This chapter focuses on widely used approximate method: the numerical integration of the Schrodinger equation, restricting ourselves to one dimension for simplicity. There is a plethora of approaches that yield highly accurate numerical eigenvalues and eigenfunctions. In quantum mechanics one sometimes seeks after bound stationary states that satisfy given physical boundary conditions. Boundary conditions at infinity play a relevant role in quantum mechanics, in which case one looks for bound states that vanish asymptotically at the end points of coordinate interval. Solving the propagation matrix equation exactly for every coordinate pair in the physical interval is as difficult as solving the original differential equation. However, sometimes the former is preferable to develop approximate numerical algorithms for the accurate calculation of eigenvalues and eigenfunctions. The perturbation method based on the Taylor expansion only requires derivatives of the potential energy function, which one obtains easily in most cases and appears to be preferable for the systematic improvement of the approximate propagation matrix.
This chapter focuses on widely used approximate method: the numerical integration of the Schrodinger equation, restricting ourselves to one dimension for simplicity. There is a plethora of approaches that yield highly accurate numerical eigenvalues and eigenfunctions. In quantum mechanics one sometimes seeks after bound stationary states that satisfy given physical boundary conditions. Boundary conditions at infinity play a relevant role in quantum mechanics, in which case one looks for bound states that vanish asymptotically at the end points of coordinate interval. Solving the propagation matrix equation exactly for every coordinate pair in the physical interval is as difficult as solving the original differential equation. However, sometimes the former is preferable to develop approximate numerical algorithms for the accurate calculation of eigenvalues and eigenfunctions. The perturbation method based on the Taylor expansion only requires derivatives of the potential energy function, which one obtains easily in most cases and appears to be preferable for the systematic improvement of the approximate propagation matrix.
As in classical mechanics the number of exactly soluble quantum mechanical problems is very limited and in most cases of practical interest numerical techniques and approximation methods are unavoidable. The availability of fast computers has turned quantum computation into a major industry. This chapter looks at the several approximation techniques. In seeking to solve the Schrodinger equation with the perturbed Hamiltonian, it is convenient to work with series expansions by introducing a dimensionless parameter λ. In many cases the system of interest differs from an exactly soluble system by only a small disturbance, enabling an approximation to be made by expanding in powers of a smallness parameter. The Variational Method provides a very simple method of approximating energy eigenvalues. It is most useful for computing ground state energies and its success depends on our being able to make a reasonable guess for the form of the ground state wave function.
In the previous chapter, the postulates helped us to use the wave function to find the expected results of experimental measurements by letting a dynamical quantity F (x, p, t) → F (xˆ, pˆ, t) and integrating over space. In the examples, however, the wave functions were typically given and not derived from first principles. We now endeavor to find the appropriate wave functions for a variety of physical systems in one dimension by solving the Schro¨dinger equation where for time-independent cases, the total energy is an eigenvalue. In subsequent chapters, we will extend the method to three dimensions and develop the appropriate operators, eigenvalues, and eigenfunctions for angular momentum. We will also develop methods for examining time-dependent systems. The latter chapters address special systems of interest that may be referred to as applications of quantum mechanics.
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. 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.
The objective of this study is to numerically solve and apply two approximate methods to investigate the Time Independent Schrödinger Equation (TISE) in one dimension for an infinite potential square well. These two numerical methods are the Numerov Method (NM) and Matrix Method (MM). As a simulation tool, MATLAB, a high-level programming language and an efficient simulation tool, is used for modeling and solving TISE in one dimension. Exact analytical solutions for these potential functions are obtained and compared with numerical solutions and computational techniques. The energy eigenvalues and Eigen functions of a particle (such as an electron) restricted to move inside this potential are discussed as an illustration. The numerically calculated energies of several states with increasing numbers of points were obtained from both methods and compared with the simulation results of the exact solution. As an exemplary case, the first five wave functions are accurately determined numerically where the discreteness is found since the wave function vanishes at the boundary. The obtained results show very good agreement and the similarity is clearly confirmed between the three cases. This agreement confirms that this approach was highly accurate and efficient. The accuracy and the convergence of the numerical obtained results were easily checked. The stability of these methods is due to the fact that there are no restrictions on the time steps to be taken. The merits of these numerical methods are to avoid a huge expense in time when solving Schrödinger equation.
Throughout our discussion of the general principles of quantum mechanics we have emphasized the importance of the eigenvalues and eigenfunctions of the operators representing physical quantities. However, we have seen that solving the basic eigenvalue equations determining these is often not straightforward. In the early chapters discussing the energy eigenvalue equation (or time-independent Schro¨dinger equation) for example, we found that this could often not be solved exactly and that, even when a solution was possible, it frequently required considerable mathematical analysis. Because of this, methods have been developed to obtain approximate solutions to eigenvalue equations. One of the most important of such techniques is known as time-independent perturbation theory and will be the first method to be discussed in the present chapter. Another, known as the variational principle, will be discussed later. We shall confine our discussion to the timeindependent Schro¨dinger equation, but generalization to other eigenvalue equations is quite straightforward. We shall mainly use the wave function representation, but we shall restate some of the important results in Dirac notation so that they can be readily transformed to a matrix representation if desired.