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On the quantum mechanical problem of a particle in two potential minima

David Southard Carter

Published Jan 1, 2012• 0 citations
Publisher

Abstract

The problem of a particle in two adjacent one-dimensional rectangular, potential "boxes" is an exactly soluble representative of a class of two-minima problems of considerable physical interest which have not been solved exactly. It therefore affords a valuable opportunity for a critical examination of the extent of applicability of perturbation theory methods to such problems. An exact implicit solution of the problem is obtained, and is reduced to explicit approximate form in two important special cases. These approximations are reproduced by perturbation theory methods, and their ranges of validity are demonstrated by comparison with the exact solution. The application of the model to a physical system is demonstrated by using the identical two-box problem as a basis for calculation of some constants of the ammonia molecule.

DOI
10.14288/1.0085420

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The problem of a particle in two adjacent one-dimensional rectangular, potential "boxes" is an exactly soluble representative of a class of two-minima problems of considerable physical interest which have not been solved exactly. It therefore affords a valuable opportunity for a critical examination of the extent of applicability of perturbation theory methods to such problems. An exact implicit solution of the problem is obtained, and is reduced to explicit approximate form in two important special cases. These approximations are reproduced by perturbation theory methods, and their ranges of validity are demonstrated by comparison with the exact solution. The application of the model to a physical system is demonstrated by using the identical two-box problem as a basis for calculation of some constants of the ammonia molecule.
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.
Abstract One of the great achievements of the Schrödinger wave-mechanics is the elegance of its perturbation theory, which has brought many problems, formerly considered intractable, within the range of a highly-developed mathematical technique. It is not necessary at this stage to review the numerous applications which have been made of this perturbation theory or to dwell upon its many advantages. The important advance towards an understanding of chemical forces which it has made possible is in itself a considerable achievement. There are, however, certain disadvantages in the perturbation theory in its present form, which limit the extent of its applications to complex problems of atomic and molecular structure. If the interaction of atoms, for instance, is to be calculated, as it most desirable, improved methods will have to be found.
The solution of most problems associated with electron quantum states in physical systems and structures (atoms, molecules, quantum nanostructure objects, and crystals) is hard to find because of the mathematical difficulties of getting exact solutions of the Schrödinger equation. Therefore, approximate methods of solving such problems are of special interest. We will consider some of these methods, such as the adiabatic approximation now and later the effective-mass method , using real physical systems as examples. In this chapter we will consider several widely used approximation methods for finding the wavefunctions and energies of quantum states as well as the probabilities of transitions between quantum states. First of all, we will consider stationary and non-stationary perturbation theories. What is common to these two theories is that it is assumed that the perturbation is weak and that it changes negligibly the state of the unperturbed system. Stationary perturbation theory is used for the approximate description of a system's behavior if the Hamiltonian of the quantum system being considered does not directly depend on time. In the opposite case, non-stationary theory is used. Then, we will briefly consider the quasiclassical approximation, which is used for the problems of quantum mechanics which are close to analogous problems of classical mechanics. Stationary perturbation theory for a system with non-degenerate states This theory is used for the approximate calculation of the energy levels and the wavefunctions of stationary states of systems that are subjected to the influence of small perturbations.