© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

European Journal of Physics

The discretized Schrödinger equation for the finite square well and its relationship to solid-state physics

Timothy B. Boykin, Gerhard Klimeck

Published Jul 19, 2005• 15 citations

Abstract

The discretized Schrödinger equation is most often used to solve one-dimensional quantum mechanics problems numerically. While it has been recognized for some time that this equation is equivalent to a simple tight-binding model and that the discretization imposes an underlying bandstructure unlike free-space quantum mechanics on the problem, the physical implications of this equivalence largely have been unappreciated and the pedagogical advantages accruing from presenting the problem as one of solid-state physics (and not numerics) remain generally unexplored. This is especially true for the analytically solvable discretized finite square well presented here. There are profound differences in the physics of this model and its continuous-space counterpart which are direct consequences of the imposed bandstructure. For example, in the discrete model the number of bound states plus transmission resonances equals the number of atoms in the quantum well.

DOI: 10.1088/0143-0807/26/5/020

Related papers

On the connection between bound and scattering states of finite square-well potentials: a unified approach

Ian A. Morrison, Robert Ekey, Ivan Biaggio +1 more

Abstract We discuss a general description of the solutions to the 1D time-independent Schrödinger equation that does not a priori distinguish between scattering states and bound states and emphasizes and reinforces their relationship and connection to each other. This manuscript also introduces the concept of transfer matrices, which it presents as a logical extension of the traditional approach to evaluating 1D potentials. Using the transfer matrix method and a finite step approximation allows for a simple and straight-forward numerical solution of arbitrary 1D potentials. It also separates the process of solving the Schrödinger equation from selecting physically relevant solutions, which is a critical skill in quantum mechanics and is at the core of physics problems in general.

European Journal of Physics•Nov 20, 2020•0 citations

Some One-Dimensional Solutions to the Schrödinger Equation

Walter Wilcox

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.

CRC Press eBooks•Apr 19, 2016•0 citations

Some One-Dimensional Solutions to the Schrödinger Equation

Walter Wilcox

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.

CRC Press eBooks•Aug 23, 2019•0 citations

Quantum Theory of a Square Well Plus Delta Function Potential

W. Edward Gettys

The s-wave Schrödinger equation is solved for a square well potential with an attractive delta function at the well edge. This potential provides a number of soluble illustrations of the techniques of quantum mechanics. The bound state eigenvalue problem is solved as is the scattering problem for the cross section. The potential is capable of producing a sharp low energy resonance. The resonances are discussed using both R-matrix and S-matrix methods. Numerical examples are presented.

American Journal of Physics•May 1, 1973•2 citations

Applications of Schrödinger Equation: Potential (Quantum) Wells

Z. Ficek

The wave nature of particles plays an important role in their physical properties that, for example, particles confined into a small bounded area can have only particular discrete energies. This chapter explains how to solve the time-independent Schrodinger equation. It considers four cases of a one-dimensional motion of particles confined in potentials rapidly changing with x : infinite potential quantum well; a potential step; square-well potential; and tunneling through a potential barrier. The chapter helps to learn the characteristic properties of solutions to this equation. It shows the differences between the predictions of quantum mechanics and classical physics. Quantum tunneling is important in the understanding of a number of physical phenomena such as thermonuclear reactions and conduction in metals and semiconductors. The most advanced application of quantum tunneling is the scanning tunneling microscope. The details of the calculations are left for the readers as a tutorial problem.

•Mar 3, 2017•0 citations

Discretized Infinite Potential Well and Their Associated Coherent States

Dušan Popov

Since, unlike “traditional” physics, computational physics uses the discretization formalism, in this paper we have focused our attention on the discrete approach in order to solve the Schrödinger as well as the Bloch equations for a free particle and the “quantum gas” of free particles in an infinite quantum well with the finite length, respectively. By applying the so called continuous quantum mechanics limit we recover the corresponding results in continuous-variable quantum mechanics. For the infinite quantum well model we formulated the associated coherent states, which lead to the expression for the coherent qubits, in counterpart to the thermal qubits, obtained as a result of using the thermofield dynamics model. Thus, a “bridge” is created between the infinite quantum well model, coherent states and thermofield dynamics, which ultimately converge towards Quantum Computing.

Applied Mathematics and Statistics•Mar 20, 2026•0 citations

The discretized Schrödinger equation is most often used to solve one-dimensional quantum mechanics problems numerically. While it has been recognized for some time that this equation is equivalent to a simple tight-binding model and that the discretization imposes an underlying bandstructure unlike free-space quantum mechanics on the problem, the physical implications of this equivalence largely have been unappreciated and the pedagogical advantages accruing from presenting the problem as one of solid-state physics (and not numerics) remain generally unexplored. This is especially true for the analytically solvable discretized finite square well presented here. There are profound differences in the physics of this model and its continuous-space counterpart which are direct consequences of the imposed bandstructure. For example, in the discrete model the number of bound states plus transmission resonances equals the number of atoms in the quantum well.

Abstract We discuss a general description of the solutions to the 1D time-independent Schrödinger equation that does not a priori distinguish between scattering states and bound states and emphasizes and reinforces their relationship and connection to each other. This manuscript also introduces the concept of transfer matrices, which it presents as a logical extension of the traditional approach to evaluating 1D potentials. Using the transfer matrix method and a finite step approximation allows for a simple and straight-forward numerical solution of arbitrary 1D potentials. It also separates the process of solving the Schrödinger equation from selecting physically relevant solutions, which is a critical skill in quantum mechanics and is at the core of physics problems in general.

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.

The wave nature of particles plays an important role in their physical properties that, for example, particles confined into a small bounded area can have only particular discrete energies. This chapter explains how to solve the time-independent Schrodinger equation. It considers four cases of a one-dimensional motion of particles confined in potentials rapidly changing with x : infinite potential quantum well; a potential step; square-well potential; and tunneling through a potential barrier. The chapter helps to learn the characteristic properties of solutions to this equation. It shows the differences between the predictions of quantum mechanics and classical physics. Quantum tunneling is important in the understanding of a number of physical phenomena such as thermonuclear reactions and conduction in metals and semiconductors. The most advanced application of quantum tunneling is the scanning tunneling microscope. The details of the calculations are left for the readers as a tutorial problem.

Since, unlike “traditional” physics, computational physics uses the discretization formalism, in this paper we have focused our attention on the discrete approach in order to solve the Schrödinger as well as the Bloch equations for a free particle and the “quantum gas” of free particles in an infinite quantum well with the finite length, respectively. By applying the so called continuous quantum mechanics limit we recover the corresponding results in continuous-variable quantum mechanics. For the infinite quantum well model we formulated the associated coherent states, which lead to the expression for the coherent qubits, in counterpart to the thermal qubits, obtained as a result of using the thermofield dynamics model. Thus, a “bridge” is created between the infinite quantum well model, coherent states and thermofield dynamics, which ultimately converge towards Quantum Computing.