Solve the quatum mechANIC Question

Problem 1.1 Consider a particle and two normalized energy eigenfunctions ψ1(x) and ψ2(x) corresponding to the eigenvalues E1 = E2. Assume that the eigenfunctions vanish outside the two non-overlapping regions 1 and 2 respectively. (a) Show that, if the particle is initially in region 1 then it will stay there forever. (b) If, initially, the particle is in the state with wave function ψ(x, 0) = √ 1 2 [ψ1(x) + ψ2(x)] show that the probability density |ψ(x, t)| 2 is independent of time. (c) Now assume that the two regions 1 and 2 overlap partially. Starting with the initial wave function of case (b), show that the probability density is a periodic function of time. (d) Starting with the same initial wave function and assuming that the two eigenfunctions are real and isotropic, take the two partially overlapping regions 1 and 2 to be two concentric spheres of radii R1 > R2. Compute the probability current that flows through 1.

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