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American Journal of Physics

Ladder operators for some spherically symmetric potentials in quantum mechanics

J. D. Newmarch, RAYMUND M. GOLDING

Published Jun 1, 1978• 17 citations
Publisher

Abstract

The energy levels of the free field, Coulomb potential, and the three-dimensional harmonic oscillator are found using the Dirac operator formalism by the construction of suitable ladder operators. The degeneracy of each level is also discussed.
DOI
10.1119/1.11225

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Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product
Synopsis: Direct product spaces are encountered in the generalization to higher dimensions, and the connection to wave function separability is discussed. The spherical coordinate basis is introduced in preparation for discussing spherically symmetric potentials. Angular momentum operators and properties are examined, and a complete set of compatible observables is determined. The angular momentum eigenvalues and eigenstates are determined using ladder operators, and the connection to spherical harmonics is made. The stage is set for understanding the Schrödinger radial equation.
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.