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Chapter 39: Q 5 Exercise (page 1136)
Make a table in which you list all possible outcomes of rolling two dice. Call the dice A and B. What is the probability of rolling
(a) a 7,
(b) any double, and
(c) a 6 or an 8?
You can give the probabilities as fractions, such as 3/36.
Therefore, the probabilities are:
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The probability density for finding a particle at position is
and zero elsewhere
Soot particles, from incomplete combustion in diesel engines, are typically in diameter and have a density of . FIGURE P39.45 shows soot particles released from rest, in vacuum, just above a thin plate with a -diameter holeroughly the wavelength of visible light. After passing through the hole, the particles fall distance d and land on a detector. If soot particles were purely classical, they would fall straight down and, ideally, all land in a -diameter circle. Allowing for some experimental imperfections, any quantum effects would be noticeable if the circle diameter were . How far would the particles have to fall to fill a circle of this diameter?
a. Starting with the expressionfor a wave packet, find an expression for the product
for a photon.
b. Interpret your expression. What does it tell you?
c. The Bohr model of atomic quantization says that an atom in an excited state can jump to a lower-energy state by emitting a photon. The Bohr model says nothing about how long this process takes. You'll learn in Chapter 41 that the time any particular atom spends in the excited state before emitting a photon is unpredictable, but the average lifetime of many atoms can be determined. You can think of as being the uncertainty in your knowledge of how long the atom spends in the excited state. A typical value is . Consider an atom that emits a photon with a wavelength as it jumps down from an excited state. What is the uncertainty in the energy of the photon? Give your answer in .
d. What is the fractional uncertainty in the photon's energy?
Physicists use laser beams to create an atom trap in which atoms are confined within a spherical region of space with a diameter of about . The scientists have been able to cool the atoms in an atom trap to a temperature of approximately , which is extremely close to absolute zero, but it would be interesting to know if this temperature is close to any limit set by quantum physics. We can explore this issue with a onedimensional model of a sodium atom in a -long box.
a. Estimate the smallest range of speeds you might find for a sodium atom in this box.
b. Even if we do our best to bring a group of sodium atoms to rest, individual atoms will have speeds within the range you found in part a. Because there's a distribution of speeds, suppose we estimate that the root-mean-square speed of the atoms in the trap is half the value you found in part a. Use this to estimate the temperature of the atoms when they've been cooled to the limit set by the uncertainty principle.
For the electron wave function shown in FIGURE Q39.3, at what position or positions is the electron most likely to be found? Least likely to be found? Explain.
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