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Description of Interaction Applying Representation of Physical Quantity at Finite Temperature from Changed Distribution Function of Quantum Statistics

Jun Iizuka

Published Sep 1, 2021• 0 citations
OpenAlex

Abstract

So far, we have proposed a method of expressing finite temperature in the form of ground state and pure state by changing the shape of the distribution function and using the step function. Then, in the second quantized system in this form, the Hamiltonian and the method of giving its eigenvalues are presented. We did this in the form that the creation and annihilation operators, whose matrix elements changed as the statistics changed, acted on the Hamiltonian in the ground state. We also discussed the basics of the parastatistics method used for that. In this paper, we apply this finite temperature expression method , express the eigen- values of the Hamiltonian of the interacting system, and try to express the energy given by the interaction as heat.

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arXiv:1308.3275•Aug 14, 2013•4 citations
So far, we have proposed a method of expressing finite temperature in the form of ground state and pure state by changing the shape of the distribution function and using the step function. Then, in the second quantized system in this form, the Hamiltonian and the method of giving its eigenvalues are presented. We did this in the form that the creation and annihilation operators, whose matrix elements changed as the statistics changed, acted on the Hamiltonian in the ground state. We also discussed the basics of the parastatistics method used for that. In this paper, we apply this finite temperature expression method , express the eigen- values of the Hamiltonian of the interacting system, and try to express the energy given by the interaction as heat.
So far, we have proposed a method of expressing finite temperature in the form of ground state and pure state by changing the shape of the distribution function and using the step function. In this paper, we calculate the Hamiltonian and its eigenval- ues in the second quantized system in this format. Here, the Hamiltonian in the ground state is operated by the creation and annihilation operator in which the matrix elements are changed according to the change in the statistics. It also shows the basics of commutation relations of parastatistics and discusses generalized mathematics.
An alternative approach to the quantum statistics of interacting particles is proposed. It consists of calculating the equilibrium thermodynamical quantities of the many-body system via the pair distribution function with the assumption that the particles interact with each other only through pair central forces. The proposed approach has some advantage over the usual treatment via the partition function in that the pair distribution function is easier to deal with than the partition function in certain circumstances particularly when the collective motion description of the system is desirable. This is because only the pair distribution function can be expressed directly in terms of the collective interaction which is closely connected with collective elementary excitation, such as a plasmon in the electron gas and a phonon in the hard-sphere Boson gas. The equation of state as well as the internal energy are obtained in the form of integrals of the pair distribution function. The close analogy between the pair distribution function and the two-body propagator, which appears in the quantum field theory, makes it possible to analyze the former by the use of Feynman diagrams identical with those usually introduced for the latter. The collective interaction, which is defined by the sum of the direct and the indirect interactions, is introduced as a particular partial sum of the perturbation series of the pair distribution function. This is used in rewriting the pair distribution function in terms of the collective interaction.It is shown that, while the simple chain approximation to the collective interaction in the electron gas is responsible for the transfer of a plasmon, the same approximation to the collective pseudo-interaction in the hard-sphere Boson system has a relation similar to that of transfer of a phonon, both cases occuring at low temperatures. The explicit calculation of the pair distribution functions for these systems at the absolute zero temperature is carried out up to the first order in the collective interaction (simple chain approximation.) These results are used to calculate the ground-state energies. For the electron gas the energy thus obtained confirms Gell-Mann-Bruckner's calculation of the correlation energy. For the hard-sphere Bosons the calculated energy reproduces the result of Lee, Huang, and Yang. The extension of the calculation to the finite temperature case is also indicated. In particular the classical Debye-Huckel equation of state for the electron gas is briefly discussed.
A systematic development of the theory of finite Fermi systems for non-zero temperatures (Tnot =0) is undertaken on the basis of the method of temperature Green's functions. The properties of compound nuclear states are considered. The variations of the density matrix deltarho, the self-energy part delta..sigma.., the mean square radius delta , the quadrupole moment delta, etc, are studied in a transition from the ground state (T=0) to an excited state with temperature T. The relationship connecting the excitation energy with temperature is obtained. Consistently taken into account is the strong interaction of the nucleons, in particular, the change in the one-particle nuclear potential as the temperature T is varied. Qualitative quasi-classical estimates are given for the effects considered, and the limits on the applicability of the statistical description of nuclei at large T are discussed.
Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.
This paper is a study of the representations of the current algebra for an infinite, free, Bose or Fermi system at finite temperature. The state of the algebra is calculated for a finite system by use of the grand canonical formalism of statistical physics, and the thermodynamic limit is used to get the representation of the infinite case. An expression of the states is given in terms of determinants of one particle operators. In a second part, the usual formulas for the occupation density, the condensation temperature for bosons, etc., are recovered. It is also found that the canonical and grand canonical formalisms are equivalent when used to study Fermi systems in the thermodynamic limit, but not so for the Bose systems. It is postulated that this has something to do with the phase transition of Bose systems. Finally, the problem of the Hamiltonian is discussed in some details. The states calculated previously are used to test a method for calculating expectation values of the Hamiltonian, given the states of the algebra; it yields the correct value for the case of the free system at finite temperature.
Summary of the lectures: 1. Creation and annihilation operators for the system of indistinguishable particles 1.1 The permutation group and the states of a system of indistinguishable particles 1.2 Dimension of the Hilbert space of a system of indistinguishable particles 1.3 Definition and properties of the creation and annihilation operators 1.4 TheFockspace 1.5 The representations of state vectors and operators 1.5.1 N-particlewave-functions 1.5.2 The second-quantization representation of operators 1.5.3 Examples 1.6 Evolution of operators in the Heisenberg picture 1.7 Statistical operators of indistinguishable particles 1.7.1 The averages of the s-particle operators 1.7.2 The general structure of the one-particle statistical operator 2 Quadratic Hamiltonian and the diagonalization 2.1 Diagionalization of the Hamiltonian quadratic in the fermion operators 2.2 Diagionalization of the Hamiltonian quadratic in the boson operators 2.2.1 Diagonalization of the quadratic bosonic Hamiltonian possessing a zero mode 2.3 Long range order, condensation and the Bogoliubov spectrum of weakly interacting Bose gas 2.3.1 The excitation spectrum of weakly interacting Bose gas inabox 2.3.2 The excitation spectrum of an interacting Bose gas in an externalpotential 2.4 The Jordan-Wigner transformation: fermionization of interacting 1D spin chains