[QM] Addition of spin for normal and identical particles

In summary, the problem involves two particles with spin ##s=1## and angular momentum ##l=0##. The total spin can take values of 0, 1, and 2 according to the theory of addition of spins. However, when the particles are distinguishable, the total spin may be restricted due to the symmetry or antisymmetry of the total wave function. For bosons, there is no restriction on the possible values of total spin, but for fermions, not all values may be observed. To solve the problem, one may consider writing out the spin wave function in terms of the old basis and see how swapping affects the expression.
  • #1
Coffee_
259
2
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]
 
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  • #2
Do you know of any rule that applies to indistinguishable particles?
 
  • #3
DrClaude said:
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.
 
  • #4
Coffee_ said:
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.
 
  • #5
DrClaude said:
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.
 
  • #6
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.
 

What is spin in quantum mechanics?

Spin is a fundamental property of particles in quantum mechanics that describes their intrinsic angular momentum. It is a quantum number that can take on discrete values, such as 1/2, 1, 3/2, etc.

How is spin added for normal and identical particles?

In quantum mechanics, the addition of spin for normal and identical particles follows the same rules as adding angular momentum for classical objects. The total spin of a system is the sum of the individual spins of each particle.

What is the difference between adding spin for normal and identical particles?

The main difference is that for identical particles, the wave function must be symmetric (for bosons) or anti-symmetric (for fermions) under particle exchange. This leads to different possible total spin values for the system.

What is the Pauli exclusion principle in relation to spin?

The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state simultaneously. This is due to the anti-symmetric nature of the wave function for fermions, which means they cannot have the same spin state.

How does spin affect the properties of particles?

The spin of a particle can affect its magnetic moment, energy levels, and interactions with other particles. It also plays a crucial role in determining the behavior of systems with multiple identical particles, such as atoms and molecules.

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