Consider a quantum-mechanical system : 3 particles in a rigid sphere.
Which of the following is the correct form of wavefunction for this system ?
(a) $\psi(r_1,r_2,r_3,t) = \psi_1(r_1,t) . \psi_2(r_2,t).\psi_3(r_3,t)$
(b) $\psi(r_1,r_2,r_3,t) = \psi_1(r_1,t) + \psi_2(r_2,t) + \psi_3(r_3,t) $
where $r_1,r_2,r_3$ are position vectors of the three particles and $t$ is the time.
If one is correct, then what is wrong with the other?
Also what is justification for the right form - experimental / theoretical ?
(B) Even though I am giving example of 3-particle system above, still I want to ask this : In Schrodinger equation, can $\psi$ refer to more than one particle?
If I am not wrong, Schrodinger himself derived his equation for a single particle only from making changes in "Hamilton-Jacobi equation for a single particle".
Here I am NOT referring to usual explanation given in textbooks - that either repeat the experiment with single particle OR do single experiment with ensemble of particles, prepared in the same state.