PHYS 2053 | HW 5 | Sarah Welch, Nicholas Fleming, Alexander Marshall
Homework V: Quantum Tunneling and Wave Packets
1. Free Particle
Using a wave packet, potential = 0
 Describe the stationary wave function 𝝓.
o It’s localized.
o It has an absolute maximum in the middle, and equal
relative minima on each side.
o It has a real and imaginary part; these elements are out of
phase with each other.
o It has a probability distribution, with the maximum
probability of the particle’s location at the peak amplitude
of the wave packet.
Figure 1: Real part of a wave packet, representing a free particle.
 Vary the energy and describe the change in 𝝓.
o As energy decreases, the local minima on the sides become
smaller until they smooth out completely. The maximum in
the middle retains its amplitude.
o As the energy increases, the wave packet adds additional
local maxima and minima.
o With E ≤ V, the wave completely flattens out.
o Things that don’t change with variation in E:
 Probability density
 Width of the wave packet
 Imaginary and real parts still out of phase
PHYS 2053 | HW 5 | Sarah Welch, Nicholas Fleming, Alexander Marshall
 Explain why this change occurs.
o As the energy increases, more k-vectors are being added, so
more waves are included in the composite wave packet, and
more local maxima/minima are observable.
 Change the potential and describe and explain the resulting
change in 𝝓.
o [Energy constant at 0.7]
o As potential increases, the packet stretches out: fewer
oscillations in the same region.
o As potential decreases, the packet squeezes more
oscillations into the same region
o The width of the packet, probability density, and out-ofphase-ness of the real and imaginary parts remains
unchanged.
 Now make a wave packet and let it evolve several fs. Look at
the magnitude as well as the real part of 𝝓 and describe the
changes. Do the wave function wave fronts move at the same
speed as the packet?
o [Potential = 0, E = 0.7, w = 0.5]
o The wave packet quickly stretches out, with the probability
density stretching as well. (It’s still centered at the greatest
magnitude of the wave packet.)
o The phase velocity is greater than the group velocity;
therefore, the resultant is the elongation of the wave packet.
Figure 2: Real and Imaginary parts of a Gaussian wave packet, illustrating the
spread through several fs.
PHYS 2053 | HW 5 | Sarah Welch, Nicholas Fleming, Alexander Marshall
 Describe the difference between the behavior of a plane wave
and a wave packet.
Plane Wave
Wave Packet
Wavelength
Well-defined
Not well-defined
Energy
Well-defined
Not well-defined
Amplitude
Well-defined
Not-defined
Position
Not localized*
Localized*
*-The position is not localized when the amplitude is well-defined because the wave
is spread through all of space, leading to the uncertainty of the particle’s position.


When the potential and total energy are manipulated, the amplitude is
constant for the plane wave, but the wavelength changes.
The envelope of the wave packet does not evolve through time, but the wave
propogates.
2. Step Potential
 Create a step potential and a deBroglie particle.
 With E ≥ V, the wave packet approaches the step. When it hits
the step, part of the wave is reflected, and part of the wave
continues through the step. The wave function and
probability density change on each side of the step, with them
both falling off more quickly inside the step than on the
reflection side of the step. With increasing energy, the
transmission probability increases and the reflection
probability correspondingly decreases.
 Given an energy value, (pick and show) calculate and verify
the (relative) amplitudes of the incident, reflected, and
transmitted stationary wave functions, and the transmission
and reflection probabilities. (You can choose to show the
summed or separated incident wave parts.)
o Chosen parameters:
 E = 0.7 eV
 V = 0.5 eV
o PhET values:
 R = 0.09
PHYS 2053 | HW 5 | Sarah Welch, Nicholas Fleming, Alexander Marshall
 T = 0.91
o Calculation:
o 𝑘1 = √
2𝑚𝐸
o 𝑘2 = √
o 𝑇=
o 𝑅=
ℏ2
=√
2𝑚
2𝑚(𝐸−𝑉0 )
ℏ2
4𝑘1 𝑘2
(𝑘1 +𝑘2 )2
(𝑘1 −𝑘2 )2
(𝑘1 +𝑘2 )2
=
=
ℏ2
(0.83666)
=√
2𝑚
(0.4472)
ℏ2
4(0.83666∗0.4472)
(0.83666+0.4472)2
(0.83666−0.4472)2
(0.83666+0.4472)2
= 0.908
= 0.092
 Look also at the imaginary parts of 𝝓. Describe and explain
the difference between the imaginary and real parts (phase).
o The imaginary and real parts exhibit the same falling off
pattern, but with slightly different phases. During the
tunneling phenomenon, the phase difference of the
imaginary and real parts decreases—they get closer to
being in sync.
 Increase the step so that V ≥ E. What happens? Compare the
real and imaginary parts. Is it similar to the other case?
Describe and explain.
o With V > E, the T = 0 and R = 1.0.
o The real and imaginary parts shift into and out of phase
with each other.
 Compare the difference in behavior between a plane wave
and a wave packet.
o On the incident side of the step, the plane wave’s real and
imaginary parts oscillate continually from more in phase to
more out of phase.
o On the transmission side of the step, the plane wave’s real
and imaginary parts become “locked together”, traveling
with constant phase shift.
