CHAPTER II. Hydrogen Atom
Part 1. Electron orbital motion
2.1 Consider an electron in the N shell. (a) What is the smallest orbital angular
momentum it could have? (b) What is the largest orbital angular momentum it could
have? Express your answers in terms of ħ and in SI units. (c) What is the largest orbital
angular momentum this electron could have in any chosen direction? Express your
answers in terms of ħ and in SI units. (d) What is the largest spin angular momentum this
electron could have in any chosen direction? Express your answers in terms of ħ and in SI
units. (e) For the electron in part (c), what is the ratio of its spin angular momentum in the
z-direction to its orbital angular momentum in the z-direction?
2.2 An electron is in the hydrogen atom with n = 5. (a) Find the possible values of L and
Lz for this electron, in units of ħ. (b) For each value of L, find all the possible angles
between ⃗ and the z-axis. (c) What are the maximum and minimum values of the
magnitude of the angle between ⃗ and the z-axis?
2.3 Using the results of quantum mechanics, calculate the magnetic moments that are
possible for an n=3 level.
2.4 Show the possible orientations of the orbital angular moment vector for l=0,1,2,3 and
4.
2.5 For l=3 calculate the possible angles that L makes with the z-axis.
2.6 Determine the normal Zeeman splitting of the calcium red line 6438 Å when atoms
are placed in a magnetic field of 0.009T.
2.7 A hydrogen atom in a 3p state is placed in a uniform external magnetic field ⃗ .
Consider the interaction of the magnetic field with the atom’s orbital magnetic dipole
moment. (a) What field magnitude B is required to split the 3p state into multiple levels
with an energy difference of 2.71×10-5 eV between adjacent levels? (b) How many levels
will there be?
2.8 Determine the normal Zeeman splitting in the mercury 419Å line when in a magnetic
field of 0.3T.
2.9 What magnetic flux density B is required to observe the normal Zeeman effect if a
spectrometer can resolve spectra lines separated by 0.5Å at 5000Å?
2.10 In a normal Zeeman experiment, the calcium 4226 Å line splits into three lines
separated by 0.25Å in magnetic field of 3 T. Determine e/m for the electron from these
data.
2.11 Transition occur in an atom between l=2 and l=1 states in a magnetic field of 0.6 T.
If the wavelength before the field was turned on was 5000Å, determine the wavelengths
that are observed.
Part 2. Electron Spin
2.12 A hydrogen atom in the n = 1, ms = -1/2 state is placed in a magnetic field with a
magnitude of 0.480 T in the +z – direction. (a) Find the magnetic interaction energy (in
electron volts) of the electron with the field. (b) Is there any orbital magnetic dipole
moment interaction for this state? Explain. Can there be an orbital magnetic dipole
moment interaction for n ≠ 1?
2.13 Determine the maximum separation of a beam of hydrogen atoms that moves a
distance of 20 cm with a speed of 2x105 m/s perpendicular to a magnetic field whose
gradient is 2x102 T/m. Neglect the magnetic moment of the proton.
2.14 Determine the energy difference between the electrons that are “aligned” and “antialigned” with a uniform magnetic field of 0.8 T when a beam of free electrons moves
perpendicular to the field.
2.15 Express ⃗ ⃗ in term of J, L, and S.
2.16 Calculate the possible values of ⃗ ⃗ for L=1 and S=1/2.
2.17 Estimate the strength of the magnetic field produced by the electron’s orbital motion
which results in the two sodium D lines (5889.95 A, 5895.92 A).
Supplementary Problems
2.16 An electron in He+ is an n=2 orbit. What is its magnetic moment due to its orbital
motion according to the Bohr theory?
Ans. 1.85 x 10-23 J/T
2.17 Do problem 2.9 using quantum-mechanical theory.
Ans. 1.31 x 10-23 J/T or 0
2.18 An electron in a circular orbit has an angular momentum of √2ђ . In a field of 0.5 T,
what is its Larmor frequency?
Ans. 6.99 x 109 Hz
2.19 For l=3 calculate the possible angles that L makes with the z-axis.
Ans. 16.8o; 35.3o; 60o.
2.20 Determine the normal Zeeman splitting in the mercury 419Å line when in a
magnetic field of 0.3T.
Ans. 3.38 x 10-2Å.
2.21 What will the separation be between adjacent normal Zeeman components for
emitted radiation of 4500Å in a magnetic field of 0.4T?
Ans. 3.78 x 10-2Å
2.22 A 5000Å line exhibits a normal Zeeman splitting of 1.1 x 10-3 Å. Find the magnetic
field.
Ans. 9.42 x 10-3 T
2.23 What is the frequency difference in the photons emitted in a normal Zeeman effect
corresponding to transitions from adjacent magnetic sublevels to the same final state in a
magnetic field of 1.2T?
Ans. 1.68 x 1010 Hz
2.24 Transition occur in an atom between an l=2 and an l=1 state in a field of 0.2 T. If the
wavelength before the field was turned on was 4000 Å, determine the final wavelength
observed.
Ans. 4000.0149Å; 4000Å, 3999.9851Å
2.25 In the Stern-Gerlach experiment, silver atoms traverse a distance of 0.1 m through
an inhomogenous magnetic field with gradient of 60 T/m. If the separation observed on
the collector plate is 0.15 mm, determine the velocity of the silver atoms. The mass of a
silver atom is 1.97x10-25 kg.
Ans. 455 m/s
2.26 What is the energy difference between the two electron spin orientations when the
electrons are in a magnetic field of 0.5 T?
Ans. 5.79x10-5 eV.
2.27 Refer to problem 2.24. Determine the wavelength of the radiation that can cause the
electrons to “flip” their spins.
Ans. 2.14 cm
2.28 The wavelength needed to cause an electron-spin “flip” is 1.5 cm. Calculate the
magnetic field the electron is in.
Ans. 0.714 T.
2.29 Calculate the possible values of J for L=3 and S=1/2.
Ans. 7/2; 5/2
2.30 For problem 2.27, calculate ⃗ ⃗.
2.31 Estimate the strength of the magnetic field produced by the electron’s orbital motion
which results in the 7664.1 A and 7699.0A lines observed in the L=1 to L=0 transition in
potassium.
Ans. 63.3 T.