How Do You Solve the Hamiltonian for a Two-Electron Ferromagnet?

[vputz]

Homework Statement

Okay, this would be easy if it hadn't been 15 years since undergrad quantum. Here goes.

I'm finding the energy spectrum of a Heisenberg "two-electron ferromagnet", if you will, with a Hamiltonian described by

$$H=-J\hat{S_1}\cdot\hat{S_2}-h(\hat{S_{z1}}+\hat{S_{z2}})$$

Homework Equations

The Attempt at a Solution

Well, after a while of dusting off my brain and groveling to fellow students, I figured out that

$$(\hat{S_1}+\hat{S_2})^2 = \hat{S_1}^2 + \hat{S_2}^2 + 2\hat{S_1}\cdot\hat{S_2} \rightarrow \hat{S_1}\cdot\hat{S_2} = \frac{1}{2}( (\hat{S_1}+\hat{S_2})^2 -\hat{S_1}^2 -\hat{S_2}^2 )$$

So my Hamiltonian is now

$$H = -\frac{1}{2}J((\hat{S_1}+\hat{S_2})^2 - \hat{S_1}^2 -\hat{S_2}^2 ) - h(\hat{S_{z1}}+\hat{S_{z2}})$$

Okay. Now, the eigenvalues of $$\hat{S}^2$$ are $$s(s+1)$$ (we're doing the usual $$\hbar=1$$ trick). And the eigenvalues of $$\hat{S_z}$$ are $$m$$. And I know that electrons have $$s=\frac{1}{2}$$ and $$m=-s...s$$ in integer steps.

So... it should just be a matter of plugging in possible values for, er, s&m, so to speak. But the $$(\hat{S_1}+\hat{S_2})^2$$ term confuses me. My gut feeling is to treat that as an $$\hat{S}^2$$ term but use values $$-1,0,1$$ as possible values of $$\hat{S_1}+\hat{S_2}$$. Is that the right way to handle it?

 

[nrqed]

Homework Statement

Okay, this would be easy if it hadn't been 15 years since undergrad quantum. Here goes.

I'm finding the energy spectrum of a Heisenberg "two-electron ferromagnet", if you will, with a Hamiltonian described by

$$H=-J\hat{S_1}\cdot\hat{S_2}-h(\hat{S_{z1}}+\hat{S_{z2}})$$

Homework Equations

The Attempt at a Solution

Well, after a while of dusting off my brain and groveling to fellow students, I figured out that

$$(\hat{S_1}+\hat{S_2})^2 = \hat{S_1}^2 + \hat{S_2}^2 + 2\hat{S_1}\cdot\hat{S_2} \rightarrow \hat{S_1}\cdot\hat{S_2} = \frac{1}{2}( (\hat{S_1}+\hat{S_2})^2 -\hat{S_1}^2 -\hat{S_2}^2 )$$

So my Hamiltonian is now

$$H = -\frac{1}{2}J((\hat{S_1}+\hat{S_2})^2 - \hat{S_1}^2 -\hat{S_2}^2 ) - h(\hat{S_{z1}}+\hat{S_{z2}})$$

Okay. Now, the eigenvalues of $$\hat{S}^2$$ are $$s(s+1)$$ (we're doing the usual $$\hbar=1$$ trick). And the eigenvalues of $$\hat{S_z}$$ are $$m$$. And I know that electrons have $$s=\frac{1}{2}$$ and $$m=-s...s$$ in integer steps.

So... it should just be a matter of plugging in possible values for, er, s&m, so to speak. But the $$(\hat{S_1}+\hat{S_2})^2$$ term confuses me. My gut feeling is to treat that as an $$\hat{S}^2$$ term but use values $$-1,0,1$$ as possible values of $$\hat{S_1}+\hat{S_2}$$. Is that the right way to handle it?

yes, that's roughly right.

Your H may be written as $-J/2 ( S_{tot}^2 - S_1^2 - S_2^2 -h S_{tot,z})$
This is diagonal if you use for basis the three spin 1 states. Applying $S_{tot}^2$will give $1 \times (1+1) \hbar^2 = 2 \hbar^2$ for any of the spin 1 states. Applying S_1^2 or S_2^2 will give $1/2(1/2+1) hbar^2 = 3/4 \hbar^2$. The only term that will distinguish between the three S=1 states is the $S_{tot,z}$ operator that gives $m_{tot} \hbar$.