Number of ways for total spin-1

[tanaygupta2000]

Homework Statement

Enumerate the number of independent ways in which 2N spin-1/2's can be com-
bined to form a total spin-1. Here, N is an integer.

Relevant Equations

Addition of spins, s1 + s2 = S
s1 = s2 = 1/2

To the extent I understood this question, we have to enumerate the total no. of ways to get a total of spin S = 1 from 2N number of spin-1/2's.
Now, I think that by spin-1/2's, the question is referring to s1 = s2 = 1/2 (and not something like 3/2, 5/2, ...).
When N = 1, we have 1/2 + 1/2 => No. of ways = 1
When N = 2, we have 1/2 + 1/2 + 1/2 + 1/2 => No. of ways = 3 + 2 + 1 = 6
Likewise,
For N = N, No. of ways should be given by = (2N-1)+(2N-2)+(2N-3)+...+3+2+1
I am sure I'm missing many things here, this question carries 15 marks.
Kindly help !

 

[haruspex]

When N = 2, we have 1/2 + 1/2 + 1/2 + 1/2

That does not add up to 1, and I don't see how it would lead to 6 anyway.

 

[tanaygupta2000]

Yes sir.
Can I follow this approach:

• For two spins => 1/2 + 1/2 = 1 way
• For four spins => (-1/2) + 1/2 + 1/2 + 1/2 = 4 ways since minus sign can be on any of the 4 spins
• For six spins => (-1/2) + (-1/2) + 1/2 + 1/2 + 1/2 + 1/2 = 15 ways of arranging minus signs
• Likewise, for 2N spins, total no. of ways = 2N[C](2N-1) (C --> Combination)

and so on?
Is this method okay as required in the question?

 

[haruspex]

Yes sir.
Can I follow this approach:

• For two spins => 1/2 + 1/2 = 1 way
• For four spins => (-1/2) + 1/2 + 1/2 + 1/2 = 4 ways since minus sign can be on any of the 4 spins
• For six spins => (-1/2) + (-1/2) + 1/2 + 1/2 + 1/2 + 1/2 = 15 ways of arranging minus signs
• Likewise, for 2N spins, total no. of ways = 2N[C](2N-1) (C --> Combination)

and so on?
Is this method okay as required in the question?

Your specific cases look good, but don’t match your general formula ($^{2N}C_{2N-1}$).

 

[tanaygupta2000]

Your specific cases look good, but don’t match your general formula ($^{2N}C_{2N-1}$).

Yes sir, sorry I meant $^{2N}C_{N-1}$ number of ways are possible for 2N spins.

 

[haruspex]

ok