线性谐振子Schrdinger方程的波包解

Wave Packet Solutions of the Schrdinger Equation for Simple Harmonic Oscillator

通过精确求解简谐振子的含时Schr dinger方程 ,分别得到相应于基态和各激发态的波包形式解 ,并且证明这些波包的质心按照经典运动的周期振动 ,而波包的形状保持不变 .在振子的力常数随时间改变的情况下 ,所得到的波包解与通常简谐振子的波包结构相同 .其质心仍遵从经典规律 ,但波包的形状会发生改变 :可以展开、收缩或脉动 .

By solving exactly the time dependent Schrdinger equation for simple harmonic oscillator, we have obtained respectively the wave packet solutions associated with the displaced ground state and the displaced excited eigenstates, and then demonstrated the centroid of the wave packet oscillates according to the classical law and that the wave packet form remains unchanged. For simple harmonic oscillator with the force constant depending on time the wave packets with the same structures as that of the usual simple harmonic oscillator can still be found to move in such a way that the centroids move according to the classical law, but their wave packets may deform in time. Depending on how the force constant varies with time, the wave packets can spread, contract or pulsate.