# Measure of Two Identical Particles

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?

• You haven't measured anything about the other particle? Commented Dec 14, 2021 at 23:10
• I think my idea is that we have some device that detected a particle at $y$. But we don't know which particle it is. Commented Dec 15, 2021 at 0:17
• What do the subscripts $a$ and $b$ represent? Spin? If $a\neq b$, then the particles are not identical. Commented Dec 15, 2021 at 3:29

The eigenstate which $$\psi(x_1,x_2)$$ collapses to is ill-posed because you provided/have information of only one particle whereas the system requires two to be described.
Collapse means the wave function $$|\psi\rangle=\sum_i a_i|\psi_i\rangle$$, sum of every eigenstate, falling into one single eigenstate $$\sum a_i|\psi_i\rangle\to1\cdot|\psi_\sphericalangle\rangle$$. So that, for instance, a measurement, say $$\hat{x}$$, reads $$x=\langle\psi_\sphericalangle|\;\hat{x}\;|\psi_\sphericalangle\rangle$$ where $$x$$ represents the eigenvalue of $$\hat{x}$$.
Just for the idea, for two-particle bosonic system, the general form of a wave function can be expressed as $$|\psi(x_1,x_2)\rangle=\sum_j\sum_i \frac{a_{ij}}{\sqrt{2}}\left(|\psi_i(x_1)\rangle\otimes|\psi_j(x_2)\rangle+|\psi_j(x_1)\rangle\otimes|\psi_i(x_2)\rangle\right)\;.$$
Regardless of its lengthy expression, the collapse just means that the wave function falls into one single eigenstate whose position eigenvalue is of the form of $$(x_1,x_2)$$.