Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1710.00509 · Journal of Physics Communications

Time-independent Green’s function of a quantum simple harmonic oscillator system and solutions with additional generic delta-function potentials

Chun-Khiang Chua, Yu-Tsai Liu, Gwo-Guang Wong

Published Dec 4, 2017• 16 citations
arXivHTMLPDF

Abstract

The one-dimensional time-independent Green's function G 0 of a quantum simple harmonic oscillator (SHO) system ( ) can be obtained by solving the equation directly. It has a compact expression, which gives correct eigenvalues and eigenfunctions easily. The Green's function G with an additional delta-function potential can be obtained readily. The same technics of solving the Green's function G 0 can be used to solve the eigenvalue problem of the SHO with an generic delta-function potential at an arbitrary site, i.e. . The Wronskians play an important and interesting role in the above studies. Furthermore, the approach can be easily generalized to solve the quantum system of a SHO with two or more generic delta-function potentials. We give the solutions of the case with two additional delta-functions for illustration.

DOI
10.1088/2399-6528/aa9eeb

Related papers

The Quantum Harmonic Oscillator with λδ' (x) Potential

M A Dalabeeh

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.

Jordan Journal of Physics•Aug 31, 2025•0 citations

Generation of Time-Independent and Time-Dependent Harmonic Oscillator-Like Potentials Using Supersymmetric Quantum Mechanics

Timothy B. Huber

The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.

IU Journal of Undergraduate Research•Dec 16, 2018•0 citations

A systematic approach for obtaining the Green functions of time-dependent Schrödinger equations by Fourier transform

Gin-yih Tsaur, Jyhpyng Wang

Abstract The Green function method is a powerful technique for solving the initial value problem in quantum mechanics. Once the Green function is solved the whole wavefunction evolution is represented in a concise form and can be evaluated conveniently by numerical integration. We present a method for constructing the Green functions systematically which is different from the conventional methods of eigenfunction expansion or path integration. By using variable changing, function substitution, and Fourier transforms, the time dependent Schrödinger equations can be simplified and the solutions for the simplified equations can be easily derived. We then obtain the Green functions for the original equations by the reverse transforms. The method is demonstrated for the linear potential, the harmonic oscillator, the centrifugal potential, and the centripetal barrier oscillator, where the Green function for the centripetal barrier oscillator has not been solved previously by conventional methods. The method and examples illustrated in this paper can be utilised to strengthen undergraduate courses on quantum mechanics and/or partial differential equation.

European Journal of Physics•May 10, 2016•1 citation

Eigenenergy values and Eigenfunctions of one-dimensional quantum mechanical harmonic oscillator

Rohit Gupta, Dinesh Verma

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.

•Mar 12, 2023•1 citation

Green’s Function of the Schrödinger Equation in the Potential Quantization Method

B. Ya. Balagurov

Abstract The problem for determining Green’s function G(r, r') for the time-independent Schrödinger equation is considered using the potential quantization method. The expression for G(r, r') is obtained in terms of the eigenfunctions and eigenvalues of this method. The equivalence of the used and traditional methods for solving the given problem has been proven.

Journal of Experimental and Theoretical Physics•Mar 1, 2019•0 citations

Operator and Factorization Methods for the Schrödinger Equation

Abstract The special and recurring role that the simple harmonic oscillator (SHO) plays in quantum mechanics can be attributed both to its physical relevance and its simple solutions. The fact that the ground state solution is the minimum uncertainty wave packet (Section 12.4) and the highly constrained connection between the position-space and momentum-space wavefunctions (P4.23 and Section 9.2.2) also indicates that this problem occupies a special niche. The symmetry between x and p present in the solutions is obviously a reflection of the fact that only for the SHO is the potential energy function quadratic in x. It is often the case in physics where systems with a high degree of symmetry are amenable to solution in a variety of ways, sometimes quite unexpected. In this chapter, we discuss a powerful method of solving the harmonic oscillator problem involving the factorization of the differential equation using differential operators; we then (briefly) discuss extensions and applications of operator methodsÉ to other physical problems.

•Apr 13, 2006•0 citations
The one-dimensional time-independent Green's function G 0 of a quantum simple harmonic oscillator (SHO) system ( ) can be obtained by solving the equation directly. It has a compact expression, which gives correct eigenvalues and eigenfunctions easily. The Green's function G with an additional delta-function potential can be obtained readily. The same technics of solving the Green's function G 0 can be used to solve the eigenvalue problem of the SHO with an generic delta-function potential at an arbitrary site, i.e. . The Wronskians play an important and interesting role in the above studies. Furthermore, the approach can be easily generalized to solve the quantum system of a SHO with two or more generic delta-function potentials. We give the solutions of the case with two additional delta-functions for illustration.
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 harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.
Abstract The Green function method is a powerful technique for solving the initial value problem in quantum mechanics. Once the Green function is solved the whole wavefunction evolution is represented in a concise form and can be evaluated conveniently by numerical integration. We present a method for constructing the Green functions systematically which is different from the conventional methods of eigenfunction expansion or path integration. By using variable changing, function substitution, and Fourier transforms, the time dependent Schrödinger equations can be simplified and the solutions for the simplified equations can be easily derived. We then obtain the Green functions for the original equations by the reverse transforms. The method is demonstrated for the linear potential, the harmonic oscillator, the centrifugal potential, and the centripetal barrier oscillator, where the Green function for the centripetal barrier oscillator has not been solved previously by conventional methods. The method and examples illustrated in this paper can be utilised to strengthen undergraduate courses on quantum mechanics and/or partial differential equation.
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.
Abstract The special and recurring role that the simple harmonic oscillator (SHO) plays in quantum mechanics can be attributed both to its physical relevance and its simple solutions. The fact that the ground state solution is the minimum uncertainty wave packet (Section 12.4) and the highly constrained connection between the position-space and momentum-space wavefunctions (P4.23 and Section 9.2.2) also indicates that this problem occupies a special niche. The symmetry between x and p present in the solutions is obviously a reflection of the fact that only for the SHO is the potential energy function quadratic in x. It is often the case in physics where systems with a high degree of symmetry are amenable to solution in a variety of ways, sometimes quite unexpected. In this chapter, we discuss a powerful method of solving the harmonic oscillator problem involving the factorization of the differential equation using differential operators; we then (briefly) discuss extensions and applications of operator methodsÉ to other physical problems.