2.11 A box containing a particle is divided into a right and a 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 |a) = |R) (R|a)+|L)(L|a), where (Ra) and (La) can be regarded as "wave functions." The particle can tunnel through the partition; this tunneling effect is characterized by the Hamiltonian H=A(|L)(R|+|R) (L|), where A is a real number with the dimension of energy. a. Find the normalized energy eigenkets. What are the corresponding energy eigenvalues? b. In the Schrödinger picture the base kets (R) and (L) are fixed, and the state vector moves with time. Suppose the system is represented by (a) as given above at t = 0. Find the state vector |α, to = 0; t) for t> 0 by applying the appropriate time-evolution operator to (a). c. Suppose at t = : 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 Schrödinger equations for the wave functions (R|a, to = 0; t) and (La, to = 0; t). Show that the solutions to the coupled Schrödinger equations are just what you expect from (b).

2.11 A box containing a particle is divided into a right and a 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 |a) = |R) (R|a)+|L)(L|a), where (Ra) and (La) can be regarded as "wave functions." The particle can tunnel through the partition; this tunneling effect is characterized by the Hamiltonian H=A(|L)(R|+|R) (L|), where A is a real number with the dimension of energy. a. Find the normalized energy eigenkets. What are the corresponding energy eigenvalues? b. In the Schrödinger picture the base kets (R) and (L) are fixed, and the state vector moves with time. Suppose the system is represented by (a) as given above at t = 0. Find the state vector |α, to = 0; t) for t> 0 by applying the appropriate time-evolution operator to (a). c. Suppose at t = : 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 Schrödinger equations for the wave functions (R|a, to = 0; t) and (La, to = 0; t). Show that the solutions to the coupled Schrödinger equations are just what you expect from (b).