14.6: Problems
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1. If a radioactive substance has a half-life of one year, does this mean that it will be completely decayed after two years? Explain.
2. What is the probability of rolling a pair of dice and getting “snake eyes,” i.e., both dice come up with ones?
3. Problem 3 has been deleted.
4. Problem 4 has been deleted.
5. Refer to the probability distribution for people's heights in figure f on page 828.
(a) Show that the graph is properly normalized.
(b) Estimate the fraction of the population having heights between 140 and 150 cm.(answer check available at lightandmatter.com)
a / Problem 6.
6. (a) A nuclear physicist is studying a nuclear reaction caused in an accelerator experiment, with a beam of ions from the accelerator striking a thin metal foil and causing nuclear reactions when a nucleus from one of the beam ions happens to hit one of the nuclei in the target. After the experiment has been running for a few hours, a few billion radioactive atoms have been produced, embedded in the target. She does not know what nuclei are being produced, but she suspects they are an isotope of some heavy element such as Pb, Bi, Fr or U. Following one such experiment, she takes the target foil out of the accelerator, sticks it in front of a detector, measures the activity every 5 min, and makes a graph (figure). The isotopes she thinks may have been produced are:
isotope | half-life (minutes) |
211Pb | 36.1 |
214Pb | 26.8 |
214 Bi |
19.7 |
223Fr | 21.8 |
239U | 23.5 |
Which one is it?
(b) Having decided that the original experimental conditions produced one specific isotope, she now tries using beams of ions traveling at several different speeds, which may cause different reactions. The following table gives the activity of the target 10, 20 and 30 minutes after the end of the experiment, for three different ion speeds.
activity (millions of decays/s) after… | |||
10 min | 20 min | 30 min | |
first ion speed | 1.933 | 0.832 | 0.382 |
second ion speed | 1.200 | 0.545 | 0.248 |
third ion speed | 7.211 | 1.296 | 0.248 |
Since such a large number of decays is being counted, assume that the data are only inaccurate due to rounding off when writing down the table. Which are consistent with the production of a single isotope, and which imply that more than one isotope was being created?
7. Devise a method for testing experimentally the hypothesis that a gambler's chance of winning at craps is independent of her previous record of wins and losses. If you don't invoke the definition of statistical independence, then you haven't proposed a test.
8. A blindfolded person fires a gun at a circular target of radius
(a) Show that the probability distribution of
(b) Determine
(c) Find the average value of
(d) Interpreting your result from part c, how does it compare with
9. We are given some atoms of a certain radioactive isotope, with half-life
(a) Find the distribution
(b) Find the average value of
(c) Interpreting your result from part b, how does it compare with
10. The speed,
(a) Sketch the distribution.
(b) Find
(c) Find the average speed in terms of
11. All helium on earth is from the decay of naturally occurring heavy radioactive elements such as uranium. Each alpha particle that is emitted ends up claiming two electrons, which makes it a helium atom. If the original
(a) How many alphas are emitted per decay chain? [Hint: Use conservation of mass.]
(b) How many electrons are emitted per decay chain? [Hint: Use conservation of charge.]
(c) How long has it been since the lava originally hardened?(answer check available at lightandmatter.com)
12. When light is reflected from a mirror, perhaps only 80% of the energy comes back. One could try to explain this in two different ways: (1) 80% of the photons are reflected, or (2) all the photons are reflected, but each loses 20% of its energy. Based on your everyday knowledge about mirrors, how can you tell which interpretation is correct? [Based on a problem from PSSC Physics.]
13. Suppose we want to build an electronic light sensor using an apparatus like the one described in the section on the photoelectric effect. How would its ability to detect different parts of the spectrum depend on the type of metal used in the capacitor plates?
14. The photoelectric effect can occur not just for metal cathodes but for any substance, including living tissue. Ionization of DNA molecules can cause cancer or birth defects. If the energy required to ionize DNA is on the same order of magnitude as the energy required to produce the photoelectric effect in a metal, which of the following types of electromagnetic waves might pose such a hazard? Explain.
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b / Problem 15.
15. (a) Rank-order the photons according to their wavelengths, frequencies, and energies. If two are equal, say so. Explain all your answers.
(b) Photon 3 was emitted by a xenon atom going from its second-lowest-energy state to its lowest-energy state. Which of photons 1, 2, and 4 are capable of exciting a xenon atom from its lowest-energy state to its second-lowest-energy state? Explain.
c / Problem 16.
16. Which figure could be an electron speeding up as it moves to the right? Explain.
17. The beam of a 100-W overhead projector covers an area of
18. In the photoelectric effect, electrons are observed with virtually no time delay (
(a) Estimate the power that would be soaked up by a single electron in a beam of light with an intensity of 1
(b) The energy,
19. In a television, suppose the electrons are accelerated from rest through a voltage difference of
20. Use the Heisenberg uncertainty principle to estimate the minimum velocity of a proton or neutron in a
21. Find the energy of a particle in a one-dimensional box of length
22. A free electron that contributes to the current in an ohmic material typically has a speed of
(a) Estimate its de Broglie wavelength, in nm.(answer check available at lightandmatter.com)
(b) If a computer memory chip contains
(c) Based on your answers from parts a and b, does an electrical engineer designing such a chip need to worry about wave effects such as diffraction?
(d) Estimate the maximum number of electric circuits that can fit on a 1
23. In classical mechanics, an interaction energy of the form
(a) Show that there is a solution to the Schrödinger equation of the form
and relate
(b) Sketch a graph showing what this wavefunction looks like.
