Web view(It’s still centered at the greatest magnitude of the wave packet.) The phase velocity is greater than the group velocity; ... When it hits the step,

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


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 regiono The width of the packet, probability density, and out-of-

phase-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.


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


T = 0.91o Calculation:

o k1=√ 2mEℏ2 = √ 2mℏ2 (0.83666)

o k 2=√ 2m (E−V 0 )ℏ2

= √ 2mℏ2 (0.4472 )

o T=4 k1k2

(k1+k2)2 =

4 (0.83666∗0.4472)(0.83666+0.4472)2

= 0.908

o R=¿¿ = ¿¿ = 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 di erence in behavior between a plane wave ffand 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.


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


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


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

packet spreads out proportionally to 1t2 , which can be

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?


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.

EA

=2i k1 e

−k2

ei k1(k1−k2)

Where:

k1=√ 2mEℏ2 , k 2=√ 2m(V 0−E)

ℏ2

Therefore:EA

=2.927i∗1e i∗0.781


Successively increase the width of the barrier until there is no transmission anymore. Plot the transmission coe cient ffias function of barrier width. Can you relate your graph to the analytic expression for the transmission coe cient?ffi

0.4 0.6 0.8 1 1.2 1.4 1.60

0.05

0.1

0.15

0.2

0.25

0.3

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

T=4 k1 k2

(k1+k2)2

Where:

k1=√ 2mEℏ2 , k 2=√ 2m(V 0−E)

ℏ2 .

This leads us to expect an exponential decay in T as barrier width increases, and the graph confirms our expectation.

Compare the di erence in behavior between a plane wave ffand 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


increased wavelength, decreased amplitude, and a constant probability density.

o When the incident/reflected waves are displayed, the simulation displays some interesting discontinuities (pictured below).

o The reflected waves also exhibit slightly decreased magnitude.

o The discontinuities occur in either real or imaginary parts of the wave function.

o 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.

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Web view(It’s still centered at the greatest magnitude of the wave packet.) The phase velocity is greater than the group velocity; ... When it hits the step,