(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

[tex]H=-J\hat{S_1}\cdot\hat{S_2}-h(\hat{S_{z1}}+\hat{S_{z2}})[/tex]

2. Relevant equations

3. The attempt at a solution

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

[tex](\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 ) [/tex]

So my Hamiltonian is now

[tex]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}})[/tex]

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

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Stupid easy question about spin.

**Physics Forums | Science Articles, Homework Help, Discussion**