This project uses the procedure of root-finding to resolve the eigenvalue problem of a rectangular quantum well. The procedure is applied to determine the first two to three energy levels of a simplified model of a hydrogen atom, represented by the rectangular quantum well. The project also explores the eigenvalue problem within various physics fields. While the project involves simple mathematical operations, it is rooted in complex physical concepts like quantum mechanics, often unfamiliar to first-year students. The fundamentals of quantum mechanics are introduced, providing enough understanding for successful project execution. The project initially focuses on the central object, the quantum state, and its probabilistic nature in quantum mechanics. The Schrödinger equation, an eigenvalue problem, is used to find state functions. This project explores eigenenergies and eigenfunctions within a rectangular finite quantum well, treating the well as a simplistic 1D model of the hydrogen atom.
This project uses the procedure of root-finding to resolve the eigenvalue problem of a rectangular quantum well. The procedure is applied to determine the first two to three energy levels of a simplified model of a hydrogen atom, represented by the rectangular quantum well. The project also explores the eigenvalue problem within various physics fields. While the project involves simple mathematical operations, it is rooted in complex physical concepts like quantum mechanics, often unfamiliar to first-year students. The fundamentals of quantum mechanics are introduced, providing enough understanding for successful project execution. The project initially focuses on the central object, the quantum state, and its probabilistic nature in quantum mechanics. The Schrödinger equation, an eigenvalue problem, is used to find state functions. This project explores eigenenergies and eigenfunctions within a rectangular finite quantum well, treating the well as a simplistic 1D model of the hydrogen atom.
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 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
As the first stage in a project aimed at finding exact bound-state solutions for the few-particle Schrodinger equation, some restricted two- and threebody systems are treated by Rayleigh-Schrddinger perturbation theory. Formal solutions for the perturbation equations are obtained by using standard techniques for solving differential equations. Particular attention is given to justifying the various boundary conditions, since the same conditions apply to formal solutions of the eigenvalue equations. It is shown that boundary conditions designed to preserve the Hermitian character of the Hamiltonian are sufficient to determine all parameters in the formal solutions. The procedure is demonstrated by application to a hydrogen atom perturbed by stationary point charges. Results are given for the ground and some excited states.
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 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