The wavefunction of identical particles is given as: $$\psi_{1,2} (x_1,x_2) = \psi_1(x_1)\psi_2(x_2) + \psi_2(x_1)\psi_1(x_2)$$ . Why is it so? Why is it the sum of the two states? What is the explanation behind this? Yes, I know that wavefunction being a linear combination of the solutions of Schrodinger's equation is a solution itself. But that does answer only mathematically.

In order to get a deep insight, I again read Feynman's introductory lectures where he wrote:

When an event can occur in several ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference. $$\phi = \phi_1 +\phi_2$$ .

Why is it so?