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CRC Press eBooks

Schrödinger Equation

L. Piccirillo

Published Jul 25, 2023• 0 citations
Publisher

Abstract

In this chapter, the authors introduce the Hamilton-Jacobi equation showing that classical mechanics can be described with a wave equation for the classical action. In order to find the equation showing that classical mechanics already contains a wave behavior in matter particles, the people need to go back to the Lagrange/Hamilton formalism. Davisson-Germer experiment has shown without any doubt that electrons behave as massive particles whose dynamics is governed by some sort of wave. A free particle is apparently the conceptually simplest system to study because the TISE will reduce to its simplest form. The simple (linear) harmonic oscillator is of paramount importance in classical as well as quantum mechanics. A proper solution to the Harmonic oscillator problem can be found by solving Schrodinger equation analytically. Charles Hermite was a French mathematician who gave important contributions in many fields of mathematics used in physics like, for example, Hermitian operators and Hermite polynomials.

DOI
10.1201/9781003145561-5

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In this chapter, the authors introduce the Hamilton-Jacobi equation showing that classical mechanics can be described with a wave equation for the classical action. In order to find the equation showing that classical mechanics already contains a wave behavior in matter particles, the people need to go back to the Lagrange/Hamilton formalism. Davisson-Germer experiment has shown without any doubt that electrons behave as massive particles whose dynamics is governed by some sort of wave. A free particle is apparently the conceptually simplest system to study because the TISE will reduce to its simplest form. The simple (linear) harmonic oscillator is of paramount importance in classical as well as quantum mechanics. A proper solution to the Harmonic oscillator problem can be found by solving Schrodinger equation analytically. Charles Hermite was a French mathematician who gave important contributions in many fields of mathematics used in physics like, for example, Hermitian operators and Hermite polynomials.
In the first part of this chapter, the Hamiltonian function and Hamilton-Jacobi equation are presented. The latter is the starting point of Schrödinger’s work to obtain his wave equation. Next, the derivation of the wave equation is given along with the foundations of Schrödinger’s wave mechanics, i.e., the Hamilton’s analogy between optics and mechanics. It is also introduced some applications of the Schrödinger’s new quantum theory to several problems (radial function of the hydrogen atom, Compton effect and quantum harmonic oscillator). Other applications are also indicated in this chapter. The Dirac notation is introduced followed by the derivation of the expectation value in quantum mechanics and its derivation with respect to time. Next, Born’s probability theory and the superposition principle are introduced followed by the derivation of the quantum linear momentum operator. Afterwards, the normalization of a wave function and the orthogonality of a wave function are given followed by the properties of the Hermitian operator. In the end of the chapter, several examples and exercises are presented (168 words).
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.
This chapter expresses the whole of quantum mechanics in terms of a small set of postulates, which embraces the behaviour of all known forms of matter. It introduces the postulates and illustrates how they are used, including how to apply them to problems of chemical interest, such as atomic and molecular structure and the properties of molecules. It also establishes the full significance of the Hψ=Eψequation and provides a foundation for its application. The chapter includes the solutions of the Schrödinger equation for a particle and the operator, which is a symbol for an instruction to carry out some action on a function. The principal mathematical difference between classical mechanics and quantum mechanics is that whereas in the former physical observables are represented by functions, in quantum mechanics they are represented by mathematical operators.
In the past few chapters, the Schrödinger equation has been developed and applied to the solution of a number of quantum mechanical problems, mainly to develop experience with the results of simple and common systems. The results from these examples can be summarized relatively simply. In general, quantization enters the problem through the non-commuting nature of conjugate operators such as position and momentum as well as from classical boundary quantization. This has led to the Schrödinger equation itself being the primary equation of motion for the wave function solution to the problem. In essence, duality is incredibly important and is the basic method by which the electron is treated as a wave, rather than as a particle, and the wave equation of interest is the Schrödinger equation. When boundary conditions are applied, either through potential barriers or through the form of the potential (as in the harmonic oscillator), the time-independent Schrödinger equation yields solutions that are often special functions. Examples of this are the sines and cosines in the rectangular-barrier case, the Airy functions in the triangular-well case, and the Hermite polynomials in the harmonic oscillator case.
In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.
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.