantum mechanics, from the beginning, in terms of a set of basis functions. 

Jprgensen P, Simmons J (1981) Second quantization-based methods in quantum chemistry. Academic, New York... 

In order to obtain the particle description required for quantum statistics, it may therefore be necessary to quantize the quantum-mechanical wave field a second time. This procedure, known as second quantization, starts from the wave field once quantized ... 

Limitation to ensembles that allow exchange of energy, but not of matter, with their environment is unnecessarily restrictive and unrealistic. What is required is an ensemble for which the particle numbers, Nj also appear as random variables. As pointed out before, the probability that a system has variable particle numbers N and occurs in a mechanical state (p, q) can not be interpreted as a classical phase density. In quantum statistics the situation is different. Because of second quantization the grand canonical ensemble, like the microcanonical and canonical ensembles, can be represented by means of a density operator in Hilbert space. 

P. R. Surjan. Second Quantized Approach to Quantum Chemistry An Elementary Introduction, Springer-Verlag, New York, 1989. 

Upon doing so, the following set of equations is obtained (early references to the derivation of such equations include A. C. Wahl, J. Chem. Phys. 4T, 2600 (1964) and F. Grein and T. C. Chang, Chem. Phys. Lett. 12, 44 (1971) a more recent overview is presented in R. Shepard, p 63, in Adv. in Chem. Phys. LXIX, K. P. Lawley, Ed., Wiley-Interscience, New York (1987) the subject is also treated in the textbook Second Quantization Based Methods in Quantum Chemistry, P. Jprgensen and J. Simons, Academic Press, New York (1981))) ... 

Remark. It should be clear that this transition to an occupation number description is a purely algebraic step. In this respect it is similar to what in quantum mechanics is denoted by the misleading term second quantization . The only difference is that here we deliberately eliminate the information about the identity of the molecules , whereas in quantum mechanical applications (e.g., to photons or... 

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. 

Most formulations of MCSCF theory are based on the second quantization formalism. We therefore review briefly in this section the basic definitions of the annihilation and creation operators, and the expansion of quantum mechanical operators in products of them. 

The symmetry properties of the quantities used in the theory of complex atomic spectra made it possible to establish new important relationships and, in a number of cases, to simplify markedly the mathematical procedures and expressions, or, at least, to check the numerical results obtained. For one shell of equivalent electrons the best known property of this kind is the symmetry between the states belonging to partially and almost filled shells (complementary shells). Using the second-quantization and quasispin methods we can generalize these relationships and represent them as recurrence relations between respective quantities (CFP, matrix elements of irreducible tensors or operators of physical quantities) describing the configurations with different numbers of electrons but with the same sets of other quantum numbers. Another property of this kind is the symmetry of the quantities under transpositions of the quantum numbers of spin and quasispin. 

The most effective way to find the matrix elements of the operators of physical quantities for many-electron configurations is the method of CFP. Their numerical values are generally tabulated. The methods of second-quantization and quasispin yield algebraic expressions for CFP, and hence for the matrix elements of the operators assigned to the physical quantities. These methods make it possible to establish the relationship between CFP and the submatrix elements of irreducible tensorial operators, and also to find new recurrence relations for each of the above-mentioned characteristics with respect to the seniority quantum number. The application of the Wigner-Eckart theorem in quasispin space enables new recurrence relations to be obtained for various quantities of the theory relative to the number of electrons in the configuration. 

Wave functions (13.1) form an orthonormal set, but their normalization factors are defined only up to a sign. The fact is that the wave function (13.1) is antisymmetric not only under coordinate permutations, but also under permutations of one-electron quantum numbers. Thus, to fix the sign of a wave function requires a way of ordering the set of quantum numbers (a) = ai, 0C2,..., ajy. There exists, however, a convenient formalism that allows us to include the constraints imposed by the requirement that the wave functions be antisymmetric in a simple operator form. This formalism became known as the second-quantization method. This chapter gives a detailed description of the fundamentals of the second-quantization method. 

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. 

In the general case, the second-quantized operator linear combination of irreducible tensorial products of electron creation operators. The combination must be selected so that a classification of states according to additional quantum numbers be provided for. Without loss of generality, all the numerical coefficients in the linear combinations can be considered real. Then, from (14.14), we can introduce the operators... 

The concept of quasispin quantum number was discussed in the Introduction and Chapter 9 (see formulas (9.22) and (9.23)). Now let us consider it in the framework of the second-quantization technique. We can introduce the following bilinear combinations of creation and annihilation operators obeying commutation relations (14.2) - the quasispin operator ... 

Since the wave functions with N > v can be found from the wave functions with N = v using (16.1), in the second-quantization representation it is necessary to construct in an explicit form only wave functions with the number of electrons minimal for given v, i.e. IolQLSMq = —Q). But even such wave functions cannot be found by generalizing directly relation (15.4) if operator cp is still defined so that it would be an irreducible tensor in quasispin space, then the wave function it produces in the general case will not be characterized by some value of quantum number Q v). This is because the vacuum state 0) in quasispin space of one shell is not a scalar, but a component of a tensor of rank Q = l + 1/2... 

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46)... 

Formula (17.16) is the most general form of the two-electron matrix element in which all four one-electron wave functions have different quantum numbers. We shall put it into general formula (13.23), whereupon the creation and annihilation operators will be rearranged to place side by side those second-quantization operators whose rank projections enter into the same Clebsch-Gordan coefficient. Summing over the projections then gives... 

The Ms and Mv quantum numbers of a given 2S+1Z/ term can be raised or lowered by using the well known shift operators JS and . The second-quantized form of these operators is as follows ... 

Surjan PR (1989) Second quantized approach to quantum chemistry. Springer-Verlag, Berlin... 

The aim of this section is to familiarize the reader with the second quantization and the many-body diagrammatic techniques which are now widely used in up-to-date quantum chemistry. These techniques are very efficient since they permit the formulation of the problem by means of diagrams from which the explicit formula can be obtained. Another advantage is that the problem of spin can be handled very simply. This approach also permits us to have a microscopic view of the problem (as will be seen in the study of ionization potentials, excitation energies, interaction of two molecular systems etc.). 

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