A particle starts out (at time t = 0) in the Nth state of the infinite square well. Now th...

A particle starts out (at time t = 0) in the Nth state of the infinite square well. Now the "floor" of the well rises temporarily (maybe water leaks in, and then drains out again), so that the potential inside is uniform but time dependent: V0(t), with V0(0) = V0(T) = 0.

(a) Solve for the exact cm (t) using Equation 1, and show that the wave function changes phase, but no transitions occur. Find the phase change, ➢(T), in terms of the function V0(t).

(b) Analyze the same problem in first-order perturbation theory, and compare your answers.

Comment: The same result holds whenever the perturbation simply adds a constant (constant in x, that is, not in t) to the potential; it has nothing to do with the infinite square well, as such. Compare Problem 1.

Equation 1

Problem 1

Suppose you add a constant V0 to the potential energy (by “constant” I mean independent of x as well as t). In classical mechanics this doesn’t change anything, but what about quantum mechanics? Show that the wave function picks up a time-dependent phase factor: exp(-i V0t/ћ). What effect does this have on the expectation value of a dynamical variable?