Exercises 159
- Show that the frequency of the photon emitted by a hydrogen
atom in going from the level n1 to the level nis always
intermediate between the frequencies of revolution of the
electron in the respective orbits.
4.7 Nuclear Motion- An antiproton has the mass of a proton but a charge ofe. If a
proton and an antiproton orbited each other, how far apart
would they be in the ground state of such a system? Why
might you think such a system could not occur? - A muon is in the n2 state of a muonic atom whose nu-
cleus is a proton. Find the wavelength of the photon emitted
when the muonic atom drops to its ground state. In what part
of the spectrum is this wavelength? - Compare the ionization energy in positronium with that in
hydrogen. - A mixture of ordinary hydrogen and tritium, a hydrogen iso-
tope whose nucleus is approximately 3 times more massive
than ordinary hydrogen, is excited and its spectrum observed.
How far apart in wavelength will the Hlines of the two kinds
of hydrogen be? - Find the radius and speed of an electron in the ground state of
doubly ionized lithium and compare them with the radius and
speed of the electron in the ground state of the hydrogen atom.
(Lihas a nuclear charge of 3e.) - (a) Derive a formula for the energy levels of a hydrogenic
atom,which is an ion such as Heor Li^2 whose nuclear
charge is Zeand which contains a single electron.
(b) Sketch the energy levels of the Heion and compare
them with the energy levels of the H atom. (c) An electron
joins a bare helium nucleus to form a Heion. Find the
wavelength of the photon emitted in this process if the
electron is assumed to have had no kinetic energy when it
combined with the nucleus.
4.9 The Laser- For laser action to occur, the medium used must have at least
three energy levels. What must be the nature of each of these
levels? Why is three the minimum number? - A certain ruby laser emits 1.00-J pulses of light whose wave-
length is 694 nm. What is the minimum number of Cr^3 ions
in the ruby?
38. Steam at 100°C can be thought of as an excited state of water
at 100°C. Suppose that a laser could be built based upon the
transition from steam to water, with the energy lost per mole-
cule of steam appearing as a photon. What would the fre-
quency of such a photon be? To what region of the spectrum
does this correspond? The heat of vaporization of water is
2260 kJkg and its molar mass is 18.02 kgkmol.
Appendix: Rutherford Scattering- The Rutherford scattering formula fails to agree with the data at
very small scattering angles. Can you think of a reason? - Show that the probability for a 2.0-MeV proton to be scattered
by more than a given angle when it passes through a thin foil is
the same as that for a 4.0-MeV alpha particle. - A 5.0-MeV alpha particle approaches a gold nucleus with an
impact parameter of 2.6 10 ^13 m. Through what angle will it
be scattered? - What is the impact parameter of a 5.0-MeV alpha particle scat-
tered by 10° when it approaches a gold nucleus? - What fraction of a beam of 7.7-MeV alpha particles incident upon
a gold foil 3.0 10 ^7 m thick is scattered by less than 1°? - What fraction of a beam of 7.7-MeV alpha particles incident
upon a gold foil 3.0 10 ^7 m thick is scattered by 90° or
more? - Show that twice as many alpha particles are scattered by a foil
through angles between 60° and 90° as are scattered through
angles of 90° or more. - A beam of 8.3-MeV alpha particles is directed at an aluminum
foil. It is found that the Rutherford scattering formula ceases to
be obeyed at scattering angles exceeding about 60°. If the
alpha-particle radius is assumed small enough to neglect here,
find the radius of the aluminum nucleus. - In special relativity, a photon can be thought of as having a
“mass” of mE c^2. This suggests that we can treat a photon
that passes near the sun in the same way as Rutherford treated
an alpha particle that passes near a nucleus, with an attractive
gravitational force replacing the repulsive electrical force. Adapt
Eq. (4.29) to this situation and find the angle of deflection for
a photon that passes bRsunfrom the center of the sun. The
mass and radius of the sun are respectively 2.0 1030 kg and
7.0 108 m. In fact, general relativity shows that this result is
exactly half the actual deflection, a conclusion supported by ob-
servations made during solar eclipses as mentioned in Sec. 1.10.
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