Eigenstates of composite spin hamiltoninan

[center o bass]

A spin-3/2 particle (labeled 1) and a spin-1/2 particle (labeled 2) are interacting via
the Hamiltonian

$$\lambda \vec S_1 \cdot \vec S_2 + g(S_{1z} + S_{2z})^2$$

where S1 is the spin operator of particle 1 (the spin-3/2) and S2 is the spin operator
of particle 2 (the spin-1/2). λ and g are positive constants (λ ≫ g).

I am asked to find the eigenstates and the corresponding energies for this hamiltonian (neglecting any spatial degrees of freedom).

Since we have a two particle system I guess one must consider addition of angular momenta and the CB-coefficients, but I am really unsure about this hamiltonian. Should one reformulate in in terms of total spin operators? Do we know how the total spin operator S acts without squaring it?

How do I get started here?

 

[susskind_leon]

Just one question: How exactly is the scalar product between S1 (spin 3/2) and S2 (spin 1/2) defined? S1 should be a vector having 4 entries, S2 a vector having 2 entries.

 

[center o bass]

S1 should be a vector having 4 entries, S2 a vector having 2 entries.

Why is that? Is'nt the hamiltonian originally defined in a classical sense so that we think of theese vectors as classical 3-dimensional spin vectors? I have the habit of doing so until they act on quantum states, then I think of them as QM operators.

I guess you could define that scalar product as the product of the direct products

$$(S \otimes I)(I \otimes S),$$

furthermore we have the relation

$$S_{tot}^2 = S_1^2 + S_2^2 + 2 \vec S_1 \cdot \vec S_2 = (S \otimes I)^2 + (I \otimes S)^2 + 2 (S \otimes I)(I\otimes S)$$

which maybe could be used to define the action of

$$(S \otimes I)(I \otimes S)$$

on a state?

 

[vela]

$$S_{tot}^2 = S_1^2 + S_2^2 + 2 \vec S_1 \cdot \vec S_2$$

Use this relation and solve for $\vec S_1 \cdot \vec S_2$.

 

[vela]

Just one question: How exactly is the scalar product between S1 (spin 3/2) and S2 (spin 1/2) defined? S1 should be a vector having 4 entries, S2 a vector having 2 entries.

You're confusing the states with the operators.

 

[susskind_leon]

Yes, you're right, sorry!

 

[center o bass]

Use this relation and solve for $\vec S_1 \cdot \vec S_2$.

Why would I want to do that? :) I'm a little bit bothered by the fact that I can not express the Hamiltonian purely in terms of the total spin and the total z spin. I could rewrite it as

$$H = \frac{\lambda}{2} ( S_{tot}^2 - S_1^2 - S_2^2 ) + g S_{ztot}^2$$

but then I have the pesky S1 and S2 operators in there. Btw is it correct to assume that any such composite spin state of particle 1 and two can be written as

$$|s m\rangle = \sum_{m1,m2} C |\frac{3}2,m_1\rangle |\frac{1}2, m_2\rangle$$

where C are the GB coeffs?

 

[vela]

Why would I want to do that? :) I'm a little bit bothered by the fact that I can not express the Hamiltonian purely in terms of the total spin and the total z spin. I could rewrite it as

$$H = \frac{\lambda}{2} ( S_{tot}^2 - S_1^2 - S_2^2 ) + g S_{ztot}^2$$

but then I have the pesky S1 and S2 operators in there.

That's not a problem because S₁², S₂², S², and S_z all commute with each other.

If you think about it, you can't really describe everything in terms of just S² and S_z. You have two quantum numbers for each particle, so the combined state needs four quantum numbers. It's not surprising that the Hamiltonian will depend on S₁² and S₂² as well.

Btw is it correct to assume that any such composite spin state of particle 1 and two can be written as

$$|s m\rangle = \sum_{m1,m2} C |\frac{3}2,m_1\rangle |\frac{1}2, m_2\rangle$$

where C are the GB coeffs?

Yes. (So far you've referred to them as CB and GB coefficients. I assume you mean the Clebsch-Gordan coefficients. )

 

[center o bass]

[QOUTE]Yes. (So far you've referred to them as CB and GB coefficients. I assume you mean the Clebsch-Gordan coefficients. )[/QUOTE]

Hehe, the _CG_ coefficients ;P If i assume that this is the case I get

$$H|sm\rangle = \left[\frac{\lambda}{2}(S_{tot}^2 - S_1^2 -S_2^2) +gS_{ztot}^2\right] |sm\rangle$$

$$= \left[\frac{\lambda}2 (\hbar^2s(s+1) - S_1^2 - S_2^2) + g\hbar^2 m\right] \sum C |\frac{3}2, m_1\rangle|\frac{1}{2} m_2\rangle$$
$$= \left[ \lambda \hbar^2 \left( \frac{2 s(s+1) - 9}{4}\right) + gm\hbar \right] |sm\rangle.$$

Is the states and the allowed energies now dictated by the allowed s and m's given s_1 and s_2?

 

[vela]

Yup.