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Progress of Theoretical Physics

Analysis of the Structure of Transformation Function in Quantum Mechanics

Smio Tani

Published Feb 1, 1954• 8 citations
Publisher

Abstract

A new device in operator calculus is introduced to treat expressions which contain non-commutable factors. By means of this technique the defining equation for the generator of transformation function of exponential type is derived. It is shown that this equation contains exclusively commutators of relevant quantities. Then, making use of this peculiarity to the exponential formula, a method is proposed for analyzing the structure of transformation function out of a given Hamiltonian of the system, and this is explained by some examples. This method consists of following procedures: First, decomposition of given Hamiltonian into parts of different character and abstraction of elementary matrices from it. Second, formation of commutators between these matrices and extension, if necessary, of the set of matrices enumerated at the beginning to include new ones which appear through operation of commutation. The matrix ring thus obtained constitutes the frame work of the transformation function. Through the course of this analysis one can find the ground why problems which are already solved exactly can be treated easily, and also can obtain some instructions how one should proceed in the solution of a given problem.

DOI
10.1143/ptp.11.190

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A new device in operator calculus is introduced to treat expressions which contain non-commutable factors. By means of this technique the defining equation for the generator of transformation function of exponential type is derived. It is shown that this equation contains exclusively commutators of relevant quantities. Then, making use of this peculiarity to the exponential formula, a method is proposed for analyzing the structure of transformation function out of a given Hamiltonian of the system, and this is explained by some examples. This method consists of following procedures: First, decomposition of given Hamiltonian into parts of different character and abstraction of elementary matrices from it. Second, formation of commutators between these matrices and extension, if necessary, of the set of matrices enumerated at the beginning to include new ones which appear through operation of commutation. The matrix ring thus obtained constitutes the frame work of the transformation function. Through the course of this analysis one can find the ground why problems which are already solved exactly can be treated easily, and also can obtain some instructions how one should proceed in the solution of a given problem.
In this paper we propose a new method of nonlinear transformation(Eigen-Mapping) to treat the eigenvalue problems in quantum mechanics as examples.The basic idea is to make the matrix of some operator we concerned diagonalization through introduction of small undetermined parameters and establishment equations for matrix elements.These problems can be devided into two types based on that there are or not 1st order approximate solution for them(in sense of the perturbation theory).In either type we consider also two cases of degenerate and nondegenerate.To obtain solution or approximate solution for eigenvalue problems,it is given the procedures of the method and usable formulas exhaustively.Seven examples in this paper are a common sight in general course of quantum mechanics.The results from them are compared with that from other approximate methods(for examples: perturbation method and variation method).After checking the results obtained,we can see that this new method has many advantages.
The transformation theory of quantum mechanics is constructed on the mathematical basis of the theory of the linear functionals and the physical notion of the wave-packet formalism of the theory. The wave-packet function, which is described by an indefinitely differentiable and rapidly decreasing function, specifies the regions of the values of some physical quantities corresponding to the physical situation. The wave functions or the transformation functions are considered to be the linear continuous functionals which have the wave-packet functions as their functional arguments and transform the wave-packet functions to the other ones. The dynamics of quantum mechanics is constructed on the former (the wave functions as functionals) but not on the latter (the wave-packet functions). The theory never contains improper functions and integrals. The present formalism is physically interpreted as a sort of the wave-packet theory of quantum mechanics. The formal theory of scattering is formulated in the framework of this theory without resort to any ε-manipulation and the adiabatic assumption, in order to investigate the conditions for potentials and the behaviors of the evolution operator at infinite past and future. Finally the definition is given for the higher-order functional derivatives. They may suggest a method of treating the overlapping singularities.
