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CRC Press eBooks

Applications of Schrödinger Equation-1

M. Srirama Rao

Published Oct 13, 2022• 0 citations
Publisher

Abstract

As an application of the postulates of quantum mechanics, this chapter discusses a simple problem that of a particle constrained to move in a one dimensional box. The structure of matter can be analyzed according to increasingly more fundamental levels of organization. Radioactive processes produce neutrinos, particles that have no detectable mass or charge. In the universe, energies at this level are available in the centers of stars and in star explosions. The wave mechanical treatment of an electron in a box gives rise to large number of discrete energy levels. Tunneling is the quantum-mechanical effect of transitioning through a classically-forbidden energy state. For a quantum particle moving against a potential energy “hill”, the wave function describing the particle can extend to the other side of the hill. This wave represents the probability of finding the particle in a certain location, meaning that the particle has the possibility of being detected on the other side of the hill.

DOI
10.1201/9781003354048-4

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As an application of the postulates of quantum mechanics, this chapter discusses a simple problem that of a particle constrained to move in a one dimensional box. The structure of matter can be analyzed according to increasingly more fundamental levels of organization. Radioactive processes produce neutrinos, particles that have no detectable mass or charge. In the universe, energies at this level are available in the centers of stars and in star explosions. The wave mechanical treatment of an electron in a box gives rise to large number of discrete energy levels. Tunneling is the quantum-mechanical effect of transitioning through a classically-forbidden energy state. For a quantum particle moving against a potential energy “hill”, the wave function describing the particle can extend to the other side of the hill. This wave represents the probability of finding the particle in a certain location, meaning that the particle has the possibility of being detected on the other side of the hill.
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 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.
Abstract A distinguishing feature of quantum mechanics, as compared with classical mechanics, is the phenomenon of matter exhibiting wave-like properties. A direct consequence of this wave property of particles is the tunneling of particles through potential energy barriers.-3 The classical particle is forbidden from penetrating the barrier region. In this chapter, the problem of one-dimensional (1D) quantum mechanical tunneling of particles through potential barriers of arbitrary shape is treated. We show that tunneling is a mixed boundary value problem, and obtain the action integral that leads to Schrodinger’s equation and at the same time gives the equations defining the boundary conditions (BCs).
Abstract Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.
The approach is to start with some of the elementary ideas of classical mechanics and to translate these concepts to their equivalents in quantum mechanics. This will be done by changing the classical components of the total energy of particle into their quantum mechanical equivalents and arriving at the Schrodinger equation. The conjugate variables, position and momentum, in the Hamiltonian functions can be determined precisely for classical mechanical applications. This chapter deals with the Schrodinger model of the behavior of single electrons as affected by potential wells. The wave mechanic approach is extended to provide an outline of the interactions of many electrons with nuclei. In terms of the Pauli exclusion principle, the four quantum numbers of the electron states must be different. As additional electrons are added, the states available for occupation can be filled only according to the rules governing the quantum numbers.
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.