What is the significance of 2S+1 in understanding bosons and fermions?

[erok81]

I have a couple questions related to different energy levels and spin. Our professor taught us one way and the TA a different in our review session. We tried to hash it out in class but had to move on. I suspect both the TA and prof were talking about different things - or maybe stuff we won't learn until a higher level quantum class (I think this is the case). Anyway...

The professor version:
Bosons can all occupy the same ground state. i.e. you have ten bosons all ten can fit in n=1 since the exclusion principle doesn't apply to bosons. Thus you'd have E=10E₀.

Fermions adhere to the exclusion principle. Spin-1/2 can only have two per energy level, spin-3/2 four per level, etc.

TA version:
He brought up 2S+1. With this he said it applies to all particles - bosons and fermions. So something with spin-1 can only have 2(1)+1 = 3 per level. This goes against what we learned from the professor. Then the TA mentioned that maybe we only deal with relativistic particles and the 2S+1 doesn't apply.

I can see 2S+1 being used to find out how many possible spin configurations there are, but it breaks down trying to use it on bosons when filling energy levels.

So...what is this 2S+1 used for? Are both people correct?

 

[G01]

2S+1 is the number of possible Sz values for both bosons and fermions.

The difference is how the particles are allowed to fill the different states. Multiple bosons can be in the same state. Thus you could have two spin one particles with the same energy and spin quantum numbers.

However, no two fermions can have the same set of quantum numbers, thus an arbitrary state can only have one fermion in it at any given time.

 

[erok81]

So it looks like the TA had it wrong then. Does non/relativistic particles make a difference how it it used?

The TA used the 2S+1 like this. If you have the same 10 spin-1 bosons you can only fit 3 per level. So with ten you'd up with n=1 has 3, n=2 has 3, n=3 has three, and n=4 has one.

Is there ever a case when that is true?

 

[G01]

The TA used the 2S+1 like this. If you have the same 10 spin-1 bosons you can only fit 3 per level. So with ten you'd up with n=1 has 3, n=2 has 3, n=3 has three, and n=4 has one.

(Still thinking in terms of an infinite well system.)

Hmm. Maybe you misunderstood the TA? In both cases there are 2s+1 distinct states for each energy.

In our square well example with a spin 1 boson:

There are three distinct states with energy E_n. (In general 2s+1 distinct states with energy E_n for spin s particles.) However, it is possible that two bosons can be in any given one of those states at the same time.

With fermions there will still be 2s+1 distinct states, but only one fermion can be in a state at any given time.

These particle statistics rules hold for relativistic particles, too. In fact, in relativistic quantum field theory, you can prove that any integer spin particle has to be a boson and that any half integer spin particle has to be a fermion. So, in fact, these rules are not only still valid in the relativistic theory, but are actually a result of combining special relativity and quantum mechanics.