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

Relevant 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!

 

[mark726]

Hello there,

Thank you for sharing your question with us. It seems like you have a good understanding of the concept of identical bosons and their wave function. However, there is a small mistake in your calculation. The exchange of two identical particles does not introduce a factor of (-1)^l, but rather (-1)^(2l) = 1. This is because the particles are identical, so their wave function must be symmetric under exchange. This means that the total wave function must remain unchanged when the particles are exchanged, and not just the space, isospin, and spin parts separately.

Therefore, the correct expression for the exchange of two identical particles would be:

$$(exchange)\psi_{\rho\rho} = (exchange) \psi_{space} \psi_{isospin} \psi_{spin} = (-1)^l (-1)^I (-1)^s (-1)^{2l} = (-1)^{l+s}$$

As you can see, this is the same expression you have obtained in your calculation. So, your conclusion is actually correct. The spin wave function must also be symmetric, which means that s must be even. Therefore, the possible values of l and s for two identical bosons are l = 0, 2, 4, ... and s = 0, 2, 4, ...

I hope this clarifies your doubt. Keep up the good work and keep exploring the fascinating world of quantum mechanics!