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Two level system: tunnel effect

  1. Sep 21, 2014 #1
    Hi!
    I'm having problems with this simple problem:

    Consider two level system: a box divided by a thin membrane. Let
     
  2. jcsd
  3. Sep 21, 2014 #2
    Hi!
    I'm having problems with this simple problem:

    Consider two level system: a box containing one particle, the box is divided by a thin membrane. Let Ψ1 and Ψ2 the probability amplitude (time-dependent only) of being on the left and right side. The particle can tunnel through the partition: so the next condition holds:

    ihΨ2=K*Ψ1 where =Temporal partial derivate
    K=constant

    Using this condition I have to write the most general hamiltonian of the system.

    Of course the wave function is:
    Ψ(t)=Ψ1⎪L〉+ Ψ1⎪R〉

    And the hamiltonian should be a 2x2 matrix. So I think that the actual problem is determine the matrix elements.

    I've tried to use the Schordinger's equation directly:

    ih Ψ(t) = ih Ψ1⎪L〉+ ih Ψ1⎪R〉= H Ψ1⎪L〉+H Ψ1⎪R〉 (Using linearity)
    And using the condition:

    ih Ψ1⎪L〉+ K*Ψ1⎪R〉= H Ψ1⎪L〉+H Ψ1⎪R〉

    And then?

    Help Please!

    Note: I know the general matrix form for the Hamiltonian, but… that is what I'm trying to look for…
     
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