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.Q:

When does quantum mechanics cease to work in a system with interacting particles?

Although the interaction between two quantum particles is weak (the cross section is very small) we can still experiment with such a system.

My question is when does the quantum mechanic cease to work in a system with interacting particles.

In the case of two particles, we can say that their wave function is represented in the two-particle state space. If we have two bound particles, then there is a possibility that we may get so many states in the case of two particles that there is no room for any other wave function to exist in a system with two or more particles.
In the case of 3 particles, how do we describe it?
Is it because the wave function is added by the wave function in the case of two particles?
Is it because we cannot make the wave function infinite as in the two-particle case?

A:

There are no problems in describing the interaction between any two particles in terms of their wave function in the common eigen-function basis. The problem of having to consider all quantum numbers is that it’s not always necessary to consider more than one eigen-function.
For example, consider a two-body system where one particle is in a state $l_1\,\l_2$ and the other is in $l’_1\,\l’_2$. Suppose there are many eigen-states that all share the same $l’_2$, but have different values of $l_1$. So the wave function is just $|l_1\,l_2\rangle\,|l_1’\,l_2’\rangle$. There are no states for $l_2$ or $l’_2$ and the wave function is just a linear combination of pure $l_1\,l_2’$ eigen-states. But you can’t do that for three particles.

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