势阱中粒子能级与波函数微扰计算的代数递推公式

Algebraic Recursion Formulas for Perturbation Calculation of Energy Levels and Wave Functions in Potential Wells

利用超位力定理 (HVT)和Hellmann -Feynman定理 (HFT) ,导出了由有精确解的势阱的能级值用微扰法直接计算一维势阱的各级近似能级的普遍代数公式 ,并导出了由能级近似值计算定态波函数近似表达式的代数公式。给出了代数公式具体应用的几个典型一维势阱实例。

Using hypervirial theorem (HVT) and Hellmann-Feynman theorem (HFT), perturbation calculation of successive order approximate values of energy levels in a potential well with power series expansion of the potential energy are processed. Algebraic recursion formulas for calculating energy levels are deduced. We use the exact energy levels for parabolic potential well (one dimensional harmonic oscillator) as zero order approximation, and derive algebraic formulas for successive order approximate energy levels for given potential energy function. The corresponding wave functions can then be written as polynomials in which coefficients are expressed in terms of the energy levels and coefficients in the power series of potential energy. In this way, tedious and cumbersome perturbation calculations in Rayleigh-Schrdinger perturbation method are avoided. Thus the present method is simple, efficient and time saving. Typical examples are illustrated with the algebraic formulas, including: energy levels for Gaussian potential well; for modified Pschel-Teller well; potential wells for anharmonic oscillators; Morse potential for vibrational energy levels of diatomic molecules and modified Morse potential for vibrational-rotational energy levels. Formulas for calculation of wave functions corresponding to calculated energy levels are given for anharmonic oscillators and for symmetric potential energy functions. The present method can be extended to two or three dimensional potential well, and can also be used in other mathematically analogous eigenvalue problem.