Let E < U_{0} .

Define k^{2}
= (2m/ħ^{2})E, k_{0}^{2}
= (2m/ħ^{2})U_{0},
ρ^{2}
= (2m/ħ^{2})(U_{0} - E) = k_{0}^{2}
- k^{2}.

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 < k

A

with tanθ(k) = ρ/k = (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) = ħk^{2}/(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

or

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 = k_{i},.

Then g’(k^{’})
has a pronounced peak at k^{ ’}= -k_{i}.

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

x_{i}(0) = -dα(k)/dk|_{ki} = 0, x_{i}(t) = dω/dk|_{k=ki}
= ħk_{i}t/m,

x_{r}(t) = dω/dk'|_{k'=-ki} + 2dθ/dk'|_{k'=-ki }=
-ħk_{i}t/m - 2dθ/dk|_{k=-ki}.

We have to evaluate dθ/dk.
We know tanθ in terms of k.

With dtanθ/dθ = (1 + tan^{2}θ) = k_{0}^{2}/k^{2}
and dtanθ/dk = -(k_{0}^{2} - k^{2})^{1/2}/k^{2}
we have

dθ/dk = -(k_{0}^{2} - k^{2})^{-1/2
}Therefore x_{r}(t) = -ħk_{i}t/m + 2/(k_{0}^{2}
- k_{i}^{2})^{1/2}.

The incident and the reflected wave packet exist only in the region x < 0.
For
times t << 0, x_{i} moves with constant velocity to the right, x_{r}
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,
x_{i} is positive and does not lie in region 1, there is no incident
wave, x_{r
}moves with constant velocity to the left. But the reflection has introduced a time
delay T. We have

x_{r}(t) = -v(t - T), v = ħk_{i}/m, T =
(2m/ħk_{i})(k_{0}^{2}
- k_{i}^{2})^{-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 U_{0} and it contains
plane wave components with E > U_{0}, then it is partially reflected and
partially transmitted. If it contains only plane wave components with E <
U_{0},
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.