Elements of Modern Physics
64
20. Use the Sommerfeld quantization condition z p x d x = nh to obtain the energy
levels of a one-dimensional harmonic oscillator.
21. A free atom undergoes a transition from an energy level E 1 to E 2 by
absorbing a photon. Show that the frequency of the photon absorbed is
2
1
2
1
2
1
(
) 1 2
E E
E E
h
m c
−
−
ν =
+
where m 1 is the mass of the initial atom. The recoil correction given by the
second term is of the order of 10
–8
, or smaller, for the hydrogen atom. In
Mössbauer effect, the atom is embedded in a crystal so that its effective
mass is that of the crystal. As a result, the second term is negligible and
one has recoil-less absorption of photons.
22. In a Franck-Hertz experiment, hydrogen atoms are bombarded with
electrons. What are the wavelengths of the emission lines observed when
the electrons are accelerated through a potential difference of 12.5V?
23. Assuming that the earth is a black body in equilibrium at a temperature of
300 K, estimate the temperature of the sun.
24. Suppose that man’s power production reaches 20% of the power received
from sunlight (this would happen in about 250 years if the present exponential
growth continues). What is the expected approximate increase in the surface
temperature of the earth?
64
20. Use the Sommerfeld quantization condition z p x d x = nh to obtain the energy
levels of a one-dimensional harmonic oscillator.
21. A free atom undergoes a transition from an energy level E 1 to E 2 by
absorbing a photon. Show that the frequency of the photon absorbed is
2
1
2
1
2
1
(
) 1 2
E E
E E
h
m c
−
−
ν =
+
where m 1 is the mass of the initial atom. The recoil correction given by the
second term is of the order of 10
–8
, or smaller, for the hydrogen atom. In
Mössbauer effect, the atom is embedded in a crystal so that its effective
mass is that of the crystal. As a result, the second term is negligible and
one has recoil-less absorption of photons.
22. In a Franck-Hertz experiment, hydrogen atoms are bombarded with
electrons. What are the wavelengths of the emission lines observed when
the electrons are accelerated through a potential difference of 12.5V?
23. Assuming that the earth is a black body in equilibrium at a temperature of
300 K, estimate the temperature of the sun.
24. Suppose that man’s power production reaches 20% of the power received
from sunlight (this would happen in about 250 years if the present exponential
growth continues). What is the expected approximate increase in the surface
temperature of the earth?