A free particle is shown to possess any non-negative energy value. Its position is described by a localized wave function called a wave packet. The solution to the particle in infinite potential well problem displays how the concept of energy quantization appears in quantum mechanics. Like the particle in an infinite well, the particle in a finite well shows discretization of energy. But unlike the particle in an infinite well, the particle in a finite well exhibits a non-zero probability of being found outside the well leading to the concept of quantum-mechanical tunneling.
A free particle is shown to possess any non-negative energy value. Its position is described by a localized wave function called a wave packet. The solution to the particle in infinite potential well problem displays how the concept of energy quantization appears in quantum mechanics. Like the particle in an infinite well, the particle in a finite well shows discretization of energy. But unlike the particle in an infinite well, the particle in a finite well exhibits a non-zero probability of being found outside the well leading to the concept of quantum-mechanical tunneling.
This paper examines the particle-in-a-box model as a simple but powerful example that highlights the fundamental differences between classical and quantum physics. In classical mechanics, a particle confined between two rigid walls moves with constant velocity and has a uniform probability of being found anywhere in the box; its energy can take any continuous value. In contrast, solving the Schrödinger equation for an infinite potential well yields standing-wave solutions whose wavelengths must satisfy fixed boundary conditions. These constraints produce discrete, quantized energy levels and non-uniform probability distributions, features with no classical analog. Through analytical derivations and numerical simulations using Python, we visualize wave functions, probability densities, and the statistical convergence of random samples toward the theoretical quantum distributions. Comparisons with the classical uniform distribution emphasize how quantum mechanics replaces certainty with probability and continuity with quantization. Finally, we connect the model to real physical systems—such as electrons confined in atoms—to show how this idealized system provides essential insight into the structure and stability of matter.
The laws of quantum mechanics differ significantly from the laws of classical mechanics. The mathematical apparatus of quantum mechanics is based on the amazing Schrödinger equation. This is a specific partial differential equation with respect to a complex-valued function called the wave function. This characteristic allows, in particular, estimating the probability of finding a quantum mechanical particle in a given volume. We establish the Schrödinger equation for a free particle and for a particle moving in an external field. Then the problem of passing a potential barrier is investigated. It is shown that, in contrast to the classical case, even a relatively slow quantum mechanical particle is capable of overcoming a sufficiently high potential barrier. The Appendix considers the concept of a normalized wave function and analyzes the movement of a particle in a potential well with infinite walls.
Abstract This chapter continues with a study of the time independent Schrödinger equation and seeks to contrast the quantum behavior of a free particle with that of a particle localized in a potential quantum well. A free particle can exist over all space or can be localized in a wave package. The wave packet is a coherent superposition of the plane waves that make up the wave function that localizes the particle because of constructive and destructive interference. The wave packet spreads out in time because the waves leading to constructive interference get out of phase. In Chapter 2, the particle was localized by a quadratic potential energy. Here, the potentials are described as piecewise constant. The approach is based on assuming a one-dimensional space, x, which is relevant to many problems in the laboratory. The solution is easily generalized to higher dimensions (x-y or x-y-z), but the physics remains the same. The objective is to understand the shape of the eigenfunctions in space and to be able to relate this to the probability density of locating the particle, as well as understanding the relevance of these systems to today’s technology.
Abstract Discusses with some rigour the properties of electrons, based on the Schrodinger equation. Introduces the concepts of wave function, quantum-mechanical operators, and wave packets. Examples cover the electron meeting an infinitely long potential barrier and the passage of electrons through a finite barrier (which leads to the phenomenon of tunnelling).The electron in a potential well is also discussed, solving the problem both for a finite and for an infinite well, and finding the permissible energy levels. The chapter is concluded with the philosophical implications that arise from the quantum-mechanical approach. Two limericks relevant to the subject are quoted.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend
The “particle-in-a-box” problem is investigated for a relativistic particle obeying the Klein–Gordon equation. To find the bound states, the standard methods known from elementary non-relativistic quantum mechanics can only be employed for “shallow” wells. For deeper wells, when the confining potentials become supercritical, we show that a method based on a scattering expansion accounts for Klein tunneling (undamped propagation outside the well) and the Klein paradox (charge density increase inside the well). We will see that in the infinite well limit, the wave function outside the well vanishes, and Klein tunneling is suppressed: Quantization is, thus, recovered, similar to the non-relativistic particle in a box. In addition, we show how wave packets can be constructed semi-analytically from the scattering expansion, accounting for the dynamics of Klein tunneling in a physically intuitive way.