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Chapter 41: Q. 20 (page 1207)
atoms are excited to an upper energy level at . At the end of , of these atoms have undergone a quantum jump to the ground state.
a. How many photons have been emitted?
b. What is the lifetime of the excited state?
a. The number of emitted photons is .
b. The lifetime of the excited state is .
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The 1997 Nobel Prize in physics went to Steven Chu, Claude Cohen-Tannoudji, and William Phillips for their development of techniques to slow, stop, and “trap” atoms with laser light. To see how this works, consider a beam of rubidium atoms (mass ) traveling at after being evaporated out of an oven. A laser beam with a wavelength of is directed against the atoms. This is the wavelength of the transition in rubidium, with being the ground state, so the photons in the laser beam are easily absorbed by the atoms. After an average time of , an excited atom spontaneously emits a 780-nm-wavelength photon and returns to the ground state.
a. The energy-momentum-mass relationship of Einstein’s theory of relativity is . A photon is massless, so the momentum of a photon is . Assume that the atoms are traveling in the positive x-direction and the laser beam in the negative x-direction. What is the initial momentum of an atom leaving the oven? What is the momentum of a photon of light?
b. The total momentum of the atom and the photon must be conserved in the absorption processes. As a consequence, how many photons must be absorbed to bring the atom to a halt?
NOTE Momentum is also conserved in the emission processes. However, spontaneously emitted photons are emitted in random directions. Averaged over many absorption/emission cycles, the net recoil of the atom due to emission is zero and can be ignored.
c. Assume that the laser beam is so intense that a ground-state atom absorbs a photon instantly. How much time is required to stop the atoms?
d. Use Newton’s second law in the form to calculate the force exerted on the atoms by the photons. From this, calculate the atoms’ acceleration as they slow.
e. Over what distance is the beam of atoms brought to a halt?
sodium atoms are excited to the state at . At what time have photons been emitted ?
There exist subatomic particles whose spin is characterized by , rather than the of electrons. These particles are said to have a spin of one.
a. What is the magnitude (as a multiple ofrole="math" localid="1650369219275" ) of the spin angular momentum for a particle with a spin of one?
b. What are the possible values of the spin quantum number?
c. Draw a vector diagram similar to Figure 41.13 to show the possible orientations of.
The ionization energy of an atom is known to be . The emission spectrum of this atom contains only the four wavelengths and . Draw an energy-level diagram with the fewest possible energy levels that agrees with these experimental data. Label each level with an appropriate quantum number. Hint: Don’t forget about the selection rule.
Show that .
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