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 = $$\left\langle$$1$$\right,0\rangle$$ and $$\vec{\varphi}$$₂ =$$\left\langle$$0$$\right,1\rangle$$ 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 $$\varphi$$₁ and $$\varphi$$₂ ? 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 Schrödinger 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!