Solving Fermions: Finding States & Energies

[guidob]

Hello. I found this forum when i was looking for some help with the following problem:

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There are 3 neutrons (s=1/2). The hamiltonian of the system is:

H = S^2 - Sz^2 - (3/2)hbar^2

I need to found the possible states and energies.
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The functions must be antisymetrical, right? But there is no spatial dependence in H.
So, the spin functions must be antisymetrical?
In that case, I don't know how do spin functions look like.
(S=S1+S2+S3 and Sz=Sz1+... are the operators of the total spin).

I'd be really greatfull if somebody help me with this.

 

[malawi_glenn]

You can start with the 2-spin (1/2) wave functions (you must know how they look like ritght?) Then using clebsh gordans, find all the possible 3--spin (1/2) wave functions.

Then you can look which are totally antisymmetric w.r.t particle exhange, and which are eigenfunctions of that hamiltionian etc.

 

[Meir Achuz]

There is no antisymmetric combination of three spin 1/2 particles.
You have to couple a mixed symmetry spin state with a mixed symmetry spatial state,
or couple symmetric spin and space states.
Assuming that H has no spatial component, you can just put S=3/2 and S=1/2 into your formula with all possible S_z.

 

[malawi_glenn]

yes, I forgot to add that, you need some orbital angular momentum to get total antisymetric.

But you don't have any spatial dependence in that hamiltonian, so just do what clem told you.

 

[guidob]

Thank you malawi and clem. I'll try that.

So, i must consider all the S=3/2 and S=1/2, with any Sz, as possible states?

You have been very helpful.

Guido.