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One-dimensional harmonic oscillator hamiltonian and wave function with ionization energy

Andrew Das Arulsamy, Kurunathan Ratnavelu

Published Feb 27, 2007• 0 citations
OpenAlex

Abstract

Abstract. 1-dimensional hamiltonian with ionization energy (ξ) is shown to be exactly the same with the total energy from the standard harmonic oscillator hamiltonian, with the harmonic oscillator potential, mω 2 x 2 /2. Commutation relations for the ladder operators, energy at the n th state and the expectation value for the potential are analyzed and shown to be consistent with the principle of ξ. We also derive 1-dimensional wave functions for Dirac delta (−αδ(x)) potential and with V (x) = 0. Formal justifications are given as to why the total energy in ξ based Fermi-Dirac statistics is exactly the same with the standard Fermi-Dirac statistics. Energy levels and Bohr radius of a Hydrogen atom are also shown to be consistent with the ionization energy concept. This concept is also applied in a free Mn atom that represent a free many-electron atom. 1.

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Abstract. 1-dimensional hamiltonian with ionization energy (ξ) is shown to be exactly the same with the total energy from the standard harmonic oscillator hamiltonian, with the harmonic oscillator potential, mω 2 x 2 /2. Commutation relations for the ladder operators, energy at the n th state and the expectation value for the potential are analyzed and shown to be consistent with the principle of ξ. We also derive 1-dimensional wave functions for Dirac delta (−αδ(x)) potential and with V (x) = 0. Formal justifications are given as to why the total energy in ξ based Fermi-Dirac statistics is exactly the same with the standard Fermi-Dirac statistics. Energy levels and Bohr radius of a Hydrogen atom are also shown to be consistent with the ionization energy concept. This concept is also applied in a free Mn atom that represent a free many-electron atom. 1.
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
A quantum state corresponds to a specific wave function. We adopt a new mathematical method [1] to improve Dirac's ladder operator method. A set of orthonormal wave functions will be used to associate the operator with the square matrix corresponding to it. These allow us to determine the matrix elements by using the operator relations without having to know the specific wave functions. As a result, we can get the direct results of matrix mechanics and wave mechanics on one-dimensional Harmonic oscillator and their descriptions will be also unified.
In this work, we study the Dirac equation and Dirac harmonic oscillator in one-dimensional via the Dunkl algebra. By using Dunkl derivative, we solve the momentum operator and Hamiltonian that include the reflection symmetry. Based on the concept of the Wigner–Dunkl algebra and the functional analysis method, we have obtained the energy eigenvalue equation and the corresponding wave function for Dirac harmonic oscillator and Dirac equation, respectively. It is shown all results in the limit state satisfied what we had expected before.
A field theoretical formulation of the rather easy problem of N identical, one-dimensional bosons, interacting pairwise via harmonic oscillator potentials of equal force constants is given, in a way similar to the way the nonlinear Schrodinger model provides a field theoretical formulation of a one-dimensional, delta -function Bose gas. The nonlinear integro-differential equation obtained for the complex scalar field of the model is treated classically, and solved in the semiclassical limit of a large number of particles. Working in the centre of mass frame of the field, a sequence of operators that would raise or lower the energy of a state by definite amounts without changing its particle number are constructed from certain bilinear expressions in the field. These operators have, in the limit of a large number of particles, commutation relations similar to those of the Fourier modes of a compact free field, except that the first mode is missing as a result of the fact that the field in the centre of mass frame is subject to the obvious constraint that its centre of mass always coincides with the coordinate origin. The quantum states can be labelled by the total centre of mass momentum, and the quasi-particle numbers in each of the independent oscillator modes. The energy spectrum of the system is calculated, giving the same results as the exact, quantum-mechanical N-body analysis.
Variational calculations play an important role in quantum mechanics, particularly in determining the ground-state energy of physical systems. Didactic examples in which one can see how the exact energy and wave function are approached by using a family of trial functions are not very numerous. An example was given in a recent book [M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, New York, 1969)] which discusses the ground state of the hydrogen atom using as trial wave function a linear combination of harmonic-oscillator states. In the present paper we carry out a similar analysis for the ground state of a particle of mass m in a three-dimensional square-well potential. We discuss not only the approach to the exact energy when we vary the frequency and the number of oscillator states but also analyze the overlap of the exact and variational wave functions and compare the exact and approximate form factors.