12.2 Spin
12.1.In the Stern-Gerlach experiment, silver atoms were used.
This was a good choice, as it turned out. Using the electron
configuration of silver atoms, explain why silver was a good
candidate for being able to observe the intrinsic angular mo-
mentum of the electron. (Hint:Don’t use the aufbau principle
to determine the electron configuration of Ag, because it’s one
of the exceptions. Look up the exact electron configuration in
a table.)
12.2.Using and labels, write two possible wavefunctions
for an electron in the 3d 2 orbital of He.
12.3.Antimatter and matter destroy each other, giving off
electromagnetic radiation as the total mass of the particles is
converted to energy. Using Einstein’s matter-energy equiva-
lence equation Emc^2 , calculate the number of joules of en-
ergy given off when (a)one electron and one positron destroy
each other (the mass of the positron is the same as the mass
of the electron), and (b)1 mole of electrons destroys 1 mole
of positrons.
12.4.Are the two spin functions and orthogonal? Why or
why not?
12.5. (a)Differentiate between the quantum numbers sand
ms. (b)What will the possible values of msbe for a particle
having an squantum number of 0, 2, and ^32 ?
12.3 He Atom
12.6.Are mathematical expressions for the following poten-
tial energies positive or negative? Explain why in each case.
(a)The attraction between an electron and a helium nucleus
(b)The repulsion between two protons in a nucleus (c)The
attraction between a north and a south magnetic pole (d)The
force of gravity between the sun and Earth (e)A rock perched
on the edge of a cliff (with respect to the base of the cliff)
12.7.Write out the complete Schrödinger equation for Li and
indicate what terms in the operator make the equation un-
solvable exactly.
12.8. (a)Assume that the electronic energy of Li was a prod-
uct of three hydrogen-like wavefunctions with principal quan-
tum number equal to 1. What would be the total energy of Li?
(b)Assume that two of the principal quantum numbers are 1
and the third principal quantum number is 2. Calculate the es-
timated electronic energy.
(c)Compare both energies with an experimental value of
3.26 10 ^17 J. Which estimate is better? Is there any reason
you might assume that this estimate would be better from the
start?
12.4 Spin Orbitals; Pauli Principle
12.9.Spin orbitals are products of spatial and spin wave-
functions, but correct antisymmetric forms of wavefunctions
for multielectron atoms are sums and differences of spatial
wavefunctions. Explain why acceptable antisymmetric wave-
functions are sums and differences (that is, combinations) in-
stead of productsof spatial wavefunctions.
12.10.Show that the correct behavior of a wavefunction for
He is antisymmetric by exchanging the electrons to show that
(1, 2) (2, 1).
12.11.Use a Slater determinant to determine the correctly
behaved wavefunction for the ground state Li.
12.12.Why does the concept of antisymmetric wavefunc-
tions not need to be considered for the hydrogen atom?
12.13. (a)Construct Slater determinant wavefunctions for Be
and B. (Hint:Although you need only include one porbital for
B, you should recognize that up to six possible determinants
can be constructed.)
(b)How many different Slater determinants can be con-
structed for C, assuming that the pelectrons spread out among
the available porbitals and have the same spin? How many
different Slater determinants are there for F?
12.14.Examples in the chapter suggest that the number of
terms in a proper antisymmetric wavefunction given by a Slater
determinant is n!, where nis the number of electrons and!
implies the factorial of n(n! 1 2 3 4 n). Coming
up with the correct terms for a proper wavefunction becomes
a difficult task very quickly; hence the extreme simplicity pro-
vided by a Slater determinant.
(a)Verify the n! relationship for the examples of He, Li, and
Be given in the text. (Hint:you may have to review rules for
evaluating determinants.)
(b)Determine the number of terms in an antisymmetric wave-
function for C, Na, Si, and P.12.5 Aufbau Principle
12.15.Using a periodic table (or Table 12.1), find the ele-
ments whose electron configurations do not follow the aufbau
principle strictly. Comment on any relationship between these
elements or their place within the periodic table.
12.16.Label each electron configuration for the listed atom
as a ground state or an excited state. (a)Li, 1s^22 p^1 (b)C,
1 s^22 s^22 p^2 (c)K, 1s^22 s^22 p^63 s^23 p^64 p^1 (d)Be, 1s^23 s^2 (e)
U (outer shells only) 7s^25 f^37 p^1
12.17.For each atomic state in exercise 12.16, determine
how many possible ways the electrons in the outermost shell
can occupy spin orbitals and satisfy Hund’s rule, and list them
explicitly. For example, in the case of Li, the outermost shell
has three more specific possibilities: 2px^1 (spin or spin ), or
2 py^1 (spin or spin ), or 2pz^1 (spin or spin ), for a total of
six possibilities.12.6 Perturbation Theory
12.18.In deriving equation 12.17, we stated that the correc-
tion to the energy is an approximation. Why can’t we simply
assume that the integral representing the first-order correction
can be solved analytically, and therefore be exact?
12.19.An anharmonicoscillator has the potential function
V^12 kx^2 cx^4 , where ccan be considered a sort of anhar-
monicity constant.Determine the energy correction to the416 Exercises for Chapter 12
EXERCISES FOR CHAPTER 12