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

Abstract

The one-dimensional time-independent Green's function G 0 of a quantum simple harmonic oscillator (2 2 ) 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 deltafunction 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.

Variational approach to the Schrödinger equation with a delta-function potential

“…A quantum-mechanical Hamiltonian operator H perturbed by a delta-function potential gδ(x) has received considerable attention [1][2][3][4][5][6][7][8][9][10][11][12][13]. In most cases H describes a free particle [1], a particle in a box [1][2][3][4] or the harmonic oscillator [1,[5][6][7][8][9][10][11][12][13].…”

“…A quantum-mechanical Hamiltonian operator H perturbed by a delta-function potential gδ(x) has received considerable attention [1][2][3][4][5][6][7][8][9][10][11][12][13]. In most cases H describes a free particle [1], a particle in a box [1][2][3][4] or the harmonic oscillator [1,[5][6][7][8][9][10][11][12][13]. Since in these cases the Schrödinger equation for H is exactly solvable one can obtain closed form expressions for the solutions to the Schrödinger equation for H g = H + gδ(x) in several different ways.…”

“…Since in these cases the Schrödinger equation for H is exactly solvable one can obtain closed form expressions for the solutions to the Schrödinger equation for H g = H + gδ(x) in several different ways. For example, from the eigenvalues and eigenfunctions of H [1,8], by solving the eigenvalue equation left and right of the origin and matching those solutions at x = 0 [2,3,5,9,10] or by means of the Green function [11,12]. In some cases the authors resorted to this kind of models to illustrate the application of approximate methods like perturbation theory [4,9], WKM method [9], or variational approaches [9,13].…”

Path-Integral Treatment of Quantum Bouncers

“…1a); infinite ramp [10][11][12] (Fig. 1b); harmonic oscillator [5,[12][13][14][15][16][17][18] (Fig. 1c); particle near a wall, a.k.a.…”

Application of Exponential Kernel to Laplace Transform

“…Finding the Laplace transform of a function and its properties is normally discussed in standard mathematical physics books [7] [8]. An interesting function (more precisely a limit of some distribution) is the Dirac delta function, which has been in use in different settings [9]- [15]. The value of the Laplace transform of Delta-function can be found in mathematical physics books [8], where it is claimed that this value is one.…”