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Time evolution (quantum systems)

  1. Nov 28, 2009 #1
    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 [tex]\vec{\varphi}[/tex]1 = [tex]\left\langle[/tex]1[tex]\right,0\rangle[/tex] and [tex]\vec{\varphi}[/tex]2 =[tex]\left\langle[/tex]0[tex]\right,1\rangle[/tex] and Eigenvalues E1 and E2. Verify this. How do these eigenstates evolve in time?

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

    3. The attempt at a solution

    I only know that [tex]\vec{\psi}[/tex] 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. jcsd
  3. Nov 28, 2009 #2

    MathematicalPhysicist

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    Gold Member

    [tex]\psi (x,t)=exp(-iHt/\hbar)\psi (x)[/tex]
    and if [tex]H\psi (x)=\lambda \psi (x)[/tex] the: [tex]exp(-iHt/\hbar)\psi (x)=exp(-it\lambda /\hbar)\psi (x)[/tex].
     
  4. Nov 28, 2009 #3
    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 [tex]\varphi[/tex]1 and [tex]\varphi[/tex]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 :blushing:
     
  5. Nov 28, 2009 #4
    you first need to verify [itex]\psi_1[/itex] and [itex]\psi_2[/itex] are eigenstates.

    what is [itex]\hat{H} \psi_1[/itex]?
     
  6. Nov 29, 2009 #5
    Ok, so now I proved that they are eigenstates. What about the time evoution then?
     
  7. Nov 29, 2009 #6
    well id suggest using the TIME DEPENDENT form of the Schrodinger eqn

    [itex]\hat{H} \psi_1 = i \hbar \frac{\partial \psi_1}{\partial t}[/itex]
    u just worked out [itex]\hat{H} \psi_1[/itex] when showing it was an energy eigenstate so subsititute that back in and rearrange it so you have a differential eqn you can solve.
     
  8. Nov 29, 2009 #7
    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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