How to Determine the Hamiltonian for a Two-Level System with Tunneling?

[Juan Carlos]

Hi!
I'm having problems with this simple problem:

Consider two level system: a box divided by a thin membrane. Let

 

[Juan Carlos]

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…