The ground-state energies and wave functions of a particle moving in one-dimensional potentials of the form V(x)=x2p, p=1,2,..., are obtained nearly exactly by iterative solution of the Schrödinger equation. These solutions are used as the ‘‘experimental benchmark’’ against which approximate solutions for the general case are tested. Both finite depth wells and infinitely deep wells are studied, as is the three-dimensional version (spherical symmetry, angular momentum quantum number l=0). The approach is aimed at student exploration of the physics of the family of potential wells, with emphasis upon connecting the familiar extreme cases, V(x)=x2 (simple harmonic oscillator) and the limiting case xk, k→∞ (square well).
The ground-state energies and wave functions of a particle moving in one-dimensional potentials of the form V(x)=x2p, p=1,2,..., are obtained nearly exactly by iterative solution of the Schrödinger equation. These solutions are used as the ‘‘experimental benchmark’’ against which approximate solutions for the general case are tested. Both finite depth wells and infinitely deep wells are studied, as is the three-dimensional version (spherical symmetry, angular momentum quantum number l=0). The approach is aimed at student exploration of the physics of the family of potential wells, with emphasis upon connecting the familiar extreme cases, V(x)=x2 (simple harmonic oscillator) and the limiting case xk, k→∞ (square well).
Abstract This article examines a particle in a one-dimensional potential well V a (x) ∝ x a , where a ≥ 2 is an even integer. The system is referred to as a harmonic oscillator when a = 2 and as a particle in a box when a → ∞. The finite difference method is applied to solve the Schrödinger equation, determining the ground-state energies and wave functions. Giving accurate description in the cases of a = 2 and a → ∞, the numerical method allows us to explore the system for intermediate values of a, revealing how the energies and wave functions evolve between these two limits. The study includes discussions on the virial theorem, the uncertainty principle, excited states, and the limitation of the numerical method.
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.
Abstract It is a common practice to approach quantum mechanics through the study of a few, exactly soluble examples using the Schrödinger equation in position-space. The number of potential energy functions for which such closed-form solutions are available is, however, quite small. Luckily, many of them actually correspond reasonably well to actual physical systems; examples include the infinite well as a model of a free particle ina “box “ (Chapters 5 and 7), the harmonic oscillator (Chapter 9), the rigid rotator (Chapter 16), and the Coulomb potential for the hydrogen atom (Chapter
By solving the Schrodinger equations in three-dimensional (3D) and two-dimensional (2D) finite depth center spherically symmetric square potential well, the eigenequations of bounded energy levels and their distributions are obtained, and the analytical solution for any bound state wave function is given. Both the eigenequations of the energy levels as well as the stationary state wave functions are closely related to the (spherical) Bessel functions. It is found that for the 3D case, there exists a bound solution when 2μa2V0/ħ2>π2/4 is present. While for the 2D case, there exists a bound state if 2μa2V0/ħ2>0.1993. Finally, numerical methods are used to analyze the characteristics of probability density distribution of different quantum states. It is found that in both three-dimensional and two-dimensional situations, the maximum number of probability density distributions inside the well corresponds exactly to the quantum number n. The larger n or the smaller α is, the higher the probability of discovering particles outside the well. In addition, the angular momentum quantum number l or m can characterize the degree to which particles deviate from the center or the uniform distribution.
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 3D infinite square well: quantum dots, wells, and wires In the preceding discussion of the Schrödinger Equation the particle of interest was assumed to be “moving in the x -direction.” Of course, it is not possible for a particle to be moving in one spatial direction only. If that were true, according to the HUP it could be anywhere in the y - and z -directions and therefore be undetectable with finite volume detectors. Now, we consider the more realistic case of motion in all three spatial directions. For this purpose, we start with the 3D infinite square well. This model provides a useful conceptual framework for understanding a burgeoning “nano-industry,” namely the production and use of “quantum dots.” The 3D infinite square well is a “rigid rectangular solid box.” The potential energy of a particle trapped inside is given by U(x,y,z) = 0 , if 0 < x < L1, 0 < y < L2, 0 < z < L3, and ∞, otherwise. The particle’s momentum now has three components and its kinetic energy is K = (px2 + py2 + pz2) /2m.