Number of ways for total spin-1

• tanaygupta2000
In summary, the question is asking for the total number of ways to get a total spin of S = 1 from 2N spin-1/2's. The correct method is to use the formula ##^{2N}C_{N-1}##, which represents the number of ways to arrange minus signs for 2N spins. This is different from the previous formula ##^{2N}C_{2N-1}## mentioned in the conversation.
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 !

tanaygupta2000 said:
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.

Yes sir.
• 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?

Last edited:
tanaygupta2000 said:
Yes sir.
• 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}##).

haruspex said:
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.

ok

1. What is the concept of "total spin-1" in physics?

In physics, "total spin-1" refers to the total angular momentum of a system composed of two or more particles with individual spin values of 1. It is a quantum number that describes the rotational symmetry of a system.

2. How is the number of ways for total spin-1 calculated?

The number of ways for total spin-1 can be calculated using the formula (2s+1)^2, where s is the spin value of each individual particle. For a system with two particles, the number of ways would be (2(1)+1)^2 = 9 ways.

3. What is the significance of the number of ways for total spin-1 in quantum mechanics?

The number of ways for total spin-1 is important in quantum mechanics because it determines the degeneracy of a system, which is the number of distinct energy states with the same energy. This is crucial in understanding the behavior of particles in a system.

4. Can the number of ways for total spin-1 be greater than 9?

Yes, the number of ways for total spin-1 can be greater than 9. It depends on the number of particles in the system and their individual spin values. For example, a system with three particles with spin values of 1 would have (2(1)+1)^3 = 27 ways.

5. How does the number of ways for total spin-1 affect the properties of a system?

The number of ways for total spin-1 can affect the properties of a system in terms of its energy levels and degeneracy. It can also impact the symmetry and stability of the system, as well as the interactions between particles. Understanding the number of ways for total spin-1 is crucial in accurately describing and predicting the behavior of a system in quantum mechanics.

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