# Solving time-independent Schrodinger equation with $V=0$

I recently came across this section in "Physical chemistry" by Peter Atkins and Julia Paula. There they discuss the solution of Schrodinger equation for a particle of mass m moving through a single dimension and potential $$V=0$$. The solution is $$\Phi=Ae^{ikx}+Be^{-ikx}$$. Then assuming $$B=0$$ probability density: $$|\Phi|^2=|A|^2= a$$ constant. But doesn't this mean the total probability diverges as it becomes non-square integrable? What am I thinking wrong? Please help.

You are right. This solution assumes that the particle has a well-defined momentum of $$p=\hbar k$$. i.e. $$\Delta p=0$$. This means that $$\Delta x=\infty$$ (excuse my poor math) by the HUP. So this is a non-physical solution
Even though $$\Phi$$ is a solution to the Schrodinger equation it is not normalizable as you mentioned. Thus it is not a valid wave-function by itself. However, since the Schrodinger equation is linear, one can build a superposition of these plane wave states such that the result is normalizable:
$$\Phi = \sum_n\Phi_n=\sum_n\left(A_ne^{ikx}+B_ne^{-ikx}\right)$$ $$H\Phi_n=E\Phi_n, \forall n\Rightarrow H\Phi=E\Phi$$