Let E < U0 .
Define k2 = (2m/ħ2)E, k02 = (2m/ħ2)U0, ρ2 = (2m/ħ2)(U0 - E) = k02 - k2.
For plane wave solutions we found a phase shift upon reflection.
Is this phase shift related to some observable effect?
Let us construct a wave packet using only k < k0, incident from the left. Let g(k) = 0 for k ≥ 0. For each plane wave we have
A1'/A1 = (k - iρ)/(k + iρ) = [(k2 + ρ2)1/2exp(-iθ(k))]/[(k2 + ρ2)1/2exp(iθ(k))] = exp(-i2θ(k)),
with tanθ(k) = ρ/k = (k02 - k2)1/2/k.
In region 1, at t = 0, we may write the wave packet as
Ψ(x,0) = (2π)-1/2∫g(k) [exp(ikx) + exp(-i2θ(k)) exp(-ikx)]dk.
In region 1 U = 0, the wave packet represents a free particle.
For a free particle we have ω(k) = ħk2/(2m) and
Ψ(x,t) = (2π)-1/2∫g(k) exp(i(kx - ω(k)t))dk + (2π)-1/2∫g(k) exp(-i(kx + ω(k)t + 2θ(k))dk
Ψ(x,t) = (2π)-1/2∫g(k) exp(i(kx - ω(k)t))dk + (2π)-1/2∫g'(k') exp(i(k'x - ω(k)t - 2θ(k'))dk'
with k = -k and g(k) = g(k). We again assume that g(k) has a pronounced peak at some k = ki,.
Then g(k) has a pronounced peak at k = -ki.
Let us choose g(k) to be real, a(k) = 0. For the centers of the incident and the reflected wave packets we then have
xi(0) = -dα(k)/dk|ki = 0, xi(t) = dω/dk|k=ki = ħkit/m,
xr(t) = dω/dk'|k'=-ki + 2dθ/dk'|k'=-ki = -ħkit/m - 2dθ/dk|k=-ki.
We have to evaluate dθ/dk. We know tanθ in terms of k.
With dtanθ/dθ = (1 + tan2θ) = k02/k2 and dtanθ/dk = -(k02 - k2)1/2/k2 we have
dθ/dk = -(k02 - k2)-1/2
Therefore xr(t) = -ħkit/m + 2/(k02 - ki2)1/2.
The incident and the reflected wave packet exist only in the region x < 0.
times t << 0, xi moves with constant velocity to the right, xr
is positive and does not lie in region 1, there is no reflected wave.
(The incident and reflected waves, by definition, only exist in region 1.) For times t >> 0,
xi is positive and does not lie in region 1, there is no incident
moves with constant velocity to the left. But the reflection has introduced a time
delay T. We have
xr(t) = -v(t - T), v = ħki/m, T = (2m/ħki)(k02 - ki2)-1/2.
This time delay is a direct consequence of the phase shift that we found for each plane wave solution. The particle is not instantaneously reflected. This is in principle a measurable effect.
When a wave packet encounters a potential step U0 and it contains plane wave components with E > U0, then it is partially reflected and partially transmitted. If it contains only plane wave components with E < U0, then it is totally reflected. But for some time ΔT the wave packet penetrates into the classically forbidden region. This results in a time delay upon reflection.