含时线性势的量子经典对应

Quantum-classical correspondence of the time-dependent linear potential

在量子力学中,Bohr对应原理指出,在大量子数近似下量子力学应过渡到经典力学;Heisenberg对应原理指出,在经典近似下量子力学中的矩阵元对应经典物理量的Fourier系数;由Heisenberg对应原理,所有可能的矩阵元之和将给出经典运动方程的解。因此,HCP提供一种从量子力学的经典极限得到经典方程的解的方法。HCP的思想应用到含时线性系统,得到含时哈密顿谐振子的经典精确解。含时线性势(TLP)的精确波函数,通过假定某种形式的波函数,可以直接从薛定谔方程中导出波函数。将HCP应用到含时线性系统,利用试探波函数方法,得到了含时线性势的薛定谔方程的一般解。根据Heisenberg对应原理,由量子矩阵元得到含时哈密顿谐振子的经典精确解。

According to the Bohr correspondence principle in the quantum mechanics, quantum mechanics should go to classical mechanics in the case of large quantum number.Due to Heisenberg correspondence principle (HCP), quantum matrix element of a Hermitian operator reduces to the coefficient of Fourier expansion of the corresponding classical quantity in the classical limit.Based upon HCP, the sum of all the possible matrix elements of the momentum operator gives the solution of the classical momentum.Therefore, HCP provides a solution to classical equations from the classical limit of quantum mechanics.The idea of HCP is applied to the time-dependent linear system, and we obtain the classical precise solution of the universal harmonic oscillator.The exact wave function of time linear potential (TLP) can be derived directly from the Schr dinger equation by assuming some form of wave function.Using the trial-function method in the time-dependent linear potential system with HCP, the solution of the Schr dinger equation for the time-dependent linear potential is obtained.Based on the HCP, the solution of the classical equation of motion is derived from the quantum matrix elements.