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Homework Statement:

You have three particles with angular momentum. How many states are there for each of the total angular momentum values (3, 2, 1, 0)?
Hint: Start by coupling two particles with angular momentum 1 and find how many states each of the possible angular momentum values have. Then couple the third particle to the previous two states.
Homework Equations:

## j = \vert j_1 + j_2 \vert , \vert j_1 + j_2  1\vert, ..., \vert j_1  j_2 \vert ##
## m = j , j+1, ..., j ##
I can solve the two particle system easily enough:
Using ##j_1 = 1## and ##j_2 = 1##, the possible total angular momentum values are ##j = 2, 1, 0##. With ## m = j , j+1, ..., j ##,
##j = 2: m = 2, 1, 0, 1, 2 ## (5 states)
##j = 1: m = 1, 0, 1## (3 states)
## j = 0: m = 0 ## (1 state)
I can imagine the second part as if there were only two particles:
For ##j_3 = 1## and ##j(combined) = 2##,
## j = 3: m = 3, 2, 1, 0, 1, 2, 3 ## (7 states)
##j = 2: m = 2, 1, 0, 1, 2 ## (5 states)
##j = 1: m = 1, 0, 1## (3 states)
For ##j_3 = 1## and ##j(combined) = 1##, (same as above)
##j = 2: m = 2, 1, 0, 1, 2 ## (5 states)
##j = 1: m = 1, 0, 1## (3 states)
## j = 0: m = 0 ## (1 state)
For ##j_3 = 1## and ##j(combined) = 0##,
##j = 1: m = 1, 0, 1## (3 states)
But I cannot figure out what to do with number of states for the three particle system. Do I multiply? Add?
I want to end up with something like:
##j(total) = 3## gives ___ states for each of the possible ##j(total)## states but I'm really confused how to get there.
Using ##j_1 = 1## and ##j_2 = 1##, the possible total angular momentum values are ##j = 2, 1, 0##. With ## m = j , j+1, ..., j ##,
##j = 2: m = 2, 1, 0, 1, 2 ## (5 states)
##j = 1: m = 1, 0, 1## (3 states)
## j = 0: m = 0 ## (1 state)
I can imagine the second part as if there were only two particles:
For ##j_3 = 1## and ##j(combined) = 2##,
## j = 3: m = 3, 2, 1, 0, 1, 2, 3 ## (7 states)
##j = 2: m = 2, 1, 0, 1, 2 ## (5 states)
##j = 1: m = 1, 0, 1## (3 states)
For ##j_3 = 1## and ##j(combined) = 1##, (same as above)
##j = 2: m = 2, 1, 0, 1, 2 ## (5 states)
##j = 1: m = 1, 0, 1## (3 states)
## j = 0: m = 0 ## (1 state)
For ##j_3 = 1## and ##j(combined) = 0##,
##j = 1: m = 1, 0, 1## (3 states)
But I cannot figure out what to do with number of states for the three particle system. Do I multiply? Add?
I want to end up with something like:
##j(total) = 3## gives ___ states for each of the possible ##j(total)## states but I'm really confused how to get there.