We address an interesting problem in elementary quantum mechanics, namely how can one obtain the momentum state eigenfunctions for the infinite square well potential as a direct solution of the time-independent Schrödinger equation in momentum space. The conventional method for obtaining the momentum state eigenfunctions for the infinite square well potential is to take the Fourier transform of the coordinate space eigenfunctions. Unfortunately, the Schrödinger equation in momentum space for the infinite well potential is not well-defined. As such we must use a potential whose Fourier transform is well-defined and then take a limit in which the potential approaches that of the infinite well potential. We present three approaches to the problem: one based on dimensional analysis, one in which the infinite well potential is viewed as the limit of a finite well whose depth approaches infinity, and one in which the infinite well potential is viewed as the limit of a power law...

中文翻译：

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动量空间中的无限方阱势

我们解决了基本量子力学中一个有趣的问题，即如何获得无穷平方阱势的动量态本征函数，作为动量空间中与时间无关的薛定equation方程的直接解。获得无限方阱势的动量态本征函数的常规方法是对坐标空间本征函数进行傅立叶变换。不幸的是，动量空间中无限势阱的Schrödinger方程并不确定。因此，我们必须使用傅立叶变换定义明确的势，然后限制该势接近无限势的势。我们提出了三种解决问题的方法：一种基于维度分析，