Problem Set 4
Due: See website for due dates
Quantization of Atomic Energy Levels
Reading: Taylor, Zafiratos, Dubson, Chapter 5; Tipler and Llewellyn, Chapter 4
Problem A
In which Bohr orbit does the electron have the largest velocity? Are we justified in
treating the electron nonrelativistically in that case?
Problem B
Explain the role of penetrating orbits and how the energy levels change as a
consequence.
Problem C
What is the basic distinction between x-ray energy levels or ordinary energy levels? Can
a hydrogen atom emit x rays? If so, how? If not, why not?
Problem D
According to the second Bohr hypothesis, the minimum value of angular momentum that
an atomic state can have is (Ln = n ). Later, when we treat quantum mechanics more
precisely, we shall see that atomic states can also have an angular momentum of zero.
What kind of orbital would that represent in classical terms? And what is there about the
orbitals we have considered in this chapter that would rule out zero angular momentum?
Problem 1
The series of hydrogen spectral lines with m = 4 is called Brackett’s series. Compute the
wavelengths of the first four lines of Brackett’s series.
Answer: 45 = 4052 nm; 46 = 2625 nm; 47 = 2166 nm; 48 = 1945 nm
Problem 2
Calculate the three longest wavelengths in the Lyman series, and indicate their position
on a horizontal linear scale. Indicate the series limit (shortest wavelength) on this scale.
Are any of these lines in the visible spectrum?
Answer: 121.57 nm, 102.57 nm, 97.25 nm
Problem 3
It is possible for a muon to be captured by a proton to form a muonic atom.
A muon is identical to an electron except for its mass, which is 105.7 MeV/c2. (a)
Calculate the radius of the first Bohr orbit of a muonic atom. (b) Calculate the magnitude
of the lowest energy. (c) What is the shortest wavelength in the Lyman series for this
atom? What sort of electromagnetic radiation is this?
Answer: 2.85 x 104 nm; 2520 eV; 0.492 nm
Problem 4
A hydrogen atom at rest in the laboratory emits the Lyman  radiation. (a) Compute the
recoil kinetic energy of the atom. (b) What fraction of the excitation energy of the n = 2
state is carried by the recoiling atom? (Hint: Use conservation of momentum.)
Answer: 5.55 x 108 eV; 5 x 109
Problem 5
An electron-positron pair can form a hydrogenlike system called positronium. Calculate
(a) the energies of the three lowest states and (b) the wavelength of the Lyman  and 
lines. (Detection of those lines is a “signature” of positronium formation.)
Answer: 6.804 eV, 1.701 eV, 0.756 eV; 243 nm, 205 nm
Problem 6
The wavelength of the K x-ray line for an element is measured to be 0.0794 nm. What
is the element? Answer: Zirconium
Problem 7
What is the minimum potential that must be applied across an x-ray tube in order to
observe the (a) K line of tungsten, (b) the K line of copper, and (c) L line of copper?
What is the min of the continuous spectrum in each case?
Answer: 72.5 keV, 0.017 nm; 10.7 keV, 0.116 nm; 1.59 keV, 0.782 nm
Problem 8
a. The total power P radiated by an accelerated charge (moving nonrelativistically) can
be shown (integral of the Poynting vector) to be
2kq2a2
P
3c 3
Find the power that would be radiated by a classical electron in the n = 1 Bohr orbit of a
hydrogen atom (r = 5.3 × 10-2nm) and use ke2 = 1.44 eV∙nm.
b. As the orbiting electron gives off energy, its orbital radius decreases, and when the
radius reaches that of the proton, the atom has collapsed completely. We can use the
rate of energy loss to estimate the time required for this collapse. With the help of the
chain rule, the rate at which the radius r shrinks depends on the radius, energy and time:
dr dr dE dr


( P)
dt dE dt dE
Evaluate this at the Bohr radius and assuming that dr/dt remains constant at this value,
make a rough estimate of the time required for the complete collapse of the classical
atom. Answer: 2.9 × 1011 eV/s, 5 × 1011 s
Problem 9
The ground state of lithium (Z = 3) has two electrons in the 1s level and one in the 2s.
Consider an excited state in which the outermost electron has been raised to the 3p
level. Since the 3p wave functions are not very penetrating, you can estimate the energy
of this electron by assuming it is completely outside both the other electrons.
a. In this approximation, what is the potential energy function felt by the outermost
electron? Write the formula for the energy of the outer electron as a function of n?
b. Estimate the energy of the 3p electron in this way and compare with the observed
value of −1.556 eV.
c. Repeat for the case that the outer electron is in the 3d level, whose observed energy
is −1.513 eV.
d. Explain why the agreement is better for the 3d level than for the 3p. Why is the
observed energy for 3p lower than that for 3d?
e. (i) Estimate the energy of Lithium electron where the electron occupies the n = 2
level (measured value is 5.4 eV); it should have about the same ionization energy as
an n = 2 electron in hydrogen (measured value is 3.4 eV). (ii) Explain why there is a
difference between the two values.