Visualizing Wave Functions

Physics 247, Briggs, Coleman, and Eriksson

February 20, 2002

Don’t start the program yet!

Don’t try to print using F6...it crashes the computer!

1 Complex Numbers in Color

A complex number can be written

z = reiφ , (1)

and we can portray both of the two pieces of information in a complexnumber using color: use the hue (red, green...) to show the phase φ, andthe brightness to show r. Take red to be φ = 0. Start with Black at r = 0,and add more red as r increases (along the ray φ = 0). r = ∞ is infinitelybright red. This we show by adding white to the red to signify brightness.Take the order of colors to be those of the visible spectrum, roygbv, using

0◦ Red60◦ Yellow90◦ Green

180◦ Cyan−90◦ Blue−60◦ Magenta.

1

1 COMPLEX NUMBERS IN COLOR 2

Using this scheme, z = 0 will be black for all φ, z = 0.5 + 0i will be a grayred, z = 1 + 0i will be a bright red, and z > 1 + 0i will be increasinglywhite red.Question 1: Describe the color, brightness and approximate phase φ ofz = 0 ,z = 1− 2i ,z = −2 + i ,z = −.5− .5i ,z = −.5 + .5i ,z = .5 + .75i .

Complex Functions

Take f(x) = eix = cos x + i sin x. Show the amplitude and label or draw thecolors in the following plot for x = 0, ±π/3, ±π/2.

Do the same forf(x) = e−x2/6e6ix (2)

for x = 0, ±π/12, ±π/6, ±π/4, ±π/3, ±5π/3, ±7π/4 .Check your answers to all the work you have done so far by starting theprogram “VQM” from the CD-ROM, and comparing your results withthose shown in the demonstrations 1.1.3 (use the “more” button), 1.8, and1.8 page 4. Ask yourselves whether you really get what you are doing asyou work. There are additional demonstrations you can explore if you havetime, but make sure you check out the function gallery 1.10 and understandthe portrayal of f(z) = z, and f(z) = 1/z. Note how Magenta, Cyan andYellow conveniently give -60, 180 and 60 degrees.

2 MOTION OF WAVE PACKETS: GROUP AND PHASE VELOCITIES3

1.1 Wave Packets

You have studied the behavior of wave packets representing free particles.Look at demonstration 1.16. Assume this is the wave packet of a particlemoving from (x, y) = (−4,−4) to (4, 4). (For a top view of this same event,look at demonstration 1.15.)Question 2: Assuming time is moving forward, what are the signs of thephase and group velocities?Question 3: Which is higher, the phase or group velocity?Question 4: What is the relation between the height of the packet and thediameter of the bright region? (The brightness of the color is equivilant tothe height.)We can make a wave packet move by making each of the constituent wavesmove. These constituent waves are the waves that make up the Fourierexpansion of the particle:

f(x, t) = Σanei(knx−ωnt) . (3)

Each of the waves n will move with a speed vn = ωn/kn, which is both thephase velocity and group velocity for the individual, pure sin waves thatcomprise the wave packet. Demonstration 3.2 shows nine such wavessumming to give a wave packet.Question 5: By putting some sketches in your lab book, show how you canmeasure the group and phase velocities of the wave packet. Note that thereare buttons that cause the demonstration to step through the movie moreslowly. Report your results, using for distance centimeters along the screen.How do you expect these numbers to compare with each other, and why?

2 Motion of Wave Packets: Group and

Phase Velocities

Question 6: Explain in your lab book the difference between group andphase velocities.Run Demonstration 3.6.Question 7: What do the different colors in the wavepacket represent?Explain why the colors appear to move backward in the wavepacket. Arethey actually moving backward? Does the particle itself move with the

3 SCATTERING OF PARTICLES BY OBSTACLES 4

colors or the packet? If we measure the particles location, will we find it atthe center of the wavepacket? Explain.

3 Scattering of Particles by Obstacles

Run example 4.15.1 and read the description.Question 8: The particle scatters off of a large circle. Do we know in whichdirection it was travelling before the scattering event? In which directionwould a classical particle travelling at the center of the wavepacket scatteroff the circle? Do we know anything about which direction the quantummechanical particle scatters? Do we know anything about the directions itdoesn’t scatter? Explain with a diagram.Run example 4.15.2.Question 9: Notice the new scattering pattern. Does the symmetry of thescattered wave tell us anything about the symmetry of the scatterer?Explain.Run examples 4.15.3 and 4.15.4.Small objects tend to scatter more uniformly than large objects. Note thatthe small circle act like a source of circular waves. This scattering is welldescribed by an approximation called the ”Born approximation,” which youwill meet again in Physics 448.Before exploring examples 4.16 and 4.17, predict the angular spread of aparticle passing through a narrow slit by using the uncertainty principle. Inparticular, how does the angle of divergence depend on the size of the slit?Now explore 4.16 and 4.17. Draw pictures of the results for several slitwidths. Don’t forget to include the reflected portion of the wave.

4 The Simple Harmonic Oscillator

Example 5.2 is the simple harmonic oscillator. Explore parts 5.2.1 and5.2.2. Notice that the odd eigenstates have asymmetric wavefunctions,while the even eigenstates have even symmetric wavefunctions. Be sure toexplore both the wavefunctions and the probability distributions.Question 10: Do the probability distributions have the same symmetryproperties as the wavefunctions?

5 TUNNELING AND THE SCANNING TUNNELING MICROSCOPE (STM)5

Question 11: What is an ”observable” in quantum mechanics. Give anexample of (i) a quantum mechanical property that is observable and (ii)one that is not.Question 12: Is the potential V(x) in the Schrodinger equation even or oddfor the case of the simple harmonic oscillator. Should observables respectthat symmetry? Is the wavefunction observable? Is the probability densityobservable?

5 Tunneling and the Scanning Tunneling

Microscope (STM)

Run parts 7.18.1, 7.18.2, and 7.18.3, which explain the physics behind ascanning tunneling microscope.Question 13: What physical process allows the electron to ”jump” from thetip to the sample? Is the electron allowed to be ”in the gap” according toclassical physics? What does this tell you about the curvature of thewavefunction ψ′′ in the gap? Is ψ zero in the gap?Question 14: In the explanation shown here, how would atoms appear in anSTM image, i.e., what quantity would be recorded and presented in grayscale to represent the atoms?Question 15: Unlike the example shown here, the STM tunnel current isoften held constant in an actual STM image of a surface. How can this beaccomplished? Your answer should include a brief explanation of theconcept of ”feedback.” When feedback is used to hold the STM currentconstant, what is the physical signal which is displayed to represent theatoms?

6 Quantum Mechanics of Ballistic Particles

- Or Baseball Meets h̄

Explore parts 6.4.1, 6.4.2, 6.4.3. Explain why the spacing between the nodesbecomes large at the top of the parabolic trajectory. Hint: Your answershould make use of the de Broglie relation for momentum and wavelength.In the remaining time, explore other parts of the software that catch youinterest. All of the topics are accessible to undergraduate physics majors,

6 QUANTUM MECHANICS OF BALLISTIC PARTICLES - OR BASEBALL MEETS H̄6

and you can learn about them in courses like Physics 249, 448, and 449(quantum mechanics). Some of these topics are also touched on in Physics322 (electromagnetism), 321 (circuits and electronics), 325 (optics), and 415(statistical physics and thermal physics).