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CRC Press eBooks

Playing with the Schrödinger Equation

Ram Yatan Prasad, Pranita

Published Jan 15, 2021• 0 citations
Publisher

Abstract

This chapter exclusively deals with the application of Schrödinger equation on various systems such as the particle in a box; the particle in a rectangular three-dimensional box; energy levels for a cubic potential box; the tunnel effect; the final equation for tunnel effect, that is, transparency factor which gives a clue to understand that the probability of transmission decreases rapidly as the width of the barrier increases; quantum mechanical explanation of emission of α -particle; the particle on a ring considering spherical polar co-ordinates; and the particle on sphere. This chapter also comprises the Legendre polynomials, associated Legendre equation, associated Legendre function and spherical harmonics, Hermite polynomials and its applications, and orthogonal properties of Hermite polynomials. It also provides the standard topics like simple linear harmonic oscillator, and its classical and quantum mechanical treatment which also includes the asymptotic and series solutions and energy levels of a linear harmonic oscillator. The Schrödinger equation also plays with rigid rotors comprising its rigorous treatment and also the F and T equations which are very important. The chapter ends with the references, standard several computational problems, questions on concepts, and also standard numerical problems for the benefits of the readers.

DOI
10.1201/9781003133605-7

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This chapter exclusively deals with the application of Schrödinger equation on various systems such as the particle in a box; the particle in a rectangular three-dimensional box; energy levels for a cubic potential box; the tunnel effect; the final equation for tunnel effect, that is, transparency factor which gives a clue to understand that the probability of transmission decreases rapidly as the width of the barrier increases; quantum mechanical explanation of emission of α -particle; the particle on a ring considering spherical polar co-ordinates; and the particle on sphere. This chapter also comprises the Legendre polynomials, associated Legendre equation, associated Legendre function and spherical harmonics, Hermite polynomials and its applications, and orthogonal properties of Hermite polynomials. It also provides the standard topics like simple linear harmonic oscillator, and its classical and quantum mechanical treatment which also includes the asymptotic and series solutions and energy levels of a linear harmonic oscillator. The Schrödinger equation also plays with rigid rotors comprising its rigorous treatment and also the F and T equations which are very important. The chapter ends with the references, standard several computational problems, questions on concepts, and also standard numerical problems for the benefits of the readers.
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.
The wave nature of particles plays an important role in their physical properties that, for example, particles confined into a small bounded area can have only particular discrete energies. This chapter explains how to solve the time-independent Schrodinger equation. It considers four cases of a one-dimensional motion of particles confined in potentials rapidly changing with x : infinite potential quantum well; a potential step; square-well potential; and tunneling through a potential barrier. The chapter helps to learn the characteristic properties of solutions to this equation. It shows the differences between the predictions of quantum mechanics and classical physics. Quantum tunneling is important in the understanding of a number of physical phenomena such as thermonuclear reactions and conduction in metals and semiconductors. The most advanced application of quantum tunneling is the scanning tunneling microscope. The details of the calculations are left for the readers as a tutorial problem.
This chapter deals with some solutions of Schrödinger's equation in situations that are relatively simple, but which nevertheless have important applications in chemistry. Most of these deal with the quantization of energy. However, there is another remarkable prediction of quantum mechanics, known as tunnelling. In classical physics, kinetic energy cannot be negative; in other words, the total energy cannot be less than the potential energy. Schrödinger's equation can, however, be solved when this is the case. In some situations, these solutions are physically significant. The chapter then looks at the harmonic oscillator, which is one of the most important elementary models in mechanics, and is especially relevant in chemistry in connection with the vibration of molecules. It also considers rotations and angular momentum.
The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.
This chapter evaluates some of the simplified Schrödinger equations obtained in the previous chapter, and presents explicit solutions for the energy levels and the corresponding wavefunctions. It begins by looking at the free particle and the particle in a box. The particle in a box is the simplest example of a confined particle, and enables us to understand the origin of many features of such systems, such as the appearance of discrete energy levels. It is a general feature of particles confined by a potential to some region of space that solutions to the Schrödinger equation can only be found at certain discrete energies. The chapter then considers the harmonic oscillator; angular momentum; rotations of a polyatomic molecule; and the hydrogen atom.
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.