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Journal of Physics B Atomic and Molecular Physics

Iterative boundary perturbation method for enclosed one-dimensional quantum systems

Jerzy Górecki, Bryson Brown

Published Nov 28, 1987• 24 citations
Publisher

Abstract

A modification of the boundary perturbation theory of Hull and Julius (1950) is proposed which lends itself to iterative solution. The method may be applied to both finite and infinite box potentials. For enclosed quantum systems described by one-dimensional Schrodinger equations it leads to the exact energy and wavefunction.
DOI
10.1088/0022-3700/20/22/008

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The authors propose a new method of solving the Schrodinger equation for a quantum system enclosed in a box with infinite potential walls. The method combines the variational technique with boundary perturbation theory and can be applied to a general non-separable case. When the Schrodinger equation for an enclosed system separates into an ordinary differential equation, the method gives the exact energy and wavefunction. As an application of the method the authors obtain the ground-state energy of the hydrogen atom placed off-centre in a spherical cavity. A generalisation of the variational boundary perturbation technique for finite potential walls is suggested.
Rayleigh–Schrödinger perturbation theory is used to find energy and wavefunction corrections for a particle confined in a box with infinite walls. The resulting perturbation equations have simple, analytic solutions. The procedure for normalization of the wavefunction corrections is demonstrated, and the perturbation energies and normalized wavefunctions are shown to approach the exact values. The method is applied to both the ground state and first excited state and can become a valuable component of a student’s introduction to perturbation analysis.
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.
The energy function E( lambda ) of a quantum system with Hamiltonian H( lambda )=H 0 + lambda H 1 is determined by an analytical iterative procedure which yields a rich variety of functional forms. The wavefunction iterates include the solutions of Rayleigh-Schrodinger perturbation theory as well as some additional functions, all of which may be obtained sequentially by standard analytical techniques. At each step, an arbitrary ( lambda -dependent) constant may be chosen freely so as to improve the results of low-order iterates, thereby enhancing the convergence of the procedure.