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arXiv:2112.03693

On the quantum-mechanical singular harmonic oscillator

Francisco Marcelo Fernández

Published Dec 5, 2021• 1 citation
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Abstract

We obtain the eigenvalues and eigenfunctions of the singular harmonic oscillator $V(x)=α/(2x^2)+x^2/2$ by means of the simple and straightforward Frobenius (power-series) method. From the behaviour of the eigenfunctions at origin we derive two branches for the eigenvalues for negative values of $α$. We discuss the well known fact that there are square-integrable solutions only for some allowed discrete values of the energy.

DOI
10.48550/arxiv.2112.03693

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We obtain the eigenvalues and eigenfunctions of the singular harmonic oscillator V(x)=α/(2x2)+x2/2V(x)=α/(2x^2)+x^2/2V(x)=α/(2x2)+x2/2 by means of the simple and straightforward Frobenius (power-series) method. From the behaviour of the eigenfunctions at origin we derive two branches for the eigenvalues for negative values of ααα. We discuss the well known fact that there are square-integrable solutions only for some allowed discrete values of the energy.
Quantum mechanics is one of the branches of physics. In this mechanics, physical problems are solved by algebraic and analytic methods. By applying a simple procedure we can find the general solutions of Schrodinger’s time - independent wave equation of one dimensional quantum mechanical harmonic oscillator without making any approximation. In this paper, we will discuss the Eigenenergy values and Eigenfunctions of one of the most important physical models of quantum mechanics, namely the one-dimensional Quantum mechanical Harmonic Oscillator by modifying the Hermite differential equation.
It is shown that the eigenvalue problem for the Hamiltonians of the standard form, H=p2/(2m)+V(x)H=p^2/(2m)+V(x)H=p2/(2m)+V(x), is equivalent to the classical dynamical equation for certain harmonic oscillators with time-dependent frequency. This is another indication of the central role played by time-dependent harmonic oscillators in quantum mechanics. The utility of the known results for eigenvalue problem in the solution of the dynamical equations of a class of time-dependent harmonic oscillators is also pointed out.
Abstract: In this work, the problem of the quantum harmonic oscillator with a delta-derivative potential λδ^' (x), where λ is a coupling constant, is solved using the Green’s function technique. A transcendental equation that governs the energy eigenvalues of the problems for each coupling constant is obtained. The eigenfunctions and their first derivative are proved to be discontinuous at the origin. The values of the discontinuity jumps are found to agree with the requirements of having a self-adjoint extension Hamiltonian. In the large coupling limit, the even energy eigenvalues and eigenfunctions for the quantum harmonic oscillator are annihilated, and only the odd parts survive. The dependence of the energy eigenvalues and eigenfunctions on the sign of λ was made clear. A mapping between the sign of λ and the positions of the particle was used to explain the discontinuity of the solution. In the large-λ regime, an educated guess for the wave functions was proposed. The proposed solutions led to the correct energy eigenvalues and obeyed the required conditions to have a self-adjoint extension Hamiltonian.
The classical solution is used as a variable for the one-dimensional quantum-mechanical oscillator, and this leads to a separable differential equation solvable in closed form which yields a set of new quantum mechanical solutions, not energy eigenfunctions, that have the property of being localized. These new solutions are shown to be eigenfunctions of a time-independent position operator. The operator algebra of that operator and an analogous momentum operator is given, as well as the relationship of the eigenfunctions to the energy eigenfunction set. The new eigenfunctions, which are not square integrable but possess a δ-function normalization, themselves represent wavepackets that at times corresponding to when the classical position is at an extremum collapse to Dirac δ functions.
This study presents a complete algebraic formulation of the quantum harmonic oscillator, demonstrating that the Hermite polynomials, H n ( x ) , emerge as the fundamental solution when the power-series method is applied to the Schrödinger equation. The physical requirement of square integrability enforces the truncation of the series, leading precisely to these polynomials, which constitute the polynomial part of the energy eigenfunctions, ψ n ( x )= N n H n ( αx ) e − α 2 x 2 /2 , where n is the quantum number. This quantum number directly determines the quantized energy levels through E n =ℏω( n+ 1 2 ) . The analysis based on creation and annihilation operators further reveals that the action of the creation operator on the ground state generates the excited states along with their corresponding Hermite functions. The orthogonality of these polynomials ensures the orthonormality of the complete set of eigenfunctions. Additionally, the geometric Berry phase emerges as a natural extension when the system undergoes adiabatic cyclic changes, revealing a deeper topological structure that complements the conventional algebraic and analytic descriptions. Therefore, Hermite polynomials encapsulate both the mathematical structure and the physical essence of quantization in this fundamental system.