量子力学中自由粒子演化的算符方法

Time-evolution operator approach to free particle in quantum mechanics

利用演化算符的抽象形式,借助于算符运算技巧,对一维自由粒子的演化问题进行了形式计算.得到了一维自由粒子波包位置概率分布的方差随时间演化的结果,证明了一维自由粒子波包随时间扩散的结论.利用谐振子问题中常用的产生算符和湮灭算符作为工具,对初始时刻中心位于原点的“静止”高斯波包随时间演化的波函数进行了计算,将得到的级数形式写为了初等表达式,从求得的概率密度可见随着时间增加该高斯波包的宽度也随之增加.

The evolution of one-dimensional free particle is calculated by using timeevolution operator.The evolution of the variance of onedimensional free particle wave packet is calculated,the results show that the wave packet spreads over time.Using the annihilation operator,the evolution of wave function of Gaussian wave packet is calculated,here the Gaussian wave packet is a“stationary”wave packet located at the origin at the initial time.A series expression is obtained,sum it up the elementary function expression is obtained,the spreading of the wave packet in time has been directly confirmed.