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arXiv:0907.3343

Quantum circuits for solving one-dimensional Schrödinger equations

Nakao, K., Matsuyama, A.

Published Jul 20, 2009• 0 citations
arXivHTMLPDF

Abstract

We construct quantum circuits for solving one-dimensional Schrödinger equations. Simulations of three typical examples, i.e., harmonic oscillator, square-well and Coulomb potential, show that reasonable results can be obtained with eight qubits. Our simulations show that simple quantum circuits can solve the standard quantum mechanical problems.

DOI
10.48550/arxiv.0907.3343

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We construct quantum circuits for solving one-dimensional Schrödinger equations. Simulations of three typical examples, i.e., harmonic oscillator, square-well and Coulomb potential, show that reasonable results can be obtained with eight qubits. Our simulations show that simple quantum circuits can solve the standard quantum mechanical problems.
Synopsis: At this point we are equipped to solve some simple but fundamental one-dimensional problems with the Schrödinger equation. After setting the stage with an introduction, four quantum mechanical problems are specified and solved. Along the way, we encounter important concepts such as parity in the case of the infinite well and harmonic oscillator problems, transmission and reflection coefficients in the case of the finite barrier, and energy bands and gaps in a model of conduction in solids. Raising and lowering ladder operators are introduced in the harmonic oscillator and square well cases. I conclude with some qualitative comments regarding expected types of solutions in one-dimensional problems.
Synopsis: At this point we are equipped to solve some simple but fundamental one-dimensional problems with the Schrödinger equation. After setting the stage with an introduction, four quantum mechanical problems are specified and solved. Along the way, we encounter important concepts such as parity in the case of the infinite well and harmonic oscillator problems, transmission and reflection coefficients in the case of the finite barrier, and energy bands and gaps in a model of conduction in solids. Raising and lowering ladder operators are introduced in the harmonic oscillator and square well cases. I conclude with some qualitative comments regarding expected types of solutions in one-dimensional problems.
Courses on undergraduate quantum mechanics usually focus on solutions of the Schrödinger equation for several simple one-dimensional examples. When the notion of a Hilbert space is introduced, only academic examples are used, such as the matrix representation of Dirac’s raising and lowering operators or the angular momentum operators. We introduce some of the same one-dimensional examples as matrix diagonalization problems, with a basis that consists of the infinite set of square well eigenfunctions. Undergraduate students are well equipped to handle such problems in familiar contexts. We pay special attention to the one-dimensional harmonic oscillator. This paper should equip students to obtain the low lying bound states of any one-dimensional short range potential.
This chapter examines some explicitly one-dimensional systems, including the finite square well, the harmonic oscillator, and the application of harmonic oscillator theory to molecular vibrations. It analyzes the computer solutions of the Erwin Schrodinger equation. Although there are no real one-dimensional square wells in nature, many real systems have potentials that resemble the square well. Near the center of force the different wave functions must differ from one another or else they would not satisfy the Schrodinger equation for different values of energy. In contemporary research almost all the manipulation of the Schrodinger equation is done not analytically but rather by computer using numerical methods. A major advantage of the dimensionless form of the Schrodinger equation is that a single solution for one kind of system is trivially easy to extend to all similar systems. The dimensionless form of the Schrödinger equation can be solved numerically if it is approximated by a so-called difference equation.
The Schrödinger equation is a significant achievement on the development of quantum mechanics. By solving the Schrödinger equation, the fundamental behaviors and properties of a microscopic particle can be found in one- to three-dimensions. The article focuses on the derivations of quantum harmonic oscillators in one dimension by solving second order-differential equation. The wave functions, probability densities under different energy levels are presented. The results can be used to estimate different forms of continuous potential experienced by an oscillator. By calculating the uncertainty relation of the oscillator under one specific excited state, the general relation can be confirmed. Based on the output, by splitting the variables, the two-dimensional harmonic oscillator can also be derived. The degenerate energy levels are presented as . Numerical simulations are made to visualize those results and suggests the relation of energy levels and number of maxima of the probability densities.