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CRC Press eBooks

Exactly Solvable Systems I: Bound States

S. Rajasekar, R. Velusamy

Published Sep 29, 2022• 0 citations
Publisher

Abstract

This chapter deals with the study of bound states of some simple one-dimensional systems. Study of bound systems and their bound states is very important because atoms, molecules and other fundamental matters are bound systems and their characteristics are understood by quantum mechanics. The chapter considers a one-dimensional linear harmonic oscillator, an anharmonic oscillator, a particle in a box, Morse oscillator, Poschl–Teller potentials and pendulum system. The time evolution of stationary state eigenfunctions is easy to obtain. The linear harmonic oscillator (LHO) is a fundamental system solved exactly in both classical and quantum mechanics. It is used in the modelling of subnuclear quarks to cosmic systems. The connection between the quantized energy eigenvalue spectrum of a bound state and the classical motions of the corresponding classical system has become increasingly interesting as the ability to experimentally probe the quantum-classical interface has dramatically improved.

DOI
10.1201/9781003172178-4

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Sciendo eBooks•Sep 1, 2018•0 citations
This chapter deals with the study of bound states of some simple one-dimensional systems. Study of bound systems and their bound states is very important because atoms, molecules and other fundamental matters are bound systems and their characteristics are understood by quantum mechanics. The chapter considers a one-dimensional linear harmonic oscillator, an anharmonic oscillator, a particle in a box, Morse oscillator, Poschl–Teller potentials and pendulum system. The time evolution of stationary state eigenfunctions is easy to obtain. The linear harmonic oscillator (LHO) is a fundamental system solved exactly in both classical and quantum mechanics. It is used in the modelling of subnuclear quarks to cosmic systems. The connection between the quantized energy eigenvalue spectrum of a bound state and the classical motions of the corresponding classical system has become increasingly interesting as the ability to experimentally probe the quantum-classical interface has dramatically improved.
In the previous chapter we focused our discussion on certain exactly solvable quantum mechanical systems possessing bound states. In the present chapter we wish to concentrate on some fascinating and interesting one-dimensional systems where the quantum states are essentially scattering states. For scattering states energy eigenvalues are continuous and energy eigenfunctions are nonnormalizable. An interesting feature of many scattering potentials is the occurrence of tunneling. It is the most striking qualitative difference between classical mechanics and quantum mechanics. In tunneling a quantum mechanical particle passes through a region where the potential energy exceeds the total energy of the particle. This phenomenon is impossible in classical mechanics. Classically, a particle is completely reflected from a barrier. Quantum tunneling has many technological applications.
On the basis of Nelson’s stochastic mechanics derivation of the Schrödinger equation, a formal mathematical structure of non-relativistic quantum mechanics equivalent to the one in classical analytical mechanics has been established in the literature. We recently were able to augment this structure by deriving quantum Hamilton equations of motion by finding the Nash equilibrium of a stochastic optimal control problem, which is the generalization of Hamilton’s principle of classical mechanics to quantum systems. We showed that these equations allow a description and numerical determination of the ground state of quantum problems without using the Schrödinger equation. We extend this approach here to deliver the complete discrete energy spectrum and related eigenfunctions for bound states of one-dimensional stationary quantum systems. We exemplify this analytically for the one-dimensional harmonic oscillator and numerically by analyzing a quartic double-well potential, a model of broad importance in many areas of physics. We furthermore point out a relation between the tunnel splitting of such models and mean first passage time concepts applied to Nelson’s diffusion paths in the ground state.
This chapter focuses on some fascinating and interesting one-dimensional systems, where the quantum states are essentially scattering states. For scattering states energy eigenvalues are continuous and energy eigenfunctions are nonnormalizable. An interesting feature of many scattering potentials is the occurrence of tunnelling. It is the most striking qualitative difference between classical mechanics and quantum mechanics. In tunnelling a quantum mechanical particle passes through a region where the potential energy exceeds the total energy of the particle. This phenomenon is impossible in classical mechanics. Classically, a particle is completely reflected from a barrier. Quantum tunnelling has many technological applications. There are number of modern opto-electronic devices involving low-dimensional heterostructure. The unbound states of semiconductor quantum wells and quantum dots play significant role in their optical properties. Quantum mechanical transmission and reflection of electrons at potential energy steps can be applied to the design of nonequilibrium transistors.
Quantum Mechanics I: The Fundamentals provides a graduate-level account of the behavior of matter and energy at the molecular, atomic, nuclear, and sub-nuclear levels. It covers basic concepts, mathematical formalism, and applications to physically important systems. This fully updated new edition addresses many topics not typically found in books at this level, including: Bound state solutions of quantum pendulum Morse oscillator Solutions of classical counterpart of quantum mechanical systems A criterion for bound state Scattering from a locally periodic potential and reflection-less potential Modified Heisenberg relation Wave packet revival and its dynamics An asymptotic method for slowly varying potentials Klein paradox, Einstein-Podolsky-Rosen (EPR) paradox, and Bell’s theorem Delayed-choice experiments Fractional quantum mechanics Numerical methods for quantum systems A collection of problems at the end of each chapter develops students’ understanding of both basic concepts and the application of theory to various physically important systems. This book, along with the authors’ follow-up Quantum Mechanics II: Advanced Topics, provides students with a broad, up-to-date introduction to quantum mechanics. Print Versions of this book also include access to the ebook version.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend