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arXiv · 1710.00509

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
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Abstract

The one-dimensional time-independent Green's function $G_0$ of a quantum simple harmonic oscillator system ($V_0(x)=m ω^2 x^2/2$) 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 simple harmonic oscillator with an generic delta-function potential at an arbitrary site, i.e. $V_1(x)\propto δ(x-a)$. 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 simple harmonic oscillator with two or more generic delta-function potentials. We give the solutions of the case with two additional delta-functions for illustration.

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Chun-Khiang Chua, Yu-Tsai Liu, Gwo-Guang Wong. 2017-12-04. Time-independent Green's Function of a Quantum Simple Harmonic Oscillator System and Solutions with Additional Generic Delta-Function Potentials. https://arxiv.org/abs/1710.00509

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