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Chapter 39: Q. 21 (page 1137)
What minimum bandwidth is needed to transmit a pulse that consists of 100 cycles of a oscillation?
The minimum bandwidth is needed to transmit a pulse that consists of 100 cycles of a oscillation is .
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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.
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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?
Ultrasound pulses with a frequency of 1.000 MHz are transmitted into water, where the speed of sound is 1500 m /s. The spatial length of each pulse is 12 mm.
a. How many complete cycles are contained in one pulse?
b. What range of frequencies must be superimposed to create each pulse?
FIGURE P39.32 shows for the electrons in an experiment.
a. Is the electron wave function normalized? Explain.
b. Draw a graph of over this same interval. Provide a numerical scale on both axes. (There may be more than one acceptable answer.)
c. What is the probability that an electron will be detected in a -wide region at ? At ? At ?
d. If electrons are detected, how many are expected to land in the interval ?
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