Concise Physical Chemistry

c16 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

264 WAVE MECHANICS OF SIMPLE SYSTEMS

The operatorhˆiis approximate because the position of each individual electronjiis

uncertain. The potential energy of repulsion between electronsViis included as an

average by integrating over the Born probability function for each electronj =iand

then summing the results:

Vi(ψi(rj))=

∑

j =i

∫ ∣∣

ψj(rj)

∣

∣^2

rij

dτ

The N equations are then solved using approximations toψ 1 ,ψ 2 ,..., ψnand enter-

ing an iterative procedure. Each of the solutionsεifor an atomic system is the energy

of an atomic orbital 1s,2s,2p, and so on. Each energy coincides with a distinct

set of self-consistent one-electron wave functionsψ 1 ,ψ 2 ,..., ψn.Theradial wave

functionslead to electron probability density functions with 1, 2, 3,...maxima at

specific distances from the nucleus corresponding to the 1s,2s,2p, and so on, “shells”

of electron density familiar from elementary chemistry.

PROBLEMS

Problem 16.1

The integral

∫∞

−∞

xe−x

2

dx

is an “improper integral because both of its limits are infinite (one would do to make

it improper). Evaluate this integral, that is, findy(x)for

y(x)=

∫∞

−∞

xe−x

2

dx

Verify your answer using a numerical integration computer routine such as Mathcad©C.

Problem 16.2

Show that the first two orbitals of a particle in a one-dimensional box of length 1 unit

are orthogonal, that is,

∫ 1

0

ψ 1 ψ 2 dτ= 0

for

ψ 1 =

√

2sin(πx)