Momentum Wavefunction of a Particle in a Box

 The momentum wavefunction fp of a particle in a box (with infinite potential walls) has been described (1) (2) in terms of many plane waves (i.e. many fp’s) even though the energy E=(hbar k)/2/2m expresses only one k. Thus, the energy E appears to be an average.  It seems one may have a plane wave exp(ipx) which collides inelastically with the wall and changes to exp(-ip1x) etc.; on average energy is conserved. In (1) and (2), the spatial wavefunction W(x)= Sum over p fp exp(ipx) is obtained by solving the time-independent Schrodinger equation, and fp from taking the Fourier transform of W(x). These are strictly math procedures it seems. In a previous note (3), the time independent Schrodinger equation is written with V(x) and W(x) both expressed in terms of Fourier transforms. This lead to an equation for each p of the form:

Sum j+k=p Vk fk  = (E-p*p/2m) fp ((1)).

In such a case, V(x) exists throughout x. For the case of an infinite wall, there is still a potential, but only at the end walls. In this note, we suggest ((1)) may be used to obtain fp without taking the Fourier transform of W(x) or solving for Vk. This approach also shows cycling is involved, even for a particle in a box. The cycling occurs at the walls.