Phys 2130 HW 8
1. Describe the limitations of the Bohr model. Which of these limitations does the
deBroglie wave model address and why?
2. Suppose you have an electron and a photon both moving through space with a kinetic
energy of 4.5eV.
a) What is the deBroglie wavelength in nm of the photon?
b) What is the deBroglie wavelength of the electron in nm?
3. You (~100 lbs) are walking briskly (~4.75 miles per hour) across campus for your
next class. What is your approximate deBroglie wavelength in meters?
4. Free electrons generally exist as wave packets. In this problem, you will be working
with the Fourier: Making Waves (see: phet.colorado.edu) simulation to explore how sine
and cosine waves add up to make wave packets and other functions.
a. Try playing with the controls on the “Amplitudes” graph. (Checking the “autoscale”
box next to the “Sum” graph may make it easier to see what’s going on.) How does
changing the “Amplitudes” graph change the “Harmonics” and “Sum” graph? Explain
how these three graphs are related.
b. Hit the “Reset” button so that there is only one sine wave shown. Think about the
definition of wavelength. Use the horizontal zoom controls next to the “Harmonics”
graph so that more than one wavelength is displayed on the graph. Check the
“Wavelength Tool” box and use the tool that appears to measure from peak to peak, from
trough to trough, and then from some arbitrary point in the wave to the same arbitrary
point one cycle farther.
Is this consistent with your definition of wavelength? How?
c. Try out the different options in the “Function” pull-down menu. For each function,
think about why each harmonic should have the approximate amplitude that it does in
order to make up that function. For example, in a triangle wave, why are all the even
harmonics zero? (You don’t have to write up anything for this part, but it may help you
think about the essay below.) Now, produce a “sawtooth” wave from the same menu.
Can you produce a “better” sawtooth wave using 3, 7, or 11 harmonics? If you wanted a
“perfect” sawtooth how many harmonics would you need? Please explain your answer.
d) Look at the “Wave Packet” function, the components that go into creating the wave
packet, and how they are interacting. What is happening near the origin(of the summed
waves) such that you get a peak? What is happening as you move away from the origin of
the summed peaks such that the wave function goes to zero?
e) What happens as you move very far away from the origin of the summed peaks. Is this
what you would expect for a true wave packet?
f) A flag at the top left of the window lets you choose “Discrete to Continuous”. Do this
and play with the different options for making wave packets. Can you correct the
“problem” you found in (e)? How did you do this and why does it help?
5. Review of Complex Numbers (this will be helpful when we get to the Schrodinger
equation)! Remember that the complex conjugate of a function (f) is written as f*.
a. What is the complex conjugate of f(x) = ieikx:
b. f(x) = asin(x) + ibcos(x) and a and b are real, what is f*f? (the complex conjugate of f,
times f)
c. Generalizations about complex conjugates: (True or False)
-
To take the complex conjugate of any number, just replace every i you see with a –i
The complex conjugate of a real number is an imaginary number.
Any complex number can be written as a+bi, where a and b are real numbers.
Any complex number can be written as Aeiθ, where A and θ are real numbers.
A complex number times its complex conjugate is always real.
A complex number times its complex conjugate is always positive.
eiθ= cos(θ) + isin(θ)
eiπ = -1
If f(t)=eiωt, where ω is real, then |f|2=f*f=1
6 A matter wave, for instance an electron (in 1 dimension for simplicity), can be
represented by the wave function ψ(x,t). You can relate the wave function to the
probability density of finding the electron, P(x,t) = |ψ|2. If the wave function, ψ, is
complex, then the absolute value signs mean that the
probability density = P(x) = |ψ|2 = ψ*(x,t) ψ (x,t), where ψ*(x,t) is the complex conjugate
of ψ (x,t). This is similar to how the relative intensity of an electromagnetic wave
interference pattern of a photon can tell you about the probability of the photon being
detected there.
a. Say an electron wave function at time=0 were a square wave between 0 and L, as
shown below (note ψ(x,t)=0 all the way out to -infinity and +infinity):
Ψ(x,t=0)
a
0
A
x
L
B
C
D
i.
How do the probabilities of finding the electron very close (within a very small
distance dx) to x=A, B, C, and D compare? (A=Probability of finding the electron
near point A)
a. A=B=C=D
b. A < B < C < D
c. A > B > C > D
d. 0 < (A=D) < (B=C)
e. (0=A=D) < (B=C)
f. (0=A=D) < B < C
ii. The integral of the probability density over all x (-infinity to infinity) tells you the
total probability of finding the electron between -infinity and infinity. What is this total
probability? (Note that probability is entered as a fraction, e.g. 0.5 if it is 50% probable.)
iii. In the graph, what does the value for “a” have to be such that when you integrated the
probability density from -infinity to infinity, you get an answer consistent with your
answer for the previous question?
iv. What is the total probability of finding the electron anywhere between x=0 and
x=L/5? (Note that probability is entered as a fraction, e.g. 0.5 if it is 50% probable.)
v. Which of the following interpretations of this wave function are valid:
- The electron’s position is higher at x=C than at x=D
- The electron is initially moving to the right, moves up then moves down, and keeps
going.
- At time=0, the electron has no chance of being found where (x,t)=0
- At time=0. the electron is equally likely to be found anywhere between x=0 and x=L.
7.
a) Write down an equation for a traveling wave ΨR which is moving to the right
with amplitude A, wave number k, and angular frequency ω. Now do the same
for the wave ΨL which has the same amplitude, magnitude of wave number, and
angular frequency, but is moving to the left. What is the speed that these waves
travel?
b) Show that the sum ΨR+ΨL is equivalent to a standing wave. What is the
amplitude and the angular frequency of this standing wave? You will need to use
a trig identity to answer this.
Additionally, please do TZD problems 6.10, 6.15, 6.19, 6.25