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}^2will 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.