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Chapter 8: Q11P (page 344)
At t =0 all of the atoms in a collection of 10000 atoms are in a excited state whose lifetime is 25 ns. Approximately how many atoms will still be in excited state at t= 12 ns.
The number of excited atoms at is .
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Energy graphs: (a) Figure 8.41 shows a graph of potential energy vs. interatomic distance for a particular molecule. What is the direction of the associated force at location A? At location B? At location C? Rank the magnitude of the force at locations A,B and C. (That is, which is greatest , which is smallest, and are any of these equal to each other?) For the energy level shown on the graph, draw a line whose height is the kinetic energy when the system is at location D.
(b) Figure 8.42 shows all of the quantized energies (bound states) for one of these molecules. The energy for each state is given on the graph, in electron volts ( ). How much energy is required to break a molecule apart, if it is initially in the ground state? (Note that the final state must be an unbound state; the unbound states are not quantized.)
(c) At high enough temperatures, in a collection of these molecules there will be at all times some molecules in each of these states, and light will be emitted. What are the energies in electron volts of the emitted light?
(d) The "inertial" mass of the molecule is the mass that appears in Newton's second law, and it determines how much acceleration will result from applying a given force. Compare the inertial mass of a molecule in the ground state and the inertial mass of a molecule in an excited state above the ground state. If there is a difference, briefly explain why and calculate the difference. If there isn't a difference, briefly explain why not.)
The mean lifetime of a certain excited atomic state is 5 ns. What is the probability of the atom staying in this excited state for t=10 ns or more?
Assume that a hypothetical object has just four quantum states, with the following energies:
(third excited state)
(second excited state)
(first excited state)
(ground state)
(a) Suppose that material containing many such objects is hit with a beam of energetic electrons, which ensures that there are always some objects in all of these states. What are the six energies of photons that could be strongly emitted by the material? (In actual quantum objects there are often 鈥渟election rules鈥� that forbid certain emissions even though there is enough energy; assume that there are no such restrictions here.) List the photon emission energies. (b) Next, suppose that the beam of electrons is shut off so that all of the objects are in the ground state almost all the time. If electromagnetic radiation with a wide range of energies is passed through the material, what will be the three energies of photons corresponding to missing (鈥渄ark鈥�) lines in the spectrum? Remember that there is hardly any absorption from excited states, because emission from an excited state happens very quickly, so there is never a significant number of objects in an excited state. Assume that the detector is sensitive to a wide range of photon energies, not just energies in the visible region. List the dark-line energies.
Summarize the differences and similarities between different energy levels in a quantum oscillator. Specifically for the first two levels in figure 8.26, compare the angular frequency , the amplitude , and the kinetic energy at the same value of . ( In a quantum-mechanical analysis the concepts of angular frequency and amplitude require reinterpretation. Nevertheless, there remain elements of the classical picture. For example, larger amplitude corresponds to a higher probability of observing a large stretch.)
Some material consisting of a collection of microscopic objects is kept at a high temperature. A photon detector capable of detecting photon energies from infrared through ultraviolet observes photons emitted with energies ofand. These are the only photon energies observed. (a) Draw and label a possible energy-level diagram for one of the microscopic objects, which has four bound states. On the diagram, indicate the transitions corresponding to the emitted photons. Explain briefly. (b) Would a spring鈥搈ass model be a good model for these microscopic objects? Why or why not? (c) The material is now cooled down to a very low temperature, and the photon detector stops detecting photon emissions. Next, a beam of light with a continuous range of energies from infrared through ultraviolet shines on the material, and the photon detector observes the beam of light after it passes through the material. What photon energies in this beam of light are observed to be significantly reduced in intensity (鈥渄ark absorption lines鈥�)? Explain briefly.
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