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CRC Press eBooks

Perturbation Theory

Shabnam Siddiqui

Published Oct 10, 2018• 0 citations
Publisher

Abstract

In the previous chapters, we have shown how to solve the Schrodinger equation for several systems as a particle in an infinite and finite potential well, scattering, harmonic oscillator, and hydrogen atom. For all these systems, we were able to solve Schrodinger's equation exactly. However, there are complicated potentials for which Schrodinger's equation cannot be solved exactly. For such situations, we resort to approximation methods for finding approximate solutions to the Schrodinger equation. In this chapter, we discuss several approximate methods for solving Schrodinger's equation.

DOI
10.1201/b22074-6

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In the previous chapters, we have shown how to solve the Schrodinger equation for several systems as a particle in an infinite and finite potential well, scattering, harmonic oscillator, and hydrogen atom. For all these systems, we were able to solve Schrodinger's equation exactly. However, there are complicated potentials for which Schrodinger's equation cannot be solved exactly. For such situations, we resort to approximation methods for finding approximate solutions to the Schrodinger equation. In this chapter, we discuss several approximate methods for solving Schrodinger's equation.
Abstract It is a common practice to approach quantum mechanics through the study of a few, exactly soluble examples using the Schrödinger equation in position-space. The number of potential energy functions for which such closed-form solutions are available is, however, quite small. Luckily, many of them actually correspond reasonably well to actual physical systems; examples include the infinite well as a model of a free particle ina “box “ (Chapters 5 and 7), the harmonic oscillator (Chapter 9), the rigid rotator (Chapter 16), and the Coulomb potential for the hydrogen atom (Chapter
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
Many physical systems cannot be solved exactly. In this case, we look for approximate methods to gain insight into their properties. Perturbation theory is one such method. When the Hamiltonian of the system doesn't change over time, one approximate method is the Time Independent Perturbation Theory, which is the subject of Chapter 10 . The infinite square well, the harmonic oscillator, and the hydrogen atom are among the few physical systems for which the Schrodinger equation can be solved exactly. For this reason, they are described in detail in most of the introductory quantum mechanics textbooks. However, most physical systems are not exactly solvable, i.e., they are described by the Schrodinger equation, whose solutions cannot be expressed in terms of commonly used functions. In order to study such systems, we try to decompose the corresponding Hamiltonian into a solvable Hamiltonian plus a small perturbation. For such cases, we say that the Hamiltonian is “close enough” to an exactly solvable Hamiltonian to warrant using perturbative methods to approximate its eigenenergies and eigenfunctions.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation can be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges. The splitting of an energy level due to the interaction of the nuclear magnetic dipole and the magnetic field Be produced by the electron orbit current is called the fine structure. The term hyperfine structure is used to describe extremely small splitting of spectral lines.
Abstract This review presents time-independent perturbative methods for solving the one-dimensional Schrödinger equation, highlighting representative cases that reveal key aspects of the theory. The focus is on their relevance to quantum computing applications, particularly in systems with finite-dimensional state spaces. The main link between these methods and quantum processing emerges when we consider systems with only a finite number of well-defined energy levels, e.g., two levels corresponding to the ‘zero’ and ‘one’ states of a quantum bit (qubit), for which the Schrödinger equation describes a quantum algorithm corresponding to the unitary evolution of the qubits. Perturbative techniques can then be applied to this evolution to gain deeper insight into the algorithm or to simplify it in a controlled manner, as described in the last sections of this review. We initially discuss representative applications using benchmark cases that exemplify the essential features of perturbation theory while avoiding unnecessary technical complications. The methods presented involve direct approaches to solutions for quantum systems with small potential perturbations, with consideration of indirect and approximation techniques. Key cases examined include the particle in a potential well, the harmonic oscillator, and systems with polynomial or Dirac delta distribution perturbations, each chosen to highlight unique aspects of perturbative analysis. Starting with an overview of the unperturbed Schrödinger equation, we derive canonical solutions and then use perturbative techniques to approximate energy shifts and wavefunction corrections in both non-degenerate and degenerate settings. By combining theoretical discussion with case studies, this review underscores how perturbative methods are applied across a range of elementary quantum mechanical problems especially for applications in the qubit model, which, in many cases of interest, can be reduced to the explicit calculations shown here.
So far attention has been limited to exactly solvable quantum systems such as the harmonic oscillator and the hydrogen atom. While this has provided important insights into the nature of the atomic world, there are many physical systems that cannot be solved exactly. This chapter will present some general techniques that can be employed to obtain the approximate properties of such systems. These are based on using the properties of systems that are assumed to be close in nature to the system under consideration.