Concise Physical Chemistry

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

BREAKING DEGENERACY 261

TABLE 16.1 The First Six Wave Functions for Hydrogen.

ψ 1 s=

1

π^1 /^2

(

1

a

) 3 / 2

e−Zr/a

ψ 2 s=

1

4 ·(2π)^1 /^2

(

2 −

Zr

a

) 3 / 2

e−Zr/^2 a

ψ 2 pz=

1

4 ·(2π)^1 /^2

(

Z

a

) 5 / 2

re−Zr/^2 acosθ

ψ 2 px=

1

4 ·(2π)^1 /^2

(

Z

a

) 5 / 2

re−Zr/^2 asinθcosφ

ψ 2 py=

1

4 ·(2π)^1 /^2

(

Z

a

) 5 / 2

re−Zr/^2 asinθsinφ

ψ 3 s=

1

81(3π)^1 /^2

(

Z

a

) 3 / 2 (

27 − 18

Zr

a

+ 2

Z^2 r^2

a^2

) 3 / 2

e−Zr/^3 a

TABLE 16.2 The First ThreesWave Functions for

Hydrogen (Simplified Form).

1 se−r

2 s ( 2 −r)e−r= 2 e−r−re−r

3 s

(

27 − 18 r+ 2 r^2

)

e−r= 27 e−r− 18 re−r+ 2 r^2 e−r

R(r) 27 18 r 2 r

2

e

.63r

l(r) 0

010155

1

0

R(r)

l(r)

r

FIGURE 16.8 Roots of the radial 3swave function of atomic hydrogen as a function of

distancer. Three nodes correspond to zeros of the function at aboutr= 2 , 7 ,and∼15. The

functionl(r) simply draws a horizontal line.