一维对称双势阱的精确求解

Exact Solution of a Symmetric Double-Well Potential in One-Dimension

讨论了一维对称简谐势的薛定鄂方程的通解,通解可用抛物线柱函数Dα(t)来表示,且可由Kummer函数表达,Hermite多项式表示的一维简谐势的解仅是该通解的一特殊形式。此通解用于一维对称双势阱模型的讨论,此模型由两相同的简谐势连接而成,得出了此模型的精确解。作为一例,本文给出了一具体模型的精确数值结果。

The general solution of the Schroedinger equation for the harmonic potential in onedimension is studied, which can be expressed by the parabolic cylinder function Dα(t) and also by the Kummer's function, rather than by the Hermite function. It is pointed out that the Hermite function is only a special form of the general solution for the potential. A doublewell potential is exactly solved by this way, which involves two symmetrically identical harmonic potentials. A precise numerical solution is given as an example for a typical doublewell potential.