Recent content by jcharles513

Homework Statement H0 = [2,0,0;0,2,0;0,0,4] H1 = [0,1,0;1,0,1;0,1,0] Find energy eigenvalues to 2nd order. Homework Equations The Attempt at a Solution I know that I need to diagonalize the perturbation in the 2x2 subspace (for my 2 degenerate eignevalues of 2 but I'm not sure...

I diagonalized the pertubation (B-part) and then added it the A-part. This made it diagonal. I'm not sure if that's the right way to do it or not? If not, can you get me started on the correct method?

I used the 3x3 subspace and got eigenvalues 0,A+B, and A-B. Is this correct? And the correct procedure?

Degen. Perturb. I'm confused about how to use degenerate perturbation on this system. My understanding of degenerate perturbation is shaky. I know that I need to take a subspace of this matrix that has the degeneracy and then transform the perturbation of that subspace to this new diagonal...

Thank you. That makes sense. I guess I wasn't searching for the right thing in google. I may have more questions after that but this gets me started.

Homework Statement Consider the so-called Spin Hamiltonian H=AS2Z+B(S2X-S2Y) for a system of spin 1. Show that the Hamiltonian in the SZ basis is the 3x3 matrix: \hbar2*[(A,0,B; 0,0,0; B,0,A)].Find the eigenvalues using degenerate pertubation theory. Homework Equations Spin Pauli MatricesThe...

Good. Thanks for your help.

Ah yes of course. So that means I would have a √2 out front for the normalization. So I would multiply each of my probabilities by two. I get 1/∏ for the ground state and 1/2 for the first excited state for t > 0.

I believe so. The normalized initial wavefunction is just A_0*(1/√2)*2*x'*exp(-x'^2/2) where x' = (xmω/hbar)^(1/4) and A_0 = (1/∏)^(1/4). Am I missing something?

I got the probability for hbarω/2 = 1/(2*∏) and then 3hbarω/2 = 1/4. What did you get?

If that's the probability amplitude than the probability would just be the magnitude of that squared correct? So now all I need is to figure out the form of the wave function before and after. t < 0 the wave function is in the groundstate of the half harmonic oscillator so for x >0 it looks...

If I was given the wavefunction ψ(x). The probability is just calculated <\phi_0|ψ(x)>. For t > 0, Ʃ<\phi_n|ψ(x)>|ψ(x)> will be the expanded wavefunction. This means that to get the different values I would just integrate and multiply by the \phi_n^* to get the probabilities where n = 0,1.

Homework Statement At time t < 0 there is an infinite potential for x<0 and for x>0 the potential is 1/2m*w^2*x^2 (harmonic oscillator potential. Then at time t = 0 the potential is 1/2*m*w^2*x^2 for all x. The particle is in the ground state. Assume t = 0+ = 0- a) what is the probability that...

That makes sense. I hate when the mistake I make is a simple integral mistake. Thanks again. You would think I should know this if I have a final soon. He taught it to us yesterday and testing monday on the final on it. So I've been working on learning it ever since. Thanks, James

Using the results from Britney Spears, 3d density of states = \frac{1}{2{\pi}^{2}}\frac{2m}{{{\hbar}^{2}}}^{3/2}E^{1/2} and integrating from 0 to ∞ of the density of states multiplied by Fermi-Dirac distribution The result is: \frac{1}{{\pi}^{2}} {\frac{2m}{{\hbar}^{2}}}^{3/2}*E_f * ln(2) I...