How Do Eigenstates of a Spin System Evolve in Time?

AI Thread Summary
The discussion centers on understanding the time evolution of eigenstates in a spin system within quantum physics. The user is struggling with a homework problem involving eigenstates and their time evolution, specifically needing to verify that certain states are eigenstates and how they evolve according to the Schrödinger equation. Key points include the necessity of solving the Schrödinger equation to determine the time dependence of the coefficients in the state vector. Participants suggest using the time-dependent form of the Schrödinger equation and substituting previously derived results to find a solvable differential equation. The user expresses gratitude for the guidance and feels more confident about passing the course.
Nafreyu
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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...
 
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\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).
 
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 :blushing:
 
you first need to verify \psi_1 and \psi_2 are eigenstates.

what is \hat{H} \psi_1?
 
Ok, so now I proved that they are eigenstates. What about the time evoution then?
 
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.
 
Great, thank you! That's easier than I thought it would be.. So maybe I can pass the course after all :wink: Thanks a lot!
 
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