量子系统的路径积分和谱的近似
Approximations to path integrals and spectra of quantum systems
摘要：
通过对路径积分的n重积分近似，得到了薛定谔方程的格林函数G（E; x_1，x_2）的表达式。在实E轴上对Re{G（E;x，x）}的近似在能级E_{j}的值附近具有峰值。一维和多维谐振子和非谐振子以及Poeschl-Teller势阱的解析和数值例子表明，近似G（E; 0，0）的这些峰值的中值以10%量级的精度对应于已经在最低阶近似n=1和n=2中的偶数能级的精确值，即当路径积分被线积分或二重积分代替时。峰的权重以相同的精度近似于x=0处的波函数的平方模的值。
英文摘要：
An expression for the Green function G(E;x_1,x_2) of the Schroedinger equation is obtained through the approximations of the path integral by n-fold multiple integrals. The approximations to Re{G(E;x,x)} on the real E-axis have peaks near the values of the energy levels E_{j}. The analytic and numerical examples for one-dimensional and multi-dimensional harmonic and anharmonic oscillators, and Poeschl-Teller potential wells, show that median values of these peaks for approximate G(E;0,0) corresponds with accuracy of order 10% to the exact values of even levels already in the lowest orders of approximation n=1 and n=2, i.e. when the path integral is replaced by a line or double integral. The weights of the peaks approximate the values of the squared modulus of the wave functions at x=0 with the same accuracy.
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