Solving Energy Levels of 2-Spin 1/2 System

ghostflame
Messages
2
Reaction score
0

Homework Statement



Find the energy levels of a 2-spin 1/2 system with spinoperators S1 and S2 in an external magnetic field. The hamiltonian is of the form,

H= A ( 1-\frac{2S_{1}}{h} . \frac{2 S_{2}}{h} )+ \frac{\mu B}{h}(S_{1,z}+S_{2,z})

The h is a h-bar, constants A, B, and S1 and S2 the spin operators

Homework Equations



I have to solve the equation H l\psi> = El\psi>

The Attempt at a Solution



The spin system is has a basis, l\uparrow\uparrow>,\left| \uparrow\downarrow>,\left|\downarrow\uparrow>,\left|\downarrow\downarrow>

so any \left| \psi> is a linear combination of the basis above, but i don't know how i can calculate the eigenvalues of the above equation. I have a feeling i have to use the Pauli matrices but iam not sure. Anyone has an idea? It should be a 3 level system...
 
Physics news on Phys.org
I know the matrices S1 and S2 commute, is also know that S1,z + S2,z = S,z

couldn't that help?
 
Why not try expanding your wavefunction into the spin basis and then using that to calculate H|\psi\rangle? Under what circumstances is your result a constant multiple of |\psi\rangle?
 
Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
The value of H equals ## 10^{3}## in natural units, According to : https://en.wikipedia.org/wiki/Natural_units, ## t \sim 10^{-21} sec = 10^{21} Hz ##, and since ## \text{GeV} \sim 10^{24} \text{Hz } ##, ## GeV \sim 10^{24} \times 10^{-21} = 10^3 ## in natural units. So is this conversion correct? Also in the above formula, can I convert H to that natural units , since it’s a constant, while keeping k in Hz ?
Back
Top