Fourier Transform for Bound States in a Step Potential

 In (1) the Fourier transform for a particle in a box with an infinite potential at the walls is calculated to show that all plane waves exp(ipx) are stirred up by collisions with the wall even though the spatial wavefunction is proportional to sin(kave x+b). In this note, we consider the case of a finite potential wall and investigate the Fourier transform. We argue that one must calculate the Fourier transform for x ranging from -infinite to + infinite. Thus, if the potential is very different in different regions, one may have W(x), the wavefunction, equaling different functions in each region. (Each is a solution to the time independent Schrodinger equation with the appropriate potential for the region in question.) If one wishes to keep the integration bounds of the Fourier transform at -/+ infinite, one may need to multiply the various W(x) functions by step like functions to ensure that a particular function is nonzero only in its appropriate region. The Fourier transform of a step function multiplied by a general function, however, is not the same as the Fourier transform of the function itself. Furthermore, one obtains an overall Fourier transform which is influenced by each spatial region. Thus, weights f(p) in W(x)= Sum over p  f(p) exp(ipx) in a damped region within a potential barrier include contributions from a region where there is no potential and one has sinusoidal behaviour (and vice versa).