Pauli Spin Operator Eigenvalues For Two Electron System

[xdrgnh]

I'm studying for a qualifying exam and I see something very strange in the answer key to one of the problems from a past qualifying exam. It appears the sigma^2 for a two electron system has eigenvalues according to the picture below of 4s(s+1) while from my understand of Sakurai it should have eigenvalue of s(s+1). Can anyone shed some light on this, I suspect it is an error.

 

[PeterDonis]

Your image didn't appear to come through. It might be better to link to a reference if you have one.

 

[xdrgnh]

Your image didn't appear to come through. It might be better to link to a reference if you have one.

Yes I see. I have a link to a google drive.

Anyway I can just upload a photo on my desktop?
https://drive.google.com/drive/folders/0B9_oicNQsA7bSFFxT1czal9hclU

if you have access to the google drive it would be under 2009 part 2, the quantum question 2.

 

[xdrgnh]

http://imgur.com/N4AxroC

here is the image. It seems like an error those eigenvalues. Do you concur with me?

 

[PeterDonis]

here is the image.

This doesn't give enough context. Do you have an actual reference?

 

[xdrgnh]

This doesn't give enough context. Do you have an actual reference?

Consider two s = 1/2 spins. Their interaction with each other is described by the Hamiltonian: Hex = A~σ1 · ~σ2 , where A is a positive constant, and ~σ1 and ~σ2 are vectors with components given by the Pauli matrices. In addition, a magnetic field B~ is applied to spin #1 only, so that the Zeeman Hamiltonian of the system is HZ = gµBB~ · ~σ1 . Here µB is the Bohr magneton and g is the g-factor. This is the problem.

 

[xdrgnh]

(a) Assume that a static field is applied, B~ = Bzˆ where ˆz is the unit vector along the z-axis. Find the eigenenergies of the system. Plot the spectrum as a function of B for fixed A, labeling all relevant features. Also find the eigenfunctions for B = 0 and in the limit of infinitely large B. (40 points) this is this is the question they are referring to. Also here is the image of the solution which I think has an error in it.

http://imgur.com/N4AxroC

 

[blue_leaf77]

It appears the sigma^2 for a two electron system has eigenvalues according to the picture below of 4s(s+1) while from my understand of Sakurai it should have eigenvalue of s(s+1). Can anyone shed some light on this, I suspect it is an error.

It's $\hat S^2$ whose eigenvalue is $\hbar^2 s(s+1)$. To get the eigenvalue of $\hat \sigma^2$, use $\hat {\mathbf S} = \frac{1}{2} \hbar \hat {\mathbf \sigma}$.