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QM: system with two lin. independent states

  1. Dec 28, 2007 #1
    [SOLVED] QM: system with two lin. independent states

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

    Imagine a system with just two linear independent states:

    [tex]|1>=(1,0)[/tex] and [tex]|2>=(0,1)[/tex] (these are actually column matrices, but I don't know how to type those in tex)

    [tex]|\Psi>=a|1>+b|2>=(a,b)[/tex], also [tex]|a|^2+|b|^2=1[/tex]

    suppose the hamiltonian is a 2x2 matrix with entries j,g above and g,j below, [tex](g,j \in R>0)[/tex].

    The time-independent schroding equation reads

    [tex]H |\Psi>=i h/(2 pi) d/dt(|\Psi>)[/tex]

    a) find the eigenvalues and eigenvectors of this hamiltonian
    b) suppose the system starts out at t=0 in |1>, what is the state at time t?

    3. The attempt at a solution

    I thought of solving the time-independent SE;

    [tex]ja+gb= i h/(2 pi) d/dt(a)[/tex]
    [tex]ga+jb= i h/(2 pi) d/dt(b)[/tex]

    but does this mean g=0? And if so, how do I get any further. I'm stuck.
     
    Last edited: Dec 28, 2007
  2. jcsd
  3. Dec 28, 2007 #2

    Dick

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    Change to the basis where H is diagonal.
     
  4. Jan 1, 2008 #3
    solved it with a friend today =)
     
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