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Chapter 39: Q2 (page 1136)
What is the difference between the probability and the probability density?
Probability is a number defining chances of and event to occur whereas probability density is distribution of probability over a region in space.
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An experiment has four possible outcomes, labeled A to D. The probability of A is PA= 40% and of B is PB = 30%. Outcome C is twice as probable as outcome D. What are the probabilities PCand PD?
A pulse of light is created by the superposition of many waves that span the frequency range , wherc is called thc center frequency of thc pulsc. Lascr technology can generate a pulse of light that has a wavelength of and lasts a mere fs femtosecond localid="1650804865678" .
a. What is the center frequency of this pulse of light?
b. How many cycles, or oscillations, of the light wave are completed during the fs pulse?
c. What range of frequencies must be superimposed to create this pulse?
d. What is the spatial length of the laser pulse as it travels through space?
e. Draw a snapshot graph of this wave packet.
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.
Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it's reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted.
a. nucleus, which decays by alpha emission, is 15fm in diameter. Model an alpha particle within a nucleus as being in a one-dimensional box. What is the maximum speed an alpha particle is likely to have?
b. The probability that a nucleus will undergo alpha decay is proportional to the frequency with which the alpha particle reflects from the walls of the nucleus. What is that frequency (reflections/s) for a maximum-speed alpha particle within a nucleus?
When 5 X 1012 photons pass through an experimental apparatus, 2.0 X 109 land in a 0.10-mm-wide strip. What is the probability density at this point?
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