用阶梯算符研究谐振子能量及相干态问题

The studies of energies and coherent state for harmonic oscillator by ladder operators

谐振子是量子力学中最基本也是十分典型和重要的问题,而在坐标表象中利用薛定谔方程的求解过程比较复杂.本文从两个无量纲的阶梯算符出发巧妙的推导出谐振子能量的本征值和本征矢,进而借用平移算符求解出谐振子的相干态.计算表明相干态表象的基矢是过完备的,同时在相干态中,坐标及其动量具有最小的不确定性.

Harmonic oscillator is not only the most basic but also a representative problem in the quantum mechanics. The discussion on it in the coordinate representation by the Schrtidinger equation is very complicated. The dimensionless ladder operators are introduced to calculate the eigenvalues and eigenvectors of Hamiltonian dexterously. The coherent states of harmonic oscillator is investigated by translation operators. It is found that the basic-vectors in the representation of coherent states are over-complete and there is the minimum uncertainty of position and its momentum in the coherent states.