Analyzing a Particle in a Box with 2D Space

kreil
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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,

|\alpha> = \alpha_R |R> + \alpha_L |L>

The particle can tunnel through the partition, described by the Hamiltonian,

H = \Delta ( |R><L| + |L>< R|)

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:

\hat H = \hat 1 \hat H \hat 1 = \Sigma |L>< L|\hat H |R>< R|

So the matrix elements are given by <L|\hat H |R> correct?
 
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Correct.
...what is your question?
 
kreil said:
So the matrix elements are given by <L|\hat H |R> 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.
 
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?
 
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.)
 
To solve this, I first used the units to work out that a= m* a/m, i.e. t=z/λ. This would allow you to determine the time duration within an interval section by section and then add this to the previous ones to obtain the age of the respective layer. However, this would require a constant thickness per year for each interval. However, since this is most likely not the case, my next consideration was that the age must be the integral of a 1/λ(z) function, which I cannot model.
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