Suppose I have two bosons with symmetric wave function (I guess there should be tensor products?): $$\psi(x_1,x_2)=\psi_a(x_1)\psi_b(x_2)+\psi_b(x_1)\psi_a(x_2).$$
Suppose now that I perform a measurement of a particle, and I find its location to be $y$. What does the wave function "collapse" to?
Since they are identical bosons, I know it should collapse to something symmetric. It seems to me that I should somehow change one of the $\psi_a$ or $\psi_b$ to be the Dirac $\delta$ function at $y$, but I also don't know which particle I measured! So how would I determine what the resulting wave funciton is, or is my problem ill-posed?