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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’s speed is no more than 10 m/s?
The probability density of the electron in a hydrogen
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Consider the electron wave function
a. Determine the normalization constant c. Your answer will be in terms of L.
b. Draw a graph of over the interval -Lx 2L.
c. Draw a graph of over the interval -L x 2L. d. What is the probability that an electron is in the interval 0 x L/3?
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?
The probability density of finding a particle somewhere along thethe probability density is . For, the probability density decreases by a factor of each time the distance from the origin is doubled. What is the probability that the particle will be found in the interval
FIGURE Q 39.1 shows the probability density for photons to be detected on the x-axis.
a. Is a photon more likely to be detected at x = 0 m or at x=1 m? Explain.
b. One million photons are detected. What is the expected number of photons in a 1 mm-wide interval at x = 0.50 m?
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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