Physical Chemistry Third Edition

736 17 The Electronic States of Atoms. I. The Hydrogen Atom

Exercise 17.4

Transform the expression forΘ 11 Φ 1 xto Cartesian coordinates. Show that this function is an

eigenfunction of the operator̂Lx, and find its eigenvalue.

PROBLEMS

Section 17.2: The Relative Schrödinger Equation.
Angular Momentum

17.5 Using formulas in Appendix F, write the formulas for the
spherical harmonic functionsY 43 andY 42.

17.6 Sketch graphs of the functions and of their squares:

a.Θ 10 (θ)

b.Θ 11 (θ)

c.Θ 20 (θ)

d.Θ 21 (θ)

e.Θ 22 (θ)

17.7 Sketch graphs of the real and imaginary parts of

a.Φ 2 (φ)

b.Φ 3 (φ)

c.Φ 4 (φ)

d.Φ 5 (φ)

17.8 a.Show that the functionsΘ 00 (θ) andΘ 10 (θ) are
orthogonal to each other.

b. Show that the functionsΘ 11 (θ) andΘ 21 (θ) are

orthogonal to each other.

17.9 Calculate the expectation value and the standard

deviation ofθfor a hydrogen atom in a state corresponding

toY 00.

17.10Calculate the expectation value and the standard deviation

ofθfor a hydrogen atom in a state corresponding toY 11.

Explain what the value means.

17.11a.Calculate the expectation value and the standard

deviation ofφfor a hydrogen atom in the 2px

state.

b. Calculate the expectation value and the standard

deviation ofφfor a hydrogen atom in the 2py

state.

17.12a.Use Eq. (16.5-5) and the expression for the commutator

[̂Lx,̂Ly] in Problem 16.16 to obtain an uncertainty

relation for̂Lxand̂Lyfor the state corresponding to the

spherical harmonic functionY 21.

b. Repeat part a using the spherical harmonic functionY 00.

Comment on your result.

17.3 The Radial Factor in the Hydrogen Atom

Wave Function. The Energy Levels of the

Hydrogen Atom

In Section 17.2, we wrote the energy eigenfunction for any central-force system as

ψ(r,θ,φ)R(r)Ylm(θ,φ)R(r)Θlm(θ)Φm(φ) (17.3-1)

The spherical harmonic functionsYlm(θ,φ)Θlm(θ)Φm(φ) are the same functions for

any system with a potential energy depending only onr. We now seek the differential

equation for the radial factor for the hydrogen atom. We replacêL^2 Ybyh ̄^2 l(l+1)Y

in Eq. (17.2-5), according to Eq. (17.2-27), and multiply the resulting equation byR

to obtain the differential equation for the radial factor:

−

d

dr

(

r^2

dR

dr

)

+

2 μr^2

h ̄^2

(V−E)R+l(l+1)R 0 (17.3-2)