A box containing a particle is divided into right and left compartments by a thin partition. We describe the position of the particle with a 2D space with basis states |R> and |L> according to whether the particle is in the right or left compartment. Thus, a generic state is written,
[tex]|\alpha> = \alpha_R |R> + \alpha_L |L> [/tex]
The particle can tunnel through the partition, described by the Hamiltonian,
[tex]H = \Delta ( |R><L| + |L>< R|)[/tex]
where delta is a real number with units of energy.
1. Write the Hamiltonian in matrix form. What are the energy eigenvalues and eigenvectors?
2. If at t=0 the particle is in the right compartment, what is the probability of finding it in the left compartment at a later time t?
The Attempt at a Solution
I don't really understand how to get it in matrix form. I think I've got to use the basis states in the following manner:
[tex]\hat H = \hat 1 \hat H \hat 1 = \Sigma |L>< L|\hat H |R>< R|[/tex]
So the matrix elements are given by [itex]<L|\hat H |R>[/itex] correct?