In Chapter 9 we applied the second quantization to the Hartree–Fock method. In this chapter we provide the basic aspects of the thermal Green function method, which is convenient for systematic perturbative expansion beyond the mean-field approximation. Several problems in quantum statistical mechanics can be elegantly and compactly described in terms of the so-called Matsubara Green functions (Matsubara (1955); Abrikosov et al. (1963)). Hence, if on one hand, the density matrix description, often used in the previous chapters, has a more direct physical interpretation, on the other hand the Green function method is far more powerful and takes advantage of a handy diagrammatic perturbation expansion. In the following chapters, we rely on the Green function for a more complete treatment of some many-body problems. To be self contained we introduce here this technical chapter with a short presentation of the by now standard Green function method. For a comprehensive treatment we refer the reader to the many books available and in particular to Abrikosov et al. (1963). The Matsubara Green function The main aim of quantum statistical mechanics is the evaluation of the partition function (see Eqs. (6.31) and (6.32)), e.g. in the grand canonical ensemble In the case of quantum gases one can evaluate Ƶ exactly as we did in Chapter 7. In general, the evaluation of the partition function is a difficult task and it is useful to use perturbation theory to obtain approximate solutions. To set up a perturbation theory, one begins by splitting the Hamiltonian as H = H 0 + H i, where H 0 represents the “easy” part, which can be solved exactly. In the second step, one could naturally think of expanding the exponential of the interacting part H i in order to write the partition function as a perturbative series. However, such a procedure requires extreme care due to the non-commutative nature of operators in quantum mechanics. The way of solving this problem is via the “ time ” ordering of operators in each term of the perturbative expansion. It can be shown, as we shall see in the present chapter, that the quantum statistical average of any time-ordered sequence of operators with respect to the unperturbed Hamiltonian H 0 can be expressed in terms of elementary quantum statistical averages of two field operators. Such elementary averages are called Green functions and in this section we introduce the basic definitions concerning them.
Abstract 1. In a recent paper Dirac* has brilliantly removed the defects before existing in the mechanics of the electron, and has shown how the phenomena usually called the “ spinning electron ” fit into place in the complete theory. He applies to he problem the method of q-numbers and, using non-commutative algebra, exhibits the properties of a free electron, and of an electron in a central field of electric force. In a second paper † he also discusses the rules of combination and the Zeeman effect. There are probably readers who will share the present writer’s feeling that the methods of non-commutative algebra are harder to follow, and certainly much more difficult to invent, than are operations of types long familiar to analysis. Wherever it is possible to do so, it is surely better to present the theory in a mathematical form that dates from the time of Laplace and Legendre, if only because the details of the calculus have been so much more thoroughly explored. So the object of the present work is to take Dirac’s system and treat it by the ordinary methods of wave calculus. The chief point of interest is perhaps the solution of the problem of the central field, which can be carried out exactly and leads to Sommerfeld’s original formula for the hydrogen levels. But it is also of some interest to exhibit the relationship of the new theory to the previous equations which were derived empirically by the present writer.* It appears that those equations were an approximation to the new ones, derived by an approximate elimination of two of Dirac’s four wave functions. We shall also review a few other points connected with the free electron, the emission of radiation from an atom and its magnetic moment, and shall outline a discussion of the Zeeman effect.
In this paper, we study within the structure of Symplectic Quantum Mechanics a bi-dimensional non-relativistic strong interaction system which represent the bound state of heavy quark-antiquark, where we consider a Cornell potential which consists of Coulomb-type plus linear potentials. First, we solve the Schrödinger equation in the phase space with the linear potential. The solution (ground state) is obtained and analyzed by means of the Wigner function related to Airy function for the cc‾c\overline{c}cc meson. In the second case, to treat the Schrödinger-like equation in the phase space, a procedure based on the Bohlin transformation is presented and applied to the Cornell potential. In this case, the system is separated into two parts, one analogous to the oscillator and the other we treat using perturbation method. Then, we quantized the Hamiltonian with the aid of stars operators in the phase space representation so that we can determine through the algebraic method the eigenfunctions of the undisturbed Hamiltonian (oscillator solution), and the other part of the Hamiltonian was the perturbation method. The eigenfunctions found (undisturbed plus disturbed) are associated with the Wigner function via Weyl product using the representation theory of Galilei group in the phase space. The Wigner function is analyzed, the non-classicality of ground state and first excited state is studied by the non-classicality indicator or negativity parameter of the Wigner function for this system. In some aspects, we observe that the Wigner function offers an easier way to visualize the non-classic nature of meson system than the wavefunction does.