Physical Chemistry Third Edition

750 17 The Electronic States of Atoms. I. The Hydrogen Atom

z

y

x

Volume element of

volumer^2 sin(
)dd
dr

dr

rsin(
)d

 rsin(
)d

rsin(
)

r d

r

d

Figure 17.10 The Volume Element in Spherical Polar Coordinates.

In spherical polar coordinates,

∫∞

0

∫π

0

∫ 2 π

0

ψ(r,θ,φ)|^2 r^2 sin(θ)dφdθdr

∫∞

0

∫π

0

∫ 2 π

0

|ψ(r,θ,φ)|^2 d^3 r1 (17.5-4)

where

d^3 rr^2 sin(θ)dφdθdr (spherical polar coordinates) (17.5-5)

The factorr^2 sin(θ), which is called aJacobian, is required to complete the element

of volume in spherical polar coordinates. The form of this Jacobian can be deduced

from the fact that an infinitesimal length in therdirection isdr, an infinitesimal arc

length in theθdirection isrdθ, and an infinitesimal arc length in theφdirection is

rsin(θ),dφif the angles are measured in radians. The element of volume is the product

of these mutually perpendicular infinitesimal lengths. Spherical polar coordinates were

depicted in Figure 17.3. The volume element is crudely depicted by finite increments

in Figure 17.10.

The Jacobian is named for Carl Gustav
Jacob Jacobi, 1804–1851, a great
German mathematician who made
numerous contributions to mathematics.

The normalization integral for the hydrogen atom orbitals can be factored in spherical

polar coordinates:

∫∞

0

R∗Rr^2 dr

∫π

0

Θ∗Θsin(θ)dθ

∫ 2 π

0

Φ∗Φdφ 1 (17.5-6)

We make the additional normalization requirement that each of the three integrals in

this equation equals unity. The constants in the formulas for the R,Θ, andΦfactors

that we have introduced correspond to this requirement. These separate normalizations

in Eq. (17.5-6) simplify the calculation of many expectation values.

EXAMPLE17.7

Calculate the expectation values〈 1 /r〉and〈V〉for a hydrogen-like atom in the 1sstate.