Analyzing a Particle in a Box with 2D Space

[kreil]

Homework Statement

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?

 

[AndyUrquijo]

Correct.
...what is your question?

 

[Chopin]

So the matrix elements are given by [itex]<L|\hat H |R>[/itex] correct?

Not just that one, though. This is a 2-state system, so the Hamiltonian will be a 2x2 matrix. One element will correspond to each of the possible transitions that can happen in the system. The one you've written down is one of them, so there are 3 others.

 

[kreil]

Thanks guys that helped a lot.

For part 2 I assume I should just apply the time evolution operator onto the initial state R and then square the resulting wave function to find the probability?

 

[Chopin]

Yes. It's easiest to decompose your initial state into a superposition of eigenstates of the Hamiltonian, because you know how those evolve (that's why the first part of the problem asked you to find them.)