Physical Chemistry , 1st ed.

pretation that this light is emitted by electronic transitions was firmly estab-

lished by Bohr, who derived the equation





1

R

n

1

 22     n

1

 (^21)  (15.2)
by assuming that the angular momentum of the electron is quantized.is the
wavelength of the light,Ris called the Rydberg constant, and n 1 and n 2 are
quantum numbers.Quantum mechanics provides a similar equation for the
spectrum of the hydrogen atom (albeit from different assumptions, namely
that the wavefunctions of electrons in hydrogen must satisfy the Schrödinger
equation). Quantum mechanics also determines that the Rydberg constant Ris

R

8

e

4

(^20)



h^2

 (15.3)

where the constants in the above expression have their usual meaning. The

relative simplicity of the spectrum of the hydrogen atom is based on equation

15.2, which is itself based on experiment (that is, observation). And so a

“selection rule” of sorts can be stated for electronic transitions in the hydro-

gen atom: allowed transitions are dictated by changes in the principal quan-

tum number.

However, this is misleading. Although electronic energy levels are dictated

by the principal quantum number, we should remember that a principal quan-

tum shell in a hydrogen atom has other quantum numbers, namely and m.

If the symmetries of the operator and wavefunctions in equation 15.1 were ex-

amined, one would find that it is the angular momentum quantum number

that dictates the selection rule. The specific selection rule for allowed electronic

transitions in the hydrogen atom (or, for that matter, hydrogen-like atoms) is

 1 (15.4)

Since photons themselves have an angular momentum, this selection rule is

consistent with the law of conservation of angular momentum. There is also a

potential effect on the mquantum number, since the change in the quan-

tum number may or may not occur in the zcomponent of the total angular

momentum. Therefore, the accompanying selection rule is

m0, 1 (15.5)

There is no restriction on the change in n, the principal quantum number. n

can have any value. Why are the selection rules, equations 15.4 and 15.5, not

obvious from the spectrum of the hydrogen atom? Because the electronic

energy of the hydrogen atom does not depend on the angular momentum

quantum number. It depends only on the principalquantum number,n.The

spatial wavefunctions of the hydrogen atom are n^2 -fold degenerate because of

variations in the and mquantum numbers, so the spectrum of the hydro-

gen atom appearsas if the energy differences are due to changes in the princi-

pal quantum number.* In reality, the spectral lines are due to electrons chang-

ing not just their principal quantum number but also, according to the selection

rule, their angular momentum quantum number.

Figure 15.1 shows some of the transitions that are possible according to the

selection rule above. For several of the possible changes in the nquantum

number, the changes are different but lead to the same E(which is what

15.3 The Hydrogen Atom 521

*They are 2n^2 -fold degenerate if the spin of the electron is included.

n  3

n  2

n  1

Energy

 0  1  2

 0 ^1

 0
Figure 15.1 Some of the allowed transitions
of the single electron in the hydrogen atom.
Despite the complexity of the diagram, the hy-
drogen atom’s electronic spectrum is relatively
simple because the subshells within the same
principal quantum number are degenerate.