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Chapter 39: Q.41 (page 1139)
What is the smallest one-dimensional box in which you can confine an electron if you want to know for certain that the electron鈥檚 speed is no more than 10 m/s?
The probability density of the electron in a hydrogen
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FIGURE Q39.4 shows the dot pattern of electrons landing on a screen. a. At what value or values of x is the electron probability density at maximum? Explain. b. Can you tell at what value or values of x the electron wave function c1x2 is most positive? If so, where? If not, why not?
3 shows the probability density for an electron that has passed through an experimental apparatus. What is the probability that the electron will land in a 0.010-mm-wide strip at (a) x = 0.000 mm, (b) x = 0.500 mm, (c) x = 1.000 mm, and (d) x = 2.000 mm?
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.
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?
An experiment finds electrons to be uniformly distributed over the interval 0 cm x 2 cm, with no electrons falling out-side this interval.
a. Draw a graph of 0 c1x2 0 2 for these electrons.
b. What is the probability that an electron will land within the interval 0.79 to 0.81 cm?
c. If 106 electrons are detected, how many will be detected in the interval 0.79 to 0.81 cm?
d. What is the probability density at x = 0.80 cm?
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