Physical Chemistry Third Edition

820 19 The Electronic States of Atoms. III. Higher-Order Approximations

the quantum number for the total orbital angular momentum, corresponds to the lowest

energy. The first rule correlates with the fact that the larger values ofScorrespond to

lower probability for small electron–electron distances, lowering the potential energy.

The form of the periodic table is determined by electron configurations. Elements

with the same number of electrons in the outer (valence) shell have similar chemical

properties. For example, all of the inert gases have eight electrons in the outer shell,

corresponding to a stable configuration with fully occupiedsandpsubshells in the

outer shell.

ADDITIONAL PROBLEMS

19.41 a.Using first-order perturbation, obtain a formula for the
ground-state energy of a particle in a one-dimensional
box with an additional potential energyĤ′bx,
wherebis a negative constant.

b.Obtain a formula for the ground-state energy of the

particle of part a, using the variation method and the

zero-order wave function as a trial function. Compare

your answer with that of part a.

c.Obtain a formula for the ground-state energy of the

particle of part a, using the variation method and the

trial functionφAx^2 (a−x). Compare your answer

with that of part a.

d.Assume that the particle is an electron in a

one-dimensional box of length 1.00 nm and that the

ends of the box are charged so thatb− 1. 60 ×

10 −^11 Jm−^1. Evaluate the energy according to parts

a and c.

19.42 a.Obtain a formula for the ground-state energy of the
particle of part a, using the variation method and the
trial functionφAxn(a−x).

b.Evaluate the energy according to part d and find the

optimum value ofn.

19.43A particle in a one-dimensional box (0<x<a) has an
additional potential energy imposed on it, given by
Ĥ′−Bsin(πx/a) whereBis a positive constant.
a.Using perturbation theory, obtain a formula for
E(0)+E(1)for the ground state (n1).

b.If the particle is an electron, if the box length is

10 .0 Å, and ifB 5. 00 × 10 −^21 J, find the values of

E(0)andE(1). Do you think thatE(0)+E(1)would be

a usable approximation to the correct energy?

c.How do you think the correct wave function would

compare with the unperturbed wave function (that for

B0).

d.Explain why a variation calculation using the

zero-order wave function as a trial function leads to

the same result asE(0)+E(1)in the perturbation

theory.

19.44If the nucleus of a hydrogen atom is atz0, and if an

electric fieldEis applied in thezdirection, the energy of

the system of the nucleus is equal to zero and the energy

of the electron is

Eel−Eez−Eercos(θ)

wherezis the vertical coordinate of the electron andr

andθare polar coordinates of the electron. An electric

dipole consists of two charges of equal magnitude,Q,

and opposite sign separated by a distanced. The dipole

moment is defined to be

|μ|Qd

When a hydrogen atom is placed in an electric field, its

electrons are shifted in one direction and its nucleus is

shifted in the other direction, creating an induced electric

dipole. The polarizabilityαof the atom is defined by

μinducedαE

For a hydrogen atom and for an electric field in thez

direction, the Hamiltonian operator is

̂HĤ(0)+eEzĤ(0)+eErcos(θ)

whereĤ(0)is the Hamiltonian in the absence of the field.

The magnitude of the dipole moment is

μe〈z〉e〈rcos(θ)〉

a.Use the variation method to find the energy of a

hydrogen atom in an electric field in thezdirection

with magnitudeE. Use the 1swave function as the

variation trial function. No minimization will be

possible. Compare your result with that of the

perturbation method in Problem 19.22.