4. Describe the setup of Experiment 1, and explain what the experimental results tell us about the nature of photons.
5. Describe the setup of Experiment 2, and explain what the experimental results tell us about the nature of photons.
Describe the setup of Experiment 3 and explain what the experimental results tell us about the nature of photons.
How do you feel about the results of these three experiments, and the
interpretations they’ve led to? There is no “correct” answer here, we’re interested in hearing your reflections about the previous three questions.
How do the results of Experiment 3 (that a photon that “should” have been acting like a particle ends up acting like a wave) provide evidence against Local Realism/Local Hidden Variable Theory?
For the following three questions you will be exploring the wave function of an electron using the Quantum Tunneling Simulation. Change the potential from barrier/well to constant, switch from wave packet to plane wave, and set the total energy to 0.5 eV (keep the potential energy set to 0 eV.)
Link: https://phet.colorado.edu/en/simulation/legacy/quantum-tunneling What is the wavelength of this wave in nm (calculate this answer)?
What is the frequency of this wave in Hz (again calculate tis answer )?
Explain what it means that the probability density is (1) constant and (2) equal to one.
For the following four questions: Consider an electron and a photon both moving through space with a kinetic energy of 3.0 eV.
12. What is the momentum of the photon (in !" #
13. What is the momentum of the electron (in !" #
14. What is the deBroglie wavelength of the photon (in nm)?
15. What is the deBroglie wavelength of the electron (in nm)?
For the following two questions: Consider a standard baseball (m = 145 g) moving through space with a kinetic energy of 3.0 eV.
The deBroglie wavelength of this baseball is:
a) Larger than that of the photon in question 14
b) About the same as that of the photon
c) Larger than that of the electron in question 15, but smaller than that of the
d) About the same as that of the electron
e) Smaller than that of the electron
Explain your answer to question 16.
For the following two questions: Consider the double slit interference experiment for individual electrons (an image of the PhET sim is shown at right).
Which slit did the electron go through?
(While there are many interpretations, there is only one answer below consistent with experimental measurement).
e) Either left or right, we just cannot know which
Explain your reasoning to the previous question.
A particle confined to move in the x direction is described by the wavefunction
y(x) shown below. Three small regions of equal size (I, II, and III) have been indicated on the graph. Rank the probabilities of finding the particle in each of these regions. Hint: Recall that the probability density function for the position of a particle is given by r(&) = |y(&)|*
a) +(,,,) > +(,) > +(,,)
b) +(,,) > +(,) > +(,,,)
c) +(,,,) > +(,,) > +(,)
d) +(,) > +(,,,) > +(,,)
e) +(,,) > +(,,,) > +(,)
For the following six questions: An electron constrained to a region of the x axis between 0 and L is described by the following wavefunction – note that this is the same kind of function you saw on the first and second long answer HW of the year:
y(&,/) = 012sin739&:;<=>? ABC 0 ≤ & ≤ 3 33
0 ABC & < 0 GHI & > 3
21. What does this wave look like at t = 0 ? (The numbers 1-5 indicate regions relevant for question 22).
22. Now consider the 5 positions labeled 1 to 5 in the diagram. At this time
(t = 0), rank the probabilities of finding the electron very close (within a very small distance dx) to each of these points. If any of these probabilities are equal to zero, state so explicitly. Explain.
23. What does the probability density r(x), look like for this wavefunction?
24. Suppose you had a bunch of electrons that were all described by this wavefunction; if you measured the position of each of these electrons, which of the following patterns would you expect to see?
25. What is the total probability of detecting this electron between 0 and L?
26.What is the total probability of detecting this electron between 0 and 2L / 3?
For the following two questions: Plots of | (x)|2, the probability density, for three free electrons are shown below.
27. Which of these electrons has the greatest uncertainty in position?
28. If the uncertainty in position is defined as the distance between the points at
which the probability density has dropped to 1⁄2 its max value (similar to a standard deviation), then what is the approximate uncertainty in position for electron B (in nm)?
29. Long Answer
Recall that in the Bohr model +J = −2 LJ, where +J = − MNO is the electrostatic potential
energy and LJ = Q RS* = TO is the non-relativistic kinetic energy. In the deBroglie model, it is O *#
suggested that the electron should be thought of as a standing wave on a ring subject to the constraint Hl = 29C where r is the Bohr radius for a given state of the atom.
Use your expression for the quantized wavelengths and the definition of total
energy in the Bohr atom to determine the quantized energies of the deBroglie model in terms of fundamental constants and the integer n.
Using the deBroglie relation U = V integer n.
, and the expressions above, determine the quantized wavelengths of the deBroglie model in terms of fundamental constants and the
Recall that the energy of the ground state of (electronic) hydrogen is measured to
be -13.6 eV. Does your expression from part b) accurately predict the energy levels of hydrogen? Explain.
Muonic hydrogen is an atom where the electron is replaced with a muon, which
has the same charge as an electron but a much greater mass (RW = 207 RN). What is the ground state energy of muonic hydrogen?
These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.
4) In experiment one we use a beam of photons generated by PM1. The beam is then split in two by the beam splitter BS1. Two mirrors MA and MB reflect the photon beams into the detectors PMA and PMB. The photons are counted by NA and NB. A 3rd counter NC gives the correlation between NA and NC.
(The correlation shows the relation between photons). Correlation C = 1 shows that the photons from the two beams are related. This demonstrates that the photons behave like waves. Correlation C=0 shows that the photons from the two beams are not related. It demonstrates that the photons behave like particles.)
In this experiment at low intensities of light, the correlation C=0. Thus photons behave like particles.
5) In experiment two we have the same setup as in the experiment one, except that we have a second beam splitter BS2. When we have the second beam splitter, the initial wave that was spilt in two (and arrived at the two detectors PMA and PMB in experiment 1) arrives at one of the two detectors.
This demonstrates the constructive interference and thus the wave like behavior of light.
6) Experiment 3) has the same setup as experiment 2). The difference is that in the path of the two beams we have 10 meters of fiber-optic, for each. Now instead of one beam arriving at one of the two detectors we have two beams, each arriving at one detector. Because we have 10 meters of fiber optic in the path of each beam, they lose coherence. When arriving at the BS2, the phases will be random. Thus the interference will be at random and instead of one beam arriving at one detector we will have two beams arriving each one at a detector.
7) The previous experiments demonstrate that light behave simultaneously as a particle and a wave. For the same arrangement we have two different conclusions: experiment 1) demonstrates that light behaves like a particle, experiment 2) demonstrates that light behaves like a wave.
8) Local realism = all characteristics of a particle pre-exist before measurement (there are no hidden variables that determine the particle state). When the photon changes its behavior from wave to particle (and from particle to wave), this demonstrates that there is no local realism, there are no preexisting characteristics for it...