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Solving the time dependant schrodinger eqn in dirac (bra ket) notation

  1. Nov 30, 2013 #1
    given:

    at t=0 |PSI(0)> = 1/2 |PSI1> + (SQRT3)/2 |PSI2>


    ---------------------------------------------------------------------

    my attempt so far:

    we can write

    |PSI1> = 1/2 |UP> + 1/2 |DOWN>

    |PSI2> = (SQRT3)/2 |UP> + (SQRT3)/2 |DOWN>

    therefore

    |PSI(0)> = 1/2 |UP> + 1/2 |DOWN> + (SQRT3)/2 |UP> + (SQRT3)/2 |DOWN>



    but then what do i do? the lecturer hasnt shown us how to solve the time DEPENDENT schrodinger eqn in ket notation, infact i dont recall her showing us how to solve the time independant eqn either, only something like this:

    |PSI(x)> = I(x)|PSI> = INTEGRAL BETWEEN -INF AND +INF of dx|x><x|PSI>

    so do i use the Identity to give some sort of integral for |PSI(O)>??

    I'm at a total loss and spent another 2+ hours going around in circles.

    If someone could point me in the right direction, or even suggest a web page that details how to solve the time dependant schrodinger eqn in dirac notation, it would REALLY be appreciated.

    edit

    I found the solution on the web, well kind of, my question needs in terms of base vectors, but;

    at t=o |PSI> = SUM Cn|PSIn>

    at t>0 |PSI> = SUM Cn|PSIn> EXP (-iEt)/h-bar

    where does the EXP term come from?? I know where it comes from in normal notation but how to get an exp term from |UP>'s and |DOWN>'s ?? and they are only vectors anyway, how can an EXP term be directional?? totally confused.
     
    Last edited: Nov 30, 2013
  2. jcsd
  3. Nov 30, 2013 #2
    it's ok found something in the book

    t=0 |PSI(0)> = SUM αi|PSIi>

    t>0 |PSI(t)> = SUM αi EXP (-iEit/h-bar) |PSIi>

    from

    ih-bar d/dt |PSI> = H |PSI>


    therefore my answer (i think) will simply be

    |PSI(t)> = (1/2 |PSI1> + (SQRT3)/2 |PSI2>)EXP (-iEit/h-bar)

    = (1/2 (|UP> + |DOWN>) + (SQRT3)/2 (|UP> + |DOWN>) ) EXP (-iEit/h-bar)


    or maybe im way off??? will update when the answers come in to help others
     
    Last edited: Nov 30, 2013
  4. Nov 30, 2013 #3

    vela

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    The homework template is there for a reason. Could you please provide the complete problem statement?
     
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