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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.

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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

The usefulness of these solutions, however, is that we can construct actual physical solutions from them through Fourier analysis.

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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:

$$\Phi = \sum_n\Phi_n=\sum_n\left(A_ne^{ikx}+B_ne^{-ikx}\right)$$ $$H\Phi_n=E\Phi_n, \forall n\Rightarrow H\Phi=E\Phi$$

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