[QM] Addition of spin for normal and identical particles

[Coffee_]

1. Problem: Consider the composed system of two particles of spin $s=1$ where their angular momenta is $l=0$. What values can the total spin take if they identical? What changes when they are distinguishable?

The Attempt at a Solution

:

The problem I have here is incorporating the fact that $l=0$ and the information about being identical or not. I know what to do if problem just stated, ''consider two particles with spin $s=1$ what values can the total spin take?''.

In that case it's kind of trivial, the theory of addition of spins or angular momenta for two particles states that the total value $j=[|j_{1} - j_{2}|,j_{1}+j_{2}]$. So applying it to these numbers would give possible total spins of 0,1 and 2.

The additional two pieces of information confuse me a bit.[/B]

 

[DrClaude]

Do you know of any rule that applies to indistinguishable particles?

 

[Coffee_]

Do you know of any rule that applies to indistinguishable particles?

Yeah, the total wave function has to be either symmetrical or antisymmetrical for fermions and bosons respectively. In this specific case in which we have two bosons this means that if I'd know that the position parts of my specific system were symmetric I could say the say that the spin part has to be symmetric as well. However the position part could be antisymmetric which would make the spin antisymmetric as well.

 

[DrClaude]

Yeah, the total wave function has to be either symmetrical or antisymmetrical for fermions and bosons respectively. In this specific case in which we have two bosons this means that if I'd know that the position parts of my specific system were symmetric I could say the say that the spin part has to be symmetric as well. However the position part could be antisymmetric which would make the spin antisymmetric as well.

Indeed, that tells you that the spin wave function must have a definite symmetry. You should check if that can restrict the possible value of the total spin.

 

[Coffee_]

Indeed, that tells you that the spin wave function must have a definite symmetry. You should check if that can restrict the possible value of the total spin.

Would you recommend writing |s,m> out in function of the old basis |s1,m1>x|s2,m2> and then somehow see how swapping things there might affect the expression? For that I guess I'd have to look up the CG coefficients.

 

[DrClaude]

I haven't worked out the solution, so I'm simply pointing out possible ways to think about the problem. If these were fermions, then there would be an obvious difference if they were indentical or not, and not all possible values of total spin would be observed. There is no such restriction for bosons, but that may be what the problem wants you to show.