PHYS 210 ASSIGNMENT 8: HELP

So who's gotten part 3??

I'm confused about the value of mu (the scalar). I'm guessing that magnetic moment is equal to a Pauli spin multiplied by some sort of combination of the basis. How exactly are they combined? I do not know. I think we're supposed to show that the 4x4 matrices given are representations of the 4 quantum states. How do we verify this, it seems like they're just combinations of the quantum states?! S44535078 01:01, 3 November 2008 (PST)

Here <math>\mu</math> is a scalar which, when multiplied by the scalar magnitude of the magnetic field B, yields the scalar Zeeman energy E_Z. This scalar value is then multiplied into the spin operator (Pauli matrix, for spin 1/2) corresponding to the field direction -- generally the z direction, to keep things simple -- so <math>\sigma_3</math> -- to give the Zeeman part of the Hamiltonian operator (a matrix), which in turn operates on state vectors representing different quantum states. In the diagonal basis, the effect of this operator will be simply to multiply the state vector by the corresponding scaler energy eigenvalue. So for a simple spin 1/2 particle in a magnetic field along the z direction, <math>H_Z = \mu B \sigma_3</math> so that <math>H_Z \vert \uparrow \rangle = \mu B \vert \uparrow \rangle</math> and <math>H_Z \vert \downarrow \rangle = - \mu B \vert \downarrow \rangle</math>. Check it out. Then build a 4x4 version for the two-spin system. Although you can make a completely general version of the Zeeman Hamiltonian that does not assume that the magnetic field is along the z direction, it gains you nothing in the calculation of the Breit-Rabi eigenvalues, so I wouldn't bother. -- Jess 09:42, 3 November 2008 (PST)