一维及二维线性谐振子的算符解法

利用Schrodinger因式分解法1]以及常用的Hamiltonian经典表式与算符表式的对应关系,得出了一维谐振子系统的升降算符.在此基础上,我们将算符技术应用到一维谐振子系统,导出升降算符所满足的方程,得到一维线性谐振子的能量本征值和本征函数,确证了零点能的存在,推导出厄米多项式及其递推公式.我们所得的结果与用常规的数理方法所得到的结论是一致的.另外,本文还将升降算符推广到二维,求出升降算符在二维中的表示形式,从而将二维问题简化成一维问题来处理,得到二维线性谐振子的能量本征值和本征函数.

Solution to the Problems of One and Two-dimensional Linear Harmonic Oscillators by Using Operators

We start investing raising and lowering operators by the schrodinger factorization of the hamiltonian in one dimensional linear harmonic oscillator system.Eigenvalue and eigenfunction of one dimensional harmonic oscillator are obtained,and zero point energy is also proved,and the multinomial of hermite and its recurrence relation are got too.The results are self consistent with the conclusions by the use of the regular method of maths and physics.In addition,raising and lowering operators of two dimensional linear harmonic oscillator are deduced from these operators of one dimensional linear harmonic oscillator ,and eigenvalue and eigenfunction of two dimensional linear harmonic oscillator are obtained.