The case of an infinite square well whose right wall expands at a constant velocity (v) ca...

The case of an infinite square well whose right wall expands at a constant velocity (v) can be solved exactly2.  A complete set of solutions is

Equation 1

where w(t) ≡ a + vt is the (instantaneous) width of the well and E'n ≡ n2π2h2/2ma2 is the nth allowed energy of the original well (width a). The general solution is a linear combination of the Φ's:

the coefficients cn are independent of t.

(a) Check that Equation 1 satisfies the time-dependent Schròdinger equation, with the appropriate boundary conditions.

(b) Suppose a particle starts out (t = 0) in the ground state of the initial well:

Show that the expansion coefficients can be written in the form

where α = mva/2 π 2h is a dimensionless measure of the speed with which the well expands. (Unfortunately, this integral cannot be evaluated in terms of elementary functions.)

(c)  Suppose we allow the well to expand to twice its original width, so the "external" time is given by w(Te) = 2a. The "internal" time is the period of the time-dependent exponential factor in the (initial) ground state. Determine Te and Ti, and show that the adiabatic regime corresponds to a « 1, so that exp(—iαz2) ≡ 1 over the domain of integration. Use this to determine the expansion coefficients, cn. Construct Ψ(x, t), and confirm that it is consistent with the adiabatic theorem.

(d) Show that the phase factor in Ψ (x, t) can be written in the form

where En(t) = n2π2h2/2mw2 is the instantaneous eigenvalue, at time t. Comment on this result.