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Chapter 8: Q5Q (page 344)
If you double the amplitude, what happens to the frequency in a classical (non quantum) harmonic oscillator? In a quantum harmonic oscillator?
Frequency will remain unchanged in case of classical (non quantum) harmonic oscillator, as well as in case of harmonic oscillator on doubling the amplitude.
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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 “selection rules” 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 (“dark”) 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.
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–mass 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 (“dark absorption lines”)? Explain briefly.
Assume that a hypothetical object has just four quantum states, with the energies shown in Figure 8.43.
(a) Suppose that the temperature is high enough that in a material containing many such objects, at any instant some objects are found in all of these states. What are all the energies of photons that could be strongly emitted by the material? (In actual quantum objects there are often “selection rules” that forbid certain emissions even though there is enough energy; assume that there are no such restrictions here.) (b) If the temperature is very low and electromagnetic radiation with a wide range of energies is passed through the material, what will be the energies of photons corresponding to missing (“dark”) lines in the spectrum? (Assume that the detector is sensitive to a wide range of photon energies, not just energies in the visible region.)
If you try to increase the energy of a quantum harmonics oscillator by adding an amount of energy , the energy doesn’t increase. Why not?
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.)
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