Physical Chemistry Third Edition

17.5 Expectation Values in the Hydrogen Atom 749

b.Sketch the orbital regions for the first few energy

eigenfunctions for a particle in a three-dimensional

spherical box.

c.Sketch the orbital regions for the first few energy

eigenfunctions for a particle in a three-dimensional

rectangular box. Compare them with the orbital

regions in part b.

17.28Sketch the nodal surfaces (cones and planes) for the real
and imaginary parts of the spherical harmonic functions:
a.Y 10 (Θ,φ)
b.Y 22 (Θ,φ)
c.Y 31 (Θ,φ)
d.Y 21 (Θ,φ)

17.29Sketch the nodal surfaces (cones and planes) for the real
and imaginary parts of the spherical harmonic functions:
a.Y 32 (Θ,φ)
b.Y 40 (Θ,φ)
c.Y 33 (Θ,φ)
d.Y 20 (Θ,φ)

17.30Transform the expression forΘ 11 Φ 1 yto Cartesian

coordinates. Show that this function is an eigenfunction

of the operator̂Lyand find its eigenvalue.

17.31Draw rough graphs of the following pairs of functions

and argue from the graphs that the functions are

orthogonal to each other:

a.Θ 00 (θ) andΘ 10 (θ)

b.Θ 11 (θ) andΘ 21 (θ)

c.Φ 2 xandΦ 3 y

17.32Calculate the angle between thezaxis and each of the

cones of possible directions of the orbital angular

momentum forl2.

17.33Find the ratio of the magnitude of the orbital angular

momentum to the maximum value of itszcomponent for

each of the casesl1, 2, 3, and 4.

17.34a.Draw a rough picture of the nodal surfaces of each of

the real 3dorbitals. From these, draw rough pictures

of the orbital regions.

b.Do the same for the complex 3dorbitals

(eigenfunctions of̂Lz).

17.5 Expectation Values in the Hydrogen Atom

For stationary states the expectation value for an error-free measurement of a mechan-

ical variableAis given by Eq. (16.4-4)

〈A〉

∫

ψ∗̂Aψdq

∫

ψ∗ψdq

(17.5-1)

If the wave function is normalized, the denominator in this equation is equal to unity

and can be omitted.

Normalization of the Hydrogen Atom Orbitals

For motion of one particle in three dimensions, normalization in Cartesian coordinates

means that

∫∞

−∞

∫∞

−∞

∫∞

−∞

ψ(x,y,z)∗ψ(x,y,z)dxdydz

∫∞

−∞

∫∞

−∞

∫∞

−∞

|ψ(x,y,z)|^2 d^3 r 1

(17.5-2)

where

d^3 rdxdydz (Cartesian coordinates) (17.5-3)