(c) Let's interpret
(d) Making
24. (a) A distance scale is shown below the wavefunctions and probability densities illustrated in figure e on page 882. Compare this with the order-of-magnitude estimate derived in subsection 13.4.4 for the radius
(b) Although we normally say the moon orbits the earth, actually they both orbit around their common center of mass, which is below the earth's surface but not at its center. The same is true of the hydrogen atom. Does the center of mass lie inside the proton, or outside it?
d / Problem 25.
25. The figure shows eight of the possible ways in which an electron in a hydrogen atom could drop from a higher energy state to a state of lower energy, releasing the difference in energy as a photon. Of these eight transitions, only D, E, and F produce photons with wavelengths in the visible spectrum.
(a) Which of the visible transitions would be closest to the violet end of the spectrum, and which would be closest to the red end? Explain.
(b) In what part of the electromagnetic spectrum would the photons from transitions A, B, and C lie? What about G and H? Explain.
(c) Is there an upper limit to the wavelengths that could be emitted by a hydrogen atom going from one bound state to another bound state? Is there a lower limit? Explain.
26. Find an equation for the wavelength of the photon emitted when the electron in a hydrogen atom makes a transition from energy level
27. Estimate the angular momentum of a spinning basketball, in units of
28. Assume that the kinetic energy of an electron the
29. Before the quantum theory, experimentalists noted that in many cases, they would find three lines in the spectrum of the same atom that satisfied the following mysterious rule:
30. The wavefunction of the electron in the ground state of a hydrogen atom is
where
(a) Calculate symbolically, without plugging in numbers, the probability that at any moment, the electron is inside the proton. Assume the proton is a sphere with a radius of
(b) Calculate the probability numerically.(answer check available at lightandmatter.com)
(c) Based on the equation for the wavefunction, is it valid to think of a hydrogen atom as having a finite size? Can
31. Use physical reasoning to explain how the equation for the energy levels of hydrogen,
should be generalized to the case of an atom with atomic number
32. A muon is a subatomic particle that acts exactly like an electron except that its mass is 207 times greater. Muons can be created by cosmic rays, and it can happen that one of an atom's electrons is displaced by a muon, forming a muonic atom. If this happens to a hydrogen atom, the resulting system consists simply of a proton plus a muon.
(a) Based on the results of section 13.4.4, how would the size of a muonic hydrogen atom in its ground state compare with the size of the normal atom?
(b) If you were searching for muonic atoms in the sun or in the earth's atmosphere by spectroscopy, in what part of the electromagnetic spectrum would you expect to find the absorption lines?
33. A photon collides with an electron and rebounds from the collision at 180 degrees, i.e., going back along the path on which it came. The rebounding photon has a different energy, and therefore a different frequency and wavelength. Show that, based on conservation of energy and momentum, the difference between the photon's initial and final wavelengths must be
34. Generalize the result of problem 33 to the case where the photon bounces off at an angle other than 180° with respect to its initial direction of motion.
35. On page 869 we derived an expression for the probability that a particle would tunnel through a rectangular barrier, i.e., a region in which the interaction energy
36. Show that the wavefunction given in problem 30 is properly normalized.
37. Show that a wavefunction of the form
38. Find the energy levels of a particle in a three-dimensional rectangular box with sides of length
39. Americium-241 is an artificial isotope used in smoke detectors. It undergoes alpha decay, with a half-life of 432 years. As discussed in example 18 on page 870, alpha decay can be understood as a tunneling process, and although the barrier is not rectangular in shape, the equation for the tunneling probability on page 870 can still be used as a rough guide to our thinking. For americium-241, the tunneling probability is about
40. As far as we know, the mass of the photon is zero. However, it's not possible to prove by experiments that anything is zero; all we can do is put an upper limit on the number. As of 2008, the best experimental upper limit on the mass of the photon is about
41. Hydrogen is the only element whose energy levels can be expressed exactly in an equation. Calculate the ratio
42. Give a numerical comparison of the number of photons per second emitted by a hundred-watt FM radio transmitter and a hundred-watt lightbulb.(answer check available at lightandmatter.com)
e / Problem 43.
43. On pp. 884-885 of subsection 13.4.4, we used simple algebra to derive an approximate expression for the energies of states in hydrogen, without having to explicitly solve the Schrödinger equation. As input to the calculation, we used the the proportionality
There are other systems of physical interest in which we have
Generalize the method used for
44. The electron, proton, and neutron were discovered, respectively, in 1897, 1919, and 1932. The neutron was late to the party, and some physicists felt that it was unnecessary to consider it as fundamental. Maybe it could be explained as simply a proton with an electron trapped inside it. The charges would cancel out, giving the composite particle the correct neutral charge, and the masses at least approximately made sense (a neutron is heavier than a proton). (a) Given that the diameter of a proton is on the order of
(b) Find the electron's minimum kinetic energy.(answer check available at lightandmatter.com)
(c) Show via
45. Suppose that an electron, in one dimension, is confined to a certain region of space so that its wavefunction is given by
Determine the constant
46. In the following,
47. (a) A radio transmitter radiates power
(b) Let the wavelength be
(c) For a 1000 kHz AM radio transmitting station, assuming reasonable values of
Benjamin Crowell (Fullerton College). Conceptual Physics is copyrighted with a CC-BY-SA license.