11.44.Calculate the difference between the Bohr radius de-
fined as aand the Bohr radius defined as a 0.
11.45.To four significant figures, the first four lines in the
Balmer series in the hydrogen atom (n 2 2) spectrum appear
at 656.5, 486.3, 434.2, and 410.3 nm. (a)From these num-
bers, calculate an average value of RH, the Rydberg constant.
(b)At what wavelengths would similar transitions appear for
He?
11.46.What would the wavelengths of the Balmer series for
deuterium be?
11.47.Construct an energy level diagram showing all orbitals
for the hydrogen atom up to n5, labeling each orbital with
its appropriate quantum numbers. How many different or-
bitals are in each shell?
11.48.What are the values of E, L, and Lzfor an F^8 atom
whose electron has the following wavefunctions, listed as
n,,m? (a)1,0,0(b)3,2,2(c)2,1, 1 (d)9,6, 3.
11.49.Why does the wavefunction 4,4,0not exist? Similarly,
why does a 3fsubshell not exist? (See exercise 11.48 for no-
tation definition.)
11.50.Calculate the total electronic energy of a mole of hy-
drogen atoms. Calculate the total electronic energy of a mole
of Heatoms. What accounts for the difference in the two to-
tal energies?
11.51.What is the probability of finding an electron in the 1s
orbital within 0.1 Å of a hydrogen nucleus?
11.52.What is the probability of finding an electron in the 1s
orbital within 0.1 Å of an Ne^9 nucleus? Compare your an-
swer to the answer to exercise 11.51 and justify the difference.
11.53.State how many radial, angular, and total nodes are in
each of the following hydrogen-like wavefunctions. (a) 2 s
(b) 3 s(c) 3 p(d) 4 f(e) 6 g(f) 7 s
11.54.Illustrate that the hydrogen wavefunctions are or-
thogonal by evaluating 2 *s 1 sd over all space.
11.55.Verify the specific value of a, the Bohr radius, by us-
ing the values of the various constants and evaluating equa-
tion 11.68.
11.56.Show that rmaxis given by equation 11.68 for 1 s.
Take the derivative of 4r^2 ^2 with respect to r, set it equal to
zero, and solve for r.
11.57.Use the forms of the wavefunctions in Table 11.4 to
determine the explicit forms for the 2pxand 2pynonimaginary
wavefunctions.
11.58.Evaluate Lzfor 3px. Compare it to the answer in
Example 11.25, and explain the difference in the answers.
11.59.Using equations 11.67 as an example, what would the
combinations for the five real 3dwavefunctions be? Use Table
11.4 to assist you.
11.60.Evaluate rfor 1 s(assume that the operator rˆ is de-
fined as “multiplication by the coordinate r”). Why does r
not equal 0.529 Å for 1 s? In this case, d 4 r^2 dr.11.61.Graph the first five wavefunctions for the harmonic os-
cillators and their probabilities. Superimpose these graphs on
the potential energy function for a harmonic oscillator and nu-
merically determine the xvalues of the classical turning points.
What is the probability that an oscillator will exist beyond
the classical turning points? Do plots of the probability begin
to show a distribution as expected by the correspondence
principle?
11.62.Construct three-dimensional plots of the first three
families of spherical harmonics. Can you identify the values of
and that correspond to nodes?
11.63.Set up and evaluate numerically the integral that shows
that Y^11 and Y^1 1 are orthogonal.
11.64.Plot the 90% surfaces of the hydrogen atom 2sand
2 pangular wavefunctions in 3-D space. Can you identify nodes
in your graphic?Exercises for Chapter 11 369Symbolic Math Exercises