Physical Chemistry , 1st ed.

which is easily shown to be, in units of wavenumbers and to four significant
figures,



8

e



4

(^20)



h^2

109,700 cm^1

This is the Rydberg constant,RH, from the hydrogen atom spectrum.* Quantum
mechanics therefore predicts the experimentally determined hydrogen atom spec-
trum.At this point, quantum mechanics predicts everything that Bohr’s the-
ory did and more, and so supersedes the Bohr theory of the hydrogen atom.
Since the spherical harmonics are part of the hydrogen atom’s wavefunc-
tions, it should come as no surprise that the total angular momentum and the
zcomponent of the total angular momentum are also observables that have
known analytic and quantized values. They are

ˆL^2 n,,m(1)^2 n,,m

Lˆzn,,mmn,,m

so that the quantized values for total angular momentum are (1) and
for the zcomponent are m. The quantum number is called the angular
momentum quantum number.The mquantum number is the z-component
angular momentum quantum number,sometimes called the magnetic quantum
number due to the differing behavior of wavefunctions having different m
values in a magnetic field (another topic for later). The angular momentum of
the hydrogen atom (due mostly to the electron) is quantized, as Bohr assumed.
However, the exact values of the quantized angular momentum are slightly dif-
ferent than what Bohr assumed. It was not possible to know this in 1913, how-
ever, and though ultimately incorrect, Bohr’s theory should be remembered as
a crucial step in the right direction.
This treatment of the hydrogen atom is also applicable to any atom that has
only one electron. In cases of other atoms, the nuclear charge is different and
the overall atom itself has a charge. The atomic number,Z, and the reduced
mass are the only changes in any of the equations from above (and the re-
duced mass approaches the mass of the electron as the nucleus gets larger). The
Schrödinger equation for these hydrogen-likeions is





2 

^2



r

1

2 



r

r^2 





r



r^2 s

1

in









sin 









r^2 si

1

n^2 









2

 (^2)  4





Z



e

0

2

r



E (11.65)

where Zshows up only in the potential energy. The only other major change
is in the expression for the quantized energy of these ions, which now has the
form

E

8

Z



2

(^20)
e
h
4
2



n^2

 (11.66)

The wavefunctions themselves also have a Zdependence on them. Table 11.4
gives the complete wavefunctions with their Zdependence already included.
(In our previous treatment of the hydrogen atom,Zwas 1.) The angular mo-
menta observables have the same forms as given above. Spectra of the hydro-
gen-like ions, which have been observed experimentally, are as simple as that

11.10 The Hydrogen Atom: The Quantum-Mechanical Solution 357

*Using modern values of the fundamental constants and to eight significant figures,
RH109,677.58 cm^1.