arXiv quant-ph/04081822004-08-30quant-ph

“弹跳”高斯波包的精确结果

Exact results for `bouncing' Gaussian wave packets

M. Belloni, M. A. Doncheski, R. W. Robinett

备注：: 14 pages, 1 embedded .eps figure
DOI：: 10.1238/Physica.Regular.071a00136

摘要：

我们考虑薛定谔方程的含时高斯波包解（对于一个自由粒子，具有任意的初始中心位置x_0和动量p_0，但在x=0处有一个无限大的壁，即所谓的弹跳波包。我们展示了如何差分或镜像解决方案的形式psi（x，t）-psi（-x，t），在这种情况下，可以被严格规范化，允许评估的一些时间依赖的期望值和其他数量的封闭形式。例如，我们明确地计算了<p^2>_t，它说明了自由粒子动能（以及总能量）如何受到远边界的影响。我们还讨论了位置，_t和动量，_t的期望值的时间依赖性<x><p>，以及它们与"碰撞“时的冲力的关系。最后，证明了x_0，p_0--> 0极限可归结为非标准自由粒子高斯解的一种特殊情况.在文献中增加这个例子，然后扩展了量子力学问题的相对少量的高斯解，并以封闭形式提供了熟悉的经典类似物（自由粒子，均匀加速度，谐振子，不稳定振子和均匀磁场）。

英文摘要：

We consider time-dependent Gaussian wave packet solutions of the Schrodinger equation (with arbitrary initial central position, x_0, and momentum, p_0, for an otherwise free-particle, but with an infinite wall at x=0, so-called bouncing wave packets. We show how difference or mirror solutions of the form psi(x,t)-psi(-x,t) can, in this case, be normalized exactly, allowing for the evaluation of a number of time-dependent expectation values and other quantities in closed form. For example, we calculate <p^2>_t explicitly which illustrates how the free-particle kinetic (and hence total) energy is affected by the presence of the distant boundary. We also discuss the time dependence of the expectation values of position, <x>_t, and momentum, <p>_t, and their relation to the impulsive force during the `collision' with the wall. Finally, the x_0,p_0 --> 0 limit is shown to reduce to a special case of a non-standard free-particle Gaussian solution. The addition of this example to the literature then expands on the relatively small number of Gaussian solutions to quantum mechanical problems with familiar classical analogs (free particle, uniform acceleration, harmonic oscillator, unstable oscillator, and uniform magnetic field) available in closed form.

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英文摘要：

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