For convenience, several of the first few wavefunctions are listed in Table 11.4
along with their respective n,, and mquantum numbers. Each characteris-
tic set (n,,m) refers to a specific wavefunction. It is easy to show that for any
n, the total number of possible wavefunctions having that value ofnis n^2.
(This will increase by a factor of 2 when we include the spin of the electron,
but that will be considered in Chapter 12.)
The eigenvalue for energy also has an analytic solution. It is
E
8 e
2
04
h (^2) n 2 (11.63)
The energy is negativehere. This is due to the convention that the interaction
of oppositely charged particles contributes to a decrease in energy. (Conversely,
the repulsion of similarly charged particles would be positive in energy.) An
energy of zero corresponds to the proton and electron at infinite distance from
each other (so that the potential energy is zero) and having no kinetic energy
with respect to each other. The energy depends on a collection of constants—
the charge on the electron,e , the reduced mass of the hydrogen atom , the
permittivity of free space 0 , Planck’s constant h—and the integer n. The en-
ergy depends on the index n;nis a quantum number and the total energy is
quantized. The energy of the hydrogen atom does not depend on the quantum
numbers or m,only on n. The index nis therefore called the principal
11.10 The Hydrogen Atom: The Quantum-Mechanical Solution 355
Table 11.4 Complete wavefunctions for hydrogen-like atomsa
n m n,,m
1 0 0 Za
3
(^3)
1/2
eZr/a
2 0 0 ^18 ^2 Za
3
(^3)
1/2
^2
Z
a
r
e
Zr/2a
2 1 1 ^18 ^2 Za
3
(^3)
1/2
ZareZr/2asin ei
2 1 0 ^18 ^2 Za
3
(^3)
1/2
ZareZr/2acos
2 1 1 ^18 ^2 Za
3
(^3)
1/2
ZareZr/2asin ei
3 0 0 2143 ^3 Za
3
(^3)
1/2
^27
18
a
Zr 2
a
Z
2
r^2
e
Zr/3a
3 1 1 811 Za
3
(^3)
1/2
Zar 6 ZareZr/3asin ei
3 1 0 811 ^2 Za
3
(^3)
1/2
Zar 6 ZareZr/3acos
3 1 1 811 Za
3
(^3)
1/2
Zar 6 ZareZr/3asin ei
3 2 2 1162 Za
3
(^3)
1/2
Za
2
2
r^2 eZr/3asin^2 e^2 i
3 2 1 811 Za
3
(^3)
1/2
Za
2
2
r^2 eZr/3asin cos ei
3 2 0 4186 ^6 Za
3
(^3)
1/2
Za
2
2
r^2 eZr/3a(3 cos^2 1)
3 2 1 811 Za
3
(^3)
1/2
Za
2
2
r^2 eZr/3asin cos ei
3 2 2 1162 Za
3
(^3)
1/2
Za
2
2
r^2 eZr/3asin^2 e^2 i
aa^4
e