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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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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.)
N=1 is the lowest electronic energy state for a hydrogen atom. (a) If a hydrogen atom is in a state N=4, what is K+U for this atom (in eV)? (b) The hydrogen atom makes a transition to state N=2, Now what is K+U in electron volts for this atom? (c) What is energy (in eV) of the photon emitted in the transition from level N=4 to N=2? (d) Which of the arrows in figure 8.40 represents this transition?
Suppose we have reason to suspect that a certain quantum object has only three quantum states. When we excite such an object we observe that it emits electromagnetic radiation of three different energies: (green), (orange), and (infrared). (a) Propose two possible energy-level schemes for this system. (b) Explain how to use an absorption measurement to distinguish between the two proposed schemes.
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.
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?
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