How Do Eigenstates of a Spin System Evolve in Time?

[Nafreyu]

Hi, I'm totally lost here...Quantum physics seems to be just incomprehensible to me! Hope someone can help me out! That would be great!

Homework Statement

(a) A spin system with 2 possible states, described by
(E1 0)=H
(0 E2)
with eigenstates \vec{\varphi}₁ = \left\langle1\right,0\rangle and \vec{\varphi}₂ =\left\langle0\right,1\rangle and Eigenvalues E₁ and E₂. Verify this. How do these eigenstates evolve in time?

(b) consider the state \vec{\psi} = a₁ \vec{\varphi}₁ + a₂ \vec{\varphi}₂ with real coefficients a₁, a₂ and total probability equal to unity. How does the state \vec{\psi} evolve in time?

The Attempt at a Solution

I only know that \vec{\psi} must solve the Schroedinger equation to show the time dependence of a₁ and a₂ and a₁² + a₂² must be equal to 1. Other than that I'm really totally lost! This is one of 4 tasks I need to finish to pass this course, I can do the other 3, but this one I just don't get. So please help! I would be very grateful...

 

[MathematicalPhysicist]

\psi (x,t)=exp(-iHt/\hbar)\psi (x)
and if H\psi (x)=\lambda \psi (x) the: exp(-iHt/\hbar)\psi (x)=exp(-it\lambda /\hbar)\psi (x).

 

[Nafreyu]

Hi, first of all thanks for your fast answer! But then.. as I said above, I'm totally lost in quantum physics, so I don't quite get your statement. I guess it's about part (a) of my assignment which shows the time evolution. But what happened to \varphi₁ and \varphi₂ ? I'm sorry for my obviously stupid questions but I guess I'm missing any understanding of this quantum system thing. I only need to pass the course and will never need it again, so I hope you could just outline your answer a little more for me! Thanks again

 

[latentcorpse]

you first need to verify \psi_1 and \psi_2 are eigenstates.

what is \hat{H} \psi_1?

 

[Nafreyu]

Ok, so now I proved that they are eigenstates. What about the time evoution then?

 

[latentcorpse]

well id suggest using the TIME DEPENDENT form of the Schrodinger eqn

\hat{H} \psi_1 = i \hbar \frac{\partial \psi_1}{\partial t}
u just worked out \hat{H} \psi_1 when showing it was an energy eigenstate so subsititute that back in and rearrange it so you have a differential eqn you can solve.

 

[Nafreyu]

Great, thank you! That's easier than I thought it would be.. So maybe I can pass the course after all Thanks a lot!