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I'm reading a paper (Rapp, 1968) that treats quantum mechanically the problem of a particle $A$ "hitting" an harmonic oscillator made of two particles, $B$ and $C$:

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In the limit $\tilde{x}\to +\infty$, the solution of the time-independent Schrodinger equation that deals with the relative energy of the system is:

$$\lim_{x\to +\infty} \psi(x,y)=H_I(\tilde{y})e^{-ik_I\tilde{x}}+\sum_{n=0}^\infty a_nH_{n}(\tilde{y})e^{ik_n\tilde{x}}$$

where $H_n(\tilde{y})$ are the spatial wavefunctions of the free harmonic oscillator. So, this solution considers the incident wave that accompanies the incident particle $A$ and also the resultant reflected waves. The authors claim that the probability of transition from the state $I$ to the state $n$ is given by the ratio between the $n^{th}$ reflected flux and the incident one:

$$P_{I\to n}=\frac{k_n}{k_I}|a_n|^2$$

But why is this probability computed like this? What is the intuition behind this relation?

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