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InTech eBooks

Quantal Cumulant Mechanics as Extended Ehrenfest Theorem

Yasuteru Shigeta

Published Apr 3, 2013• 2 citations
Publisher

Abstract

Since Schrodinger proposed wave mechanics for quantum phenomena in 1926 [1-4], referred as Schrodinger equation named after his name, this equation has been applied to atom-mol‐ ecules, condensed matter, particle, and elementary particle physics and succeeded to repro‐ duce various experiments. Although the Schrodinger equation is in principle the differential equation and difficult to solve, by introducing trial wave functions it is reduced to matrix equations on the basis of the variational principle. The accuracy of the approximate Schro‐ dinger equation depends strongly on the quality of the trial wave function. He also derived the time-dependent Schrodinger equation by imposing the time-energy correspondence. This extension opened to describe time-dependent phenomena within quantum mechanics. However there exist a few exactly solvable systems so that the methodology to solve Schro‐ dinger equation approximately is extensively explored, yet.

DOI
10.5772/53703

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Since Schrodinger proposed wave mechanics for quantum phenomena in 1926 [1-4], referred as Schrodinger equation named after his name, this equation has been applied to atom-mol‐ ecules, condensed matter, particle, and elementary particle physics and succeeded to repro‐ duce various experiments. Although the Schrodinger equation is in principle the differential equation and difficult to solve, by introducing trial wave functions it is reduced to matrix equations on the basis of the variational principle. The accuracy of the approximate Schro‐ dinger equation depends strongly on the quality of the trial wave function. He also derived the time-dependent Schrodinger equation by imposing the time-energy correspondence. This extension opened to describe time-dependent phenomena within quantum mechanics. However there exist a few exactly solvable systems so that the methodology to solve Schro‐ dinger equation approximately is extensively explored, yet.
In 1926, Schrödinger [365] recognized that the variational theory of elliptical differential equations with fixed boundary conditions could produce a discrete eigenvalue spectrum in agreement with the energy levels of Bohr's model of the hydrogen atom. This conceptually startling amalgam of classical ideas of particle and field turned out to be correct. Within a few years, the new wave mechanics almost completely replaced the ad hoc quantization of classical mechanics that characterized the “old” quantum theory initiated by Bohr. Although the matrix mechanics of Heisenberg was soon shown to be logically equivalent, the variational wave theory became the standard basis of methodology in the physics of electrons. The nonrelativistic Schrödinger theory is readily extended to systems of N interacting electrons. The variational theory of finite N -electron systems (atoms and molecules) is presented here. In this context, several important theorems that follow from the variational formalism are also derived.
In the previous chapter, the postulates helped us to use the wave function to find the expected results of experimental measurements by letting a dynamical quantity F (x, p, t) → F (xˆ, pˆ, t) and integrating over space. In the examples, however, the wave functions were typically given and not derived from first principles. We now endeavor to find the appropriate wave functions for a variety of physical systems in one dimension by solving the Schro¨dinger equation where for time-independent cases, the total energy is an eigenvalue. In subsequent chapters, we will extend the method to three dimensions and develop the appropriate operators, eigenvalues, and eigenfunctions for angular momentum. We will also develop methods for examining time-dependent systems. The latter chapters address special systems of interest that may be referred to as applications of quantum mechanics.
During 1925 three different but equivalent versions of quantum theory were proposed, Schrodinger proposed wave mechanics; Heisenberg developed matrix mechanics and Dirac-introduced operator theory. Considering the de Broglie's matter waves Erwin Schrödinger, an Austrian physicist, argued that if a particle like an electron behaves as a wave then the equation of wave motion could be successfully applied to it. He postulated a function varying in both space and time in a wave-like manner. Schrodinger set up a linear and time-dependent wave-like equation, called Schrodinger wave equation, to describe the wave aspect of a particle taking account of de Broglie's relation for wavelength. This chapter discusses the Schrodinger equation and explores the general properties of the solution of the Schrodinger equation. It verifies the Ehrenfest's theorem in certain interesting systems and solves equation for linear harmonic oscillator and shows that their solutions are identical to the solutions of classical equation of motion.
We have seen that to determine the possible energies En (energy eigenvalues) which a physical system can have and the corresponding eigenfunctions (wave functions) ψn, we have to set up and solve the time-independent Schro¨dinger equation Hψn = Enψn . (8.1.1) In this equation, the wave function ψn, representing the n-th eigenstate of the system in the coordinate representation, is a function of the continuously varying eigenvalues of the position observables of the system, the number of position observables being equal to number of degrees of freedom of the system. The Hamiltonian operator H, in the coordinate representation, is a differential operator involving the continuously varying eigenvalues of the position observables and the derivatives with respect to them. Since we will be dealing with the operators mostly in the coordinate representation, we shall denote them without the caret.
A general procedure for solving the time-dependent Schroumloumloumldinger equation
The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. We obtain the Schrödinger equation from a mathematical identity by a slight generalization of the formulation of classical statistical mechanics based on the Hamilton-Jacobi equation. This approach brings out most clearly the fact that the linearity of quantum mechanics is intimately connected to the strong coupling between the amplitude and phase of a quantum wave.