A box containing a particle is divided into a right and left compartment by a thin partition. If the particle is known to be on the right (left) side with certainty, the state is represented by the position eigenket | R > (| L>), where we have neglected spatial variations within each half of the box. The most general state vector can then be written as | α >) = |R> <R| α > + | L) (L| α >, where (R | α > and < L | α > can be regarded as “wave functions.” The particle can tunnel through the partition; this tunneling effect is characterized by the Hamiltonian H = ∆ (| L) (R | + | R > (L |), where ∆ is a real number with the dimension of energy.

a. Find the normalized energy eigenkets. What are the corresponding energy eigenvalues?

b. In the Schrodinger picture the base kets | R > and | L > are fixed, and the state vector moves with time. Suppose the system is represented by | α > as given above at t = 0. Find the state vector | α, t₀ = 0; t> for r > 0 by applying the appropriate time-evolution operator to | α >.

c. Suppose at r = 0 the particle is on the right side with certainty. What is the probability for observing the particle on the left side as a function of time?

d. Write down the coupled Schrodinger equations for the wave functions <R | α, t₀ = 0; t) and (L | α, t₀ = 0; t). Show that the solutions to the coupled Schrodinger equations are just what you expect from (b).

e. Suppose the printer made an error and wrote H as H= ∆ | L > < R|. By explicitly solving the most general time-evolution problem with this Hamiltonian, show that probability conservation is violated.