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Neutrino/schrodinger eq problem

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  1. Apr 9, 2013 #1
    Hey,

    I'm a high school student from Europe and my final paper is on Neutrino oscillations.

    I practiced some basic quantum states(qbit), but i find it much harder for this neutrino problem.

    I translated it in hope that some of you could give me some pointers. I left some parts of the theory(not needed to solve eq's) out so hope it is still clear.

    I ATTACHED IT!

    I hope someone can help,

    thanx

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



    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

    • phy.png
      phy.png
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  2. jcsd
  3. Apr 9, 2013 #2

    DrClaude

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    Staff: Mentor

    Hi Jenny!

    Could you please post the Hamiltionian you are studying?

    Also, what have you worked out up now ("The attempt at a solution")?
     
  4. Apr 9, 2013 #3
    Hey drClaude,

    There is given that we start with an statonairy eigenstate of the Hamiltonian,

    H|v1} = E1|v1}

    I will try to scan in how i tried the problem
     
    Last edited: Apr 9, 2013
  5. Apr 9, 2013 #4
    for a: this is what i have so far(attachment)

    and then i would fill in a(0)=1

    so,

    1=a(t)*e^0=a(t)
    a(t)=1
     

    Attached Files:

    • ant.png
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    Last edited: Apr 9, 2013
  6. Apr 9, 2013 #5

    DrClaude

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    Staff: Mentor

    I see, you are not told what the Hamiltonian looks like!

    Your start is good, although I don't see why you are considering that the wave function is a two-component vector.

    To continue, you should use the initial condition for ##a##, and probably also the time-independent Schrödinger equation in the quote above.
     
  7. Apr 9, 2013 #6
    Thanx! I'm not sure why i used the two component vector... but i see your point it;s unnecessary
    as there is only one component

    i don;t get tho what i'm suppost to do with the time-independent Schrödinger equation from the quote
     
  8. Apr 9, 2013 #7

    DrClaude

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    What is the right-hand-side of the time-dependent Schrödinger equation?
     
  9. Apr 9, 2013 #8
    H|ψ(t)}?
    is that equal to H|v1}

    I don't get tho why i just can't say after i found a(t)=a_0 e^(-i(A/h_bar)t)
    with the given condition condition a(0)=1

    1=a*e^(0)=a

    so a(t)=1

    hmmm, maybe i have to take a better look at your hint before i keep asking questions

    i will gve it an other go
     
  10. Apr 9, 2013 #9

    DrClaude

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    $$
    i \hbar \frac{\partial | \psi(t) \rangle}{\partial t} = \hat{H} \psi(t) = \hat{H}( a(t) | \nu_1 \rangle) = a(t) \hat{H}| \nu_1 \rangle
    $$

    You have two unknowns in there that need to be determined, ##a_0## and ##A##. This is done using the equation I just wrote above and the condition that ##a(0) = 1##.
     
  11. Apr 9, 2013 #10
    thanx DrClaude,

    i'm going to give it another try!
     
  12. Apr 10, 2013 #11
    this is what i have so far
     

    Attached Files:

    Last edited: Apr 10, 2013
  13. Apr 10, 2013 #12

    DrClaude

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    Staff: Mentor

    Great, you have solved the first part:
    $$
    a(t) =e^{-i E_1 t / \hbar}
    $$
    You can move on to B.
     
  14. Apr 10, 2013 #13
    YEY! thanx
     
  15. Apr 10, 2013 #14
    is it ok if i used the norming conditing to solve b?
     

    Attached Files:

  16. Apr 10, 2013 #15
    no longer upside down
     

    Attached Files:

  17. Apr 10, 2013 #16

    DrClaude

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    I think you have to.

    In part B, it is not clear what ##\alpha## and ##\beta## are. I know what you mean, but I don't think it is a rigourous way of solving the problem. You should start by the definition of the probability of finding the system in a given state.

    I will have a look at part C later.
     
  18. Apr 10, 2013 #17
    I see what you mean with alpha and beta

    thanx!
     
  19. Apr 10, 2013 #18
    I think i solve c, or almost solved c

    the thing i'm a bit worried about is that in my answer i get a -sin(alpha)

    Btw i mistyped the assignment of c, the condition is |ψ(0)} = |v(μ)}
    (instead of |ψ(0)} = |v(e)})
     

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