低阶量子波函数与Hamilton-Jacobi方程解的关系
Relations Between Low-lying Quantum Wave Functions and Solutions of the Hamilton-Jacobi Equation
摘要：
本文讨论了在一维势V中粒子的低阶薛定谔波函数与相应的以-V为势的Hamilton-Jacobi方程的解之间的一种新的关系。函数V是$\geq 0$，可以有几个最小值（V=0）。我们假设这个问题的特点是一个小的非谐参数$g^{-1}$和一个小得多的量子隧穿参数$ε$之间的这些不同的最小值。将波函数或其能量作为$g^{-1}$和$ε$中的形式双幂级数展开，我们证明了在这样的展开式中$g^{-m}ε^n$的系数可以用定积分表示，首阶项由Hamilton-Jacobi方程的经典解确定.对四次势$V={1/2}g^2(x^2-a^2)^2$的特例进行了详细的分析。
英文摘要：
We discuss a new relation between the low lying Schroedinger wave function of a particle in a one-dimentional potential V and the solution of the corresponding Hamilton-Jacobi equation with -V as its potential. The function V is $\geq 0$, and can have several minina (V=0). We assume the problem to be characterized by a small anhamornicity parameter $g^{-1}$ and a much smaller quantum tunneling parameter $ε$ between these different minima. Expanding either the wave function or its energy as a formal double power series in $g^{-1}$ and $ε$, we show how the coefficients of $g^{-m}ε^n$ in such an expansion can be expressed in terms of definite integrals, with leading order term determined by the classical solution of the Hamilton-Jacobi equation. A detailed analysis is given for the particular example of quartic potential $V={1/2}g^2(x^2-a^2)^2$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
