Synopsis: It is rare that exact solutions for physical quantum systems are available, and likely that one or more types of perturbation theories need to be applied. ree time-independent, or static, methods are discussed. Using exact energy shift formulas, a result correct to second order for the perturbed system energy is derived and applied to a two-state system and a harmonic oscillator example. e WKB semiclassical approximation is motivated, and applications to barrier penetration and various types of bound-state problems are made. e third class of methods gives upper limit estimates of system ground-state properties based on energy functional variations.
Synopsis: It is rare that exact solutions for physical quantum systems are available, and likely that one or more types of perturbation theories need to be applied. ree time-independent, or static, methods are discussed. Using exact energy shift formulas, a result correct to second order for the perturbed system energy is derived and applied to a two-state system and a harmonic oscillator example. e WKB semiclassical approximation is motivated, and applications to barrier penetration and various types of bound-state problems are made. e third class of methods gives upper limit estimates of system ground-state properties based on energy functional variations.
Synopsis: It is rare that exact solutions for physical quantum systems are available, and likely that one or more types of perturbation theories need to be applied. Three time-independent, or static, methods are discussed. Using exact energy shift formulas, a result correct to second order for the perturbed system energy is derived and applied to a two-state system and a harmonic oscillator example. The WKB semiclassical approximation is motivated, and applications to barrier penetration and various types of bound-state problems are made. The third class of methods gives upper limit estimates of system ground-state properties based on energy functional variations.
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.
In the previous chapters, we have solved a series of problems that all have an exact solution. However, there are only a select few problems in quantum mechanics that provide exact analytical solutions. Those problems that can be solved exactly are the free particle, the one-dimensional barrier potential, the finite and the infinite square wells, the infinite triangular well, the harmonic oscillator, and the hydrogen atom, including both its nonrelativistic and relativistic solutions. However, there are many problems different from the above-mentioned few as well as problems that cannot be approximated by these solutions. What happens then if we want to solve a different problem, such as the helium atom or the hydrogen atom in an applied electric field? To solve these problems it is necessary to use approximation techniques. In this chapter, we consider some of these techniques, the most important being the time-independent and the time-dependent perturbation theories. We begin with a discussion of time-independent disturbances in a nondegenerate system. Time-independent Perturbation Theory: The Nondegenerate Case We seek to determine the solution to a problem in which a small disturbance is applied to an otherwise determinable system. Perturbation theory is used most often in cases for which the solution to the undisturbed (unperturbed) system is known and the perturbation is small. For example, perturbation theory is useful in solving for the energy levels in a hydrogen atom in the presence of a small external electric field.
Perturbation theory for non-degenerate and degenerate systems, variational methods, and WKB approximation are covered in this chapter. First, second, and kth order corrections to energy and wave function ket of non-degenerate states, energy eigenvalues, and wave functions kets of anharmonic oscillator, first-order perturbation theory for degenerate states and linear stark effect are written under perturbation theory. Second part of the chapter includes the variational methods for computation of the energy for ground state of helium atom, Rayleigh-Ritz variational method and its application to hydrogen molecule ion, calculation of energy of excited states, and its application to 1D harmonic oscillator. The topics which are covered under WKB approximation are the classical and non-classical regions of particle motion, alternative derivation of WKB formulae, connecting formulae, and quantum condition for bound state. The kth order corrections to energy and wave function kets for non-degenerate perturbation theory, treatment of WKB approximation from two different approaches and the detailed derivation of connecting formulae, application of variational method to compute energy for excited states are topics which are not generally found in existing text books on quantum mechanics. A good number of solved examples illustrating the importance of approximation methods in contemporary fields are added at the end of chapter.
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.
With a limit on the small number of problems that admit exact solutions, we turn in this chapter to approximation methods that can be used for bound state problems. The most commonly used method is the perturbation theory for which one use a small parameter to form the basis of a systematic expansion, or a variational method in which one hopes to set the best limit on the energy of a system. We will consider both of these approximation methods with applications to problems in atomic and molecular physics.