# Quantum numbers of a system of particles

#### Aleolomorfo

Homework Statement
Find the possible values of isospin, parity, charge conjugation, G-parity and totoal angular momentum J, up to J=2, for $\rho^0 \rho^0$ state
Homework Equations
$(exchange) \psi_{space} = (parity) \psi_{space}$
$(exchange) \psi_{spin} = (-1)^s for bosons$
$(exchange) \psi_{isospin} = (-1)^s for bosons$
Hello everybody!

I have a problem with this exercise when I have to find the possible angular momentum.

Since $\rho^0 \rho^0$ are two identical bosons, their wave function must be symmetric under exchange.

$$(exchange)\psi_{\rho\rho} = (exchange) \psi_{space} \psi_{isospin} \psi_{spin} = (-1)^l (-1)^I (-1)^s$$

Since I have still calculated that $I = 0$ or $I = 2$:

$$(exchange)\psi_{\rho\rho} = (-1)^{l+s}$$

To be symmetric $\rightarrow$ l+s must be even.

However, the solution of the exercise states that:
The two particles are identical bosons, hence l must be even (l=0, 2, 4, ...). The spin wave function must be symmetrical too, hence s=0, 2, 4, ...
This is different from my conclusion.
I'd like to ask you where is my mistake.

Thanks in advance!

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