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…