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Chapter 7: Q12CQ (page 278)
Question: A particle is trapped in a spherical infinite well. The potential energy is 0for r < a and infinite for r > a . Which, if any, quantization conditions would you expect it to share with hydrogen, and why?
Answer:
The particle will have the same quantization which is found in hydrogen.
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A comet of mass describes a very elliptical orbit about a star of mass , with its minimum orbit radius, known as perihelion, being role="math" localid="1660116418480" and its maximum, or aphelion, times as far. When at these minimum and maximum
radii, its radius is, of course, not changing, so its radial kinetic energy is , and its kinetic energy is entirely rotational. From classical mechanics, rotational energy is given by , where is the moment of inertia, which for a “point comet” is simply .
(a) The comet’s speed at perihelion is . Calculate its angular momentum.
(b) Verify that the sum of the gravitational potential energy and rotational energy are equal at perihelion and aphelion. (Remember: Angular momentum is conserved.)
(c) Calculate the sum of the gravitational potential energy and rotational energy when the orbit radius is times perihelion. How do you reconcile your answer with energy conservation?
(d) If the comet had the same total energy but described a circular orbit, at what radius would it orbit, and how would its angular momentum compare with the value of part (a)?
(e) Relate your observations to the division of kinetic energy in hydrogen electron orbits of the same but different .
Question: Verify the correctness of the normalization constant of the radial wave function given in Table 7.4 as
Calculate the electric dipole moment p and estimate the transition time for a hydrogen atom electron making an electric dipole transition from the
to the state.
In Appendix G. the operator for the square of the angular momentum is shown to be
Use this to rewrite equation (7-19) as
At heart, momentum conservation is related to the universe being "translationally invariant," meaning that it is the same if you shift your coordinates to the right or left. Angular momentum relates to rotational invariance. Use these ideas to explain at least some of the differences between the physical properties quantized in the cubic three-dimensional box versus the hydrogen atom.
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