摘要
本文以严格的数学推导，求解了一特定势能下的量子力学波动方程。得出了此二阶微分方程的本征函数和本征值量子能级，以级数法得出的方程解是一新的正交多项式。此势能模型既有“无限深势阱”模型的特点，也有“简谐势振动”模型的特征。此方程的解在一定的条件下能退化为这两个模型的解。以严格的数学推导，得出了这一方程可化为“斯特姆—刘维”型微分方程。这就证明了：该方程的本征函数是正交的和完备的。
A quantum Schrodinger's equation for square tangent functional potentialis exactly solved, anel both the eigenialue and eigenfunction, i. e. a new kind of polynomial, are obtained, by means of power-series solutions.. This model is similar toboth quantum oscillator and the square well model, and the physical meaning of themodel is also discussed.It is proved that the equation belongs to the sturm-Liouvilledifferential equation. It is verified, therefore, that the eigenfuctions are orthogonaland complete.
关键词
量子力学
波动方程
解
薛定谔方程
弹性势
Quantum Mechanics
solution of the Schrodinger's equation
quantumenergy level
