Understanding Spin States of Particles with Spin-1 and Spin-1/2

[Solarmew]

Ok, so for some reason this section of the GRE book makes 0 sense to me ... maybe because i haven't taken the class yet, maybe I'm missing something ...
It says "If you have a spin-1 particle with m = +1 and a spin-1/2 particle with m = +1/2, then m_tot = +3/2 (this part makes sense, you just add them), and the system must be in the state with s_tot = 3/2"

why is s_tot just 3/2? I thought the possible spin states of the system are s1+s2, s1-s2, and everything in-between in intervals of 1, so 3/2 and 1/2 ...
i don't get it ... it even says here "We can get any spin between |s-s'| and s+s'. "

and the next part says "On the other hand, if the spin-1/2 particle had m=-1/2, then m_tot = +1/2 and either s_tot = 3/2, or s_tot = 1/2 are allowed"

so i guess 1-(-1/2)=3/2, 1+(-1/2) = 1/2
but isn't the first part also 1-1/2=1/2, 1+1/2 = 3/2
?

so confused

 

[king vitamin]

Watch out - remember that in general the spin must satisfy $$s_{tot} \geq m.$$

So you can have m = 1/2 and s = 3/2 or 1/2, but the m = 3/2, s = 1/2 violates the rule above. Recall that m is like a component of the vector with magnitude s, and a single component of a vector is always smaller than the magnitude.

 

[Solarmew]

Watch out - remember that in general the spin must satisfy $$s_{tot} \geq m.$$

So you can have m = 1/2 and s = 3/2 or 1/2, but the m = 3/2, s = 1/2 violates the rule above. Recall that m is like a component of the vector with magnitude s, and a single component of a vector is always smaller than the magnitude.

oooooh, geez, I wish the book would've stated that explicitly = -______- =
thank you!

 

[Solarmew]

how about this one?

"A meson is a bound state of a quark and an antiquark, both with spin 1/2. Which of the following is a possible value of total angular momentum j for a meson with orbital angular momentum l=2?"

I've got the formula J=L+S. And I guess we can add or subtract the spins for a total s=0,1. But then it says:
"Adding l=2 to s=0 gives only j=2, and adding l=2 to s=1 gives j=3,2,1"
How did they get the 3,2,1?

 

[king vitamin]

Hmm, I think you should consider getting a book or finding some online notes about this if it's mostly new to you.

The key is to understand that S, L and J are magnitudes of vectors related by

$$ \vec{S} + \vec{L} = \vec{J} $$

Now think about vector addition. It's completely possible to add two vectors with magnitude S=1 and L=2 to get the magnitude of J to be 3; the two vectors S and L must be parallel. Correspondingly, it's possible to add the two vectors to get 1; the vectors must simply be antiparallel. Finally, by playing with angles, you can make the magnitude of J be any number between 1 and 3.

But then there's quantum mechanics. Angular momentum must be quantized according to a set of rules, and in the above case, only the intermediate value J=2 is allowed. If you aren't familiar with the rules, it'll be easy to get stumped on some of these problems. In this case, the rule is that J must take the quantized values |L-S|, |L-S|+1, |L-S|+2, ..., L+S.