PHYS 2053 | HW 5 | Sarah Welch, Nicholas Fleming, Alexander Marshall
Figure 3: Phase shifts in a plane wave with a step potential.
o The probability density of the plane wave is periodic on the
incident side of the step, and constant on the transmission
side.
o Raising the energy relative to the height of the step shortens
the wavelength, raises the transmission coefficient, and
decreases the amplitude of the probability density as well as
the amplitude of the wave on the incident side of the step.
o An interesting phenomenon happens when E < V: the
transmission coefficient drops to 0, and the real and
imaginary parts on the incident side of the step shift into
and out of phase with each other as the reflected wave
interferes with the incident wave. The probability density is
zero at the nodes, but isn’t immediately zero on the
transmission side of the step (inside the step). It drops off
quickly, but there’s a short distance during which the
PHYS 2053 | HW 5 | Sarah Welch, Nicholas Fleming, Alexander Marshall
deBroglie particle penetrates the step: quantum tunneling.
This phenomenon is also reflected in the graph of the wave
function.
Figure 4: Plane waves exhibiting shifting phases and quantum tunneling with a step
barrier.
3. Barrier Potential
PHYS 2053 | HW 5 | Sarah Welch, Nicholas Fleming, Alexander Marshall
 With E > V, describe how 𝝓 varies over and beyond the
barrier, and explain why.
o As the wave packet crosses the barrier, its wavelength is
stretched. Some of the wave packet is transmitted, and
some is reflected.
o On the other side of the barrier, the transmitted wave
continues and eventually decays, and the probability
density decreases as the wave packet delocalizes.
o This stretching or spreading occurs because the wave
1
packet spreads out proportionally to 2, which can be
𝑡
demonstrated through manipulating integrals deriving from
the Taylor series expansion about zero of 𝜙, resulting in a
probability function distributed over more space to
maintain constant probability.
Figure 5: Behavior of a wave packet across a barrier.
 Now set E ≤ V and describe the change. What is T?
PHYS 2053 | HW 5 | Sarah Welch, Nicholas Fleming, Alexander Marshall
o With E ≤ V, the form of the wave doesn’t look very much
different than with E ≥ V. However, T is now 0, so the wave
on the far side of the barrier has tunneled through and,
consequently, falls off quickly.
Figure 6: Represents the behavior of a wave packet with a barrier, while E ≤ V
 Can you make T ≥ 0? Under which conditions, keeping E ≤ V?
o When the barrier is thin enough, even E < V produces T > 0.
 Construct yourself a barrier like Fig. 3.6 in Morrison.
Determine the ratio of the amplitudes E/A and verify your
expression with the output from the program. You may want
to choose a barrier of 0.8 eV and a width of 1 nm. Choose as
electron energy 0.6 eV.
o Note: the program wouldn’t let us choose energy of 0.6 eV,
so we used 0.61 eV.
𝐸
2𝑖𝑘1 𝑒 −𝑘2
= 𝑖𝑘
𝐴
𝑒 1 (𝑘1 − 𝑘2 )
Where:
𝑘1 = √
2𝑚𝐸
ℏ2
, 𝑘2 = √
Therefore:
𝐸
2.927𝑖 ∗ 1
=
𝐴
𝑒 𝑖∗0.781
2𝑚(𝑉0 −𝐸)
ℏ2
PHYS 2053 | HW 5 | Sarah Welch, Nicholas Fleming, Alexander Marshall
 Successively increase the width of the barrier until there is
no transmission anymore. Plot the transmission coefficient
as function of barrier width. Can you relate your graph to the
analytic expression for the transmission coefficient?
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
Figure 7: A graph of the relationship between the barrier width in nm (x-axis) and
the transmission coefficient (y-axis).
The expression for the transmission coefficient is
4𝑘1 𝑘2
𝑇=
(𝑘1 + 𝑘2 )2
Where:
𝑘1 = √
2𝑚𝐸
ℏ2
, 𝑘2 = √
2𝑚(𝑉0 −𝐸)
ℏ2
.
This leads us to expect an exponential decay in T as barrier width
increases, and the graph confirms our expectation.
 Compare the difference in behavior between a plane wave
and a wave packet when they scatter from the barrier
potential.
o Plane waves act in a similar way when scattering from this
barrier as they do when scattering from the step potential
barrier: they approach and reflect to make a composite
wave on the incident side whose real and imaginary parts
oscillate into and out of phase; on the far side of the step,
the waves which have successfully tunneled through exhibit
PHYS 2053 | HW 5 | Sarah Welch, Nicholas Fleming, Alexander Marshall
o
o
o
o
increased wavelength, decreased amplitude, and a constant
probability density.
When the incident/reflected waves are displayed, the
simulation displays some interesting discontinuities
(pictured below).
The reflected waves also exhibit slightly decreased
magnitude.
The discontinuities occur in either real or imaginary parts
of the wave function.
The probabilities and reflections are smooth leading to
them being observables, which then means they are
measurable.
Figure 8: Discontinuities in reflected plane waves from a potential barrier.