Second-quantization method

In the second-quantization method it is convenient to use the following designations of the JV-electron function instead of (10.9), often omitting the subscripts of coordinates, if this does not lead to any confusion ... 

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. 

Note, that this phase factor differs by (—l)2 from the phase factor that defines the standard phase system (Eq. (4) in the Introduction). This difference manifests itself only at half-integer values of momentum j. This change in the definition of the standard phase system is caused by the presence of commonly accepted phase relations in the second-quantization method [12, 96]. 

The traditional description of the wave function of a shell of equivalent electrons was presented in Chapter 9. Here we shall utilize the second-quantization method for this purpose. In fact, the one-electron wave function is... 

An important step was made by applying the second-quantization method to theoretical atomic spectroscopy [12, 30, 31]. It turned out... 

Abstract. Calculations of the non-linear wave functions of electrons in single wall carbon nanotubes have been carried out by the quantum field theory method namely the second quantization method. Hubbard model of electron states in carbon nanotubes has been used. Based on Heisenberg equation for second quantization operators and the continual approximation the non-linear equations like non-linear Schroedinger equations have been obtained. Runge-Kutt method of the solution of non-linear equations has been used. Numerical results of the equation solutions have been represented as function graphics and phase portraits. The main conclusions and possible applications of non-linear wave functions have been discussed. 

On the other hand, Cederbaum et al. (24) have apphed a second quantization method, the Green s function technique, to the N2 problem, with excellent results. Unfortunately, second quantization methods are ahen to the one-electron orbital concept for example, it is clear that Cederbaum s successive approximations do not correspond to ever larger chunks of correlation energy being taken into account. 

It is interesting to draw a distinction between the two aspects of correlation which we have considered so far in terms of the second-quantization method for systems of N identical Fermi particles. Those methods (which are but a more effective and general way of formulating Cl) rest upon the occupation-number representation given the set of all possible single-particle states (spinorbitals), one builds a complete set of N-particle states. .. > by constructing Slater determinants or... 

This differs only by sign from irreducible tensorial product (14.22) which makes especially convenient the use of tensor W kK in the second-quantization method. By its definition, and by (14.15)—(14.17), (14.23) and (14.25), we can always go from (14.30) to other unit tensors defined above ... 

By expanding the factor to second order in the vibrational coordinates Qk it is possible to bring the hamiltonian into a form for which a quantum mechanical treatment within the second quantization method can be performed. Although the RP methodology is of course not restricted to a second order truncation in the normal mode coordinates, this approximation makes the formulation of the method especially compact. 

A Russian team of scientists has been working on a parallel track for years, preferring the evaluation of ab initio effects to model calculations. Their initial interest was in d elements. Their techniques are described in the papers by Eremin and Kornienko (1972, 1975, 1977), Levin and Cherepanov (1979) and Eremin (1981). The contributions of two-particle operators are calculated with the help of the second quantization method. 

We begin by recording something of the historical origins of the second quantization method which should help the reader understand its importance in the study of many-body systems. ... 

What combinations of creation and annihilation operators would produce, when acting on the closed-shell ground state, the singlet and triplet excited functions in Problem 3.8 Use second-quantization methods to obtain the singlet and triplet excitation energies (cf. Problem 3.9). [Hint Use ap , Upp, etc. to indicate explicitly the operators that create or annihilate an electron in [Pg.86]

The existence of a convenient reference state means that when using second-quantization methods we do not need to start always from the vacuum state vac) we can instead use... 

All the above calculations were done for 51 sites or domains and 1 = 10. More data points could be used in our present simulation, but here we are primarily foeused to study nonlinearity or impurity induced critical behavior of DB motion and its quantum origin. To treat the problem analytically we take the help of second-quantization method. By quantizing the Hamiltonian in Eq. (1) in a number eonserving quantized form with [38]... 

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