How can the bound states of two spin 1/2 particles be split into two equations?

[kreil]

Homework Statement

Two spin 1/2 particles interact through the spin-dependent potential

$$V(r)=V_1(r)+V_2(r) \sigma ^{(1)} \cdot \sigma^{(2)}$$

Show that the equation determining the bound states can be split into two equations, one having the effective potential $V_1(r)+V_2(2)$ and the other $V_1(r)-3V_2(r)$.

The Attempt at a Solution

I'm really having trouble figuring out how to solve these spin problems, but I think for this one I should use the Schrödinger equation for two particles?

$$\left [ \frac{-\hbar^2}{2m}\left ( \nabla_1^2+\nabla_2^2 \right ) + V(r) \right ] |s,m \rangle = E |s,m\rangle$$

If so, I need to figure out how to quantify $$\sigma ^{(1)} \cdot \sigma^{(2)}$$, which I am unsure how to do.

In addition to this specific problem, any more general remarks/resources about how to solve these spin problems would be appreciated (my book is terrible in this respect)

 

[vela]

You probably saw this trick when calculating the spin-orbit interaction for the hydrogen atom. For this problem, the total spin S is equal to S=S₁+S₂. If you square this equation, you get

$$\mathbf{S}^2 = (\mathbf{S}_1+\mathbf{S}_2)^2 = \mathbf{S}_1^2+\mathbf{S}_2^2+2\mathbf{S}_1\cdot\mathbf{S}_2$$

Solving for the cross term, you get

$$\mathbf{S}_1\cdot\mathbf{S}_2 = \frac{\mathbf{S}^2-\mathbf{S}_1^2-\mathbf{S}_2^2}{2}$$