Calculating Energies for a 3 Particle Spin System with the Clebsh-Gordon Table

[Gunthi]

Homework Statement

Find the energies for a 3 spin-1/2 particles with the Hamiltonean:
H=\frac{E_0}{\hbar^2}(\vec{S_1}.\vec{S_3}+\vec{S_2}.\vec{S_3})

The Attempt at a Solution

From the Clebsh-Gordon table one gets all the spin functions:
|\frac{3}{2},\frac{3}{2}>...|\frac{1}{2},\frac{1}{2}>...|\frac{3}{2},-\frac{3}{2}> (6 states in total)

So, to get the matrix elements for the Hamiltonian I tried developing the dot product so I could work directly with the operators i.e.:

\vec{S_1}.\vec{S_3}=S_{1x}S_{3x}+S_{1y}S_{3y}+S_{1z}S_{3z}

Now the problem is that the states as they are defined represent only one particle and the spin operators act on each particle individually and there's no CG table for a 1/2x1/2x1/2 spin addition.

What's the best way to approach this?

 

[vela]

Hint: You can write the Hamiltonian as
$$\hat{H} = \frac{E_0}{\hbar^2}(\vec{S}_1+\vec{S}_2)\cdot\vec{S}_3.$$

 

[Gunthi]

Hint: You can write the Hamiltonian as
$$\hat{H} = \frac{E_0}{\hbar^2}(\vec{S}_1+\vec{S}_2)\cdot\vec{S}_3.$$

Yes but my question is how do these operators act on these states? The basis I'm using only shows the m numbers of two particles because I've already summed the spins of the first two...

 

[CFede]

The usual trick here is to write

S_{1}\cdot S_{2}=\frac{1}{2}((S_1+S_2)^2-S_1^2-S_2^2)

Those operators you DO know how they act.

S_i^2\mid s,m\rangle = s(s+1)\mid s,m\rangle=\frac{3}{4}\mid s,m\rangle

The same goes for (S_1+S_2)^2, since you now its possible values from Clebsch-Gordan coefficients

 

[finitepenguin]

The usual trick here is to write

S_{1}\cdot S_{2}=\frac{1}{2}((S_1+S_2)^2-S_1^2-S_2^2)

Those operators you DO know how they act.

S_i^2\mid s,m\rangle = s(s+1)\mid s,m\rangle=\frac{3}{4}\mid s,m\rangle

The same goes for (S_1+S_2)^2, since you now its possible values from Clebsch-Gordan coefficients

Sorry to revive this thread but I am also trying to work out 3 spin 1/2 particle addition and this comment confused me. How can you find (S_1+S_2)^2 in the |S,m> basis using C.G. coefficients when S represents the total spin S_1+S_2+S_3?

 

[finitepenguin]

I think I partly answered my own question: I guess you can treat (S_1+S_2) + S_3 as an addition of two spin systems? If that is the case then (S_1+S_2)^2|S,m>=h^2(S_1+S_2)(S_1+S_2+1)|S,m>. But then how do we treat S_1 and S_2? And I don't seem to need any C.G. coefficients using this method?

 

[TSny]

You can plow through it by using the 8 "product states" as basis states:

$|+++\rangle\equiv |\uparrow\rangle_1|\uparrow\rangle_2|\uparrow \rangle_3$

$|-++\rangle\equiv |\downarrow\rangle_1|\uparrow\rangle_2|\uparrow \rangle_3$

etc. where $|\uparrow\rangle_1$ is spin up along z for particle 1.

Find the matrix representation of H in this basis and then find the eigenvalues of H. It's not too tedius. Most matrix elements are zero and H will be in block diagonal form. Not very elegant, but it works.

 

[finitepenguin]

I agree that is probably the most straight forward way of going about this, I wanted to try to avoid explicit construction of the Hamiltonian in matrix form though with the hopes of possibly being able to generalize the process. Any hints on the route I was working on the previous post?

Thanks!