760 17 The Electronic States of Atoms. I. The Hydrogen AtomThe angular functionsYlm(θ,φ) are called spherical harmonic functions. These
functions are eigenfunctions of the operator for the square of the orbital angular momen-
tum and itszcomponent, with eigenvalues given by
̂L^2 Ylmhl ̄(l+1)Ylmand
̂LzYlmhmY ̄ lmThe radial factors are a set of functions with two quantum numbers:n, the principal
quantum number, andl, the same quantum number as in the spherical harmonic func-
tions. The hydrogen-like atom has a single electron andZprotons in the nucleus. The
energy eigenvalues of the hydrogen-like atom depend only on the principal quantum
number:EEn−(13.60 eV)Z^2
n^2
The intrinsic electronic angular momentum of the electron was introduced. It cor-
responds to fixed magnitude and two possiblezprojections,h/ ̄ 2 and−h/ ̄ 2. It is com-
monly visualized as due to a spinning motion of the electron in addition to its orbital
motion.ADDITIONAL PROBLEMS
17.43a. Calculate the de Broglie wavelength for an electron
moving with the speed found in Example 17.9.
b. Compare your result in part a with the circumference
of the first Bohr orbit for the hydrogen atom.
17.44Consider a three-dimensional harmonic oscillator, with
the potential energy function
V
k
2(x^2 +y^2 +z^2 )wherekis the force constant.
a.Using the solution for a one-dimensional oscillator,
write the formula representing the ground-state wave
function in Cartesian coordinates.
b.Using the solution for a one-dimensional oscillator
write the formula representing the energy eigenvalues.
Give the degeneracy of each of the first three energy
levels. Draw an energy level diagram for these energy
levels.
c.Draw a sketch of the orbital region for the ground
state and one of the states of the first excited level
(specify the state).
d.Explain why the solution could also be carried out in
spherical polar coordinates. Do not carry out thesolution. What can you say about the functions ofθ
andφthat occur in the solution? What can you say
about the quantum numbers that would occur? What
can you say about the energy levels? What can you
say about the degeneracies of the first three energy
levels? What other variables other than the energy
will have predictable values for the states
corresponding to the solutions in spherical polar
coordinates?
17.45Calculate the expectation value and the standard
deviation ofLzfor a hydrogen atom in the 2pxstate.
Explain what the values mean.
17.46a.For a hydrogen atom in the 2pzstate, give the value of
〈Lz〉and〈L^2 z〉.
b.From your result of part a, give the value of〈Lx〉,
〈L^2 x〉,〈Ly〉, and〈L^2 y〉.
17.47For a hydrogen atom in the 2pxstate, give the value of
〈Lx〉and〈L^2 x〉. What can you say about the value of〈L^2 y〉
for this state?
17.48Give the numerical value of each of the following:a.The reduced mass of the two particles in a hydrogen
atom.