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Physical Review

Theory of Nonlinear Effects in Crystals in the Second-Quantization Representation

Dragutin Lalović, Bratislav S. Tosic, R. B. Žakula

Published Feb 15, 1969• 33 citations
Publisher

Abstract

A method for an exact theoretical treatment of nonlinear (anharmonic) effects in crystals in the second quantization representation is presented. The method represents essentially a generalization of Bogolyubov's method of approximate second quantization. In addition, the Fermi representation of elementary excitation creation and annihilation operators is given.

DOI
10.1103/physrev.178.1472

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A method for an exact theoretical treatment of nonlinear (anharmonic) effects in crystals in the second quantization representation is presented. The method represents essentially a generalization of Bogolyubov's method of approximate second quantization. In addition, the Fermi representation of elementary excitation creation and annihilation operators is given.
In introducing second quantization for fermions, Jordan and Wigner (1927/1928) observed that the algebra of a single pair of fermion creation and annihilation operators in quantum mechanics is closely related to the algebra of quaternions H. For the first time, here we exploit this fact to study nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for a single fermionic mode. By means of these transformations, a class of fermionic Hamiltonians in an external field is related to the standard Fermi oscillator.
In introducing second quantization for fermions, Jordan and Wigner (1927, 1928) observed that the algebra of a single pair of fermion creation and annihilation operators in quantum mechanics is closely related to the algebra of quaternions H . For the first time, here we exploit this fact to study nonlinear Bogolyubov–Valatin transformations (canonical transformations for fermions) for a single fermionic mode. By means of these transformations, a class of fermionic Hamiltonians in an external field is related to the standard Fermi oscillator.
The Hamiltonian of a dynamical system of Fermi-particles is transformed by means of the general unitary transformation proposed by N. N. Bogolubov. From this Hamiltonian we obtain the approximate second quantization (a. s. q.) Hamiltonian, introducing the Bose-amplitudes with two indices. This Hamiltonian is diagonalized and the collective oscillations considered, especially for the pairs of particles with parallel spins. For the forced collective oscillations the paramagnetic term in the Hamiltonian is also taken into account. Thie term leads to the additional “spin” current, connected with the elementary excitations with spin-moment ±1. If the transfer momentum tends to zero, this current vanishes as the terms omitted in obtaining the Meissner-Ochsenfeld effect.
The formalism of the second quantization is developed entirely within the framework of the Dirac-Jordan representation theory. The equivalence of this formalism and the methods of ordinary coordinate-spin space is shown, and the equations of "connection" are fully developed. It is shown that the "creation" and "annihilation" operators, and indeed all operators which do not commute with the quantized Hamiltonian and hence whose representatives cannot be explicitly given in a configuration space of a fixed number of dimensions are easily treated in terms of the general representation theory and arise quite naturally. Both types of statistics are considered, and the Fock-Dirac density matrix and self-consistent field are treated as a simple illustration.
A review of the second quantization formulation of quantum mechanics is given based on the idea of a graded Hilbert space whose elements are represented by column matrices containing zero, one, two, ···, particle functions. The creation and annihilation operators are defined in terms of intuitive ideas about adding a particle to or removing a particle from the system. The (anti)commutation relations are shown to follow from the (anti)symmetry of the wavefunctions. The field operators which describe the creation or annihilation of a particle at a point are then obtained and their properties discussed. These operators are then used to express general operators in second quantization. Finally, a basis is introduced and the equations of occupation-number space are obtained.