# Time evolution (quantum systems)

1. Nov 28, 2009

### 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!

1. The problem statement, all variables and given/known data

(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?

3. 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....

2. Nov 28, 2009

### MathematicalPhysicist

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

3. Nov 28, 2009

### 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

4. Nov 28, 2009

### latentcorpse

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

what is $\hat{H} \psi_1$?

5. Nov 29, 2009

### Nafreyu

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

6. Nov 29, 2009

### 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.

7. Nov 29, 2009

### 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!