Time-dependent Schrodinger equation for many particles

[AxiomOfChoice]

If you've got, say, three particles, then the time-dependent Schrodinger equation (in units where $\hbar = 1$) for the system reads

$$i \frac{\partial \psi}{\partial t} = -\sum_{i=1}^3 \frac{1}{2m_i} \Delta_i \psi + \sum_{i<j} V(r_i - r_j)\psi,$$

right? And of course $\psi = \psi(r_1,r_2,r_3;t)$. But there isn't just ONE solution to this equation, right? There are MANY. And don't they correspond to, say, all particles being independent for large times, or one particle bound to another and the remaining one free, etc.? And I'm guessing this is at the heart of scattering theory - kind of examining the variety of long-time behaviors that can be exhibited in this case. Do I have this right?

 

[jfy4]

Yes that's right. The wave function as $t\rightarrow \pm \infty$ and it's corresponding probabilities are what we can measure. Not only $t$, but also as $r\rightarrow \infty$ which in a collider experiment is on the order of meters.