bei48482_FM

136 Chapter Four

If the quantum number of the initial (higher-energy) state is niand the quantum

number of the final (lower-energy) state is nf, we are asserting that

Initial energyfinal energyphoton energy

EiEfh (4.16)

where is the frequency of the emitted photon. From Eq. (4.15) we have

EiEfE (^1)    E (^1)   
We recall that E 1 is a negative quantity (13.6 eV, in fact), so E 1 is a positive quan-
tity. The frequency of the photon released in this transition is therefore
   (4.17)
Since c, 1cand
    (4.18)
Equation (4.18) states that the radiation emitted by excited hydrogen atoms
should contain certain wavelengths only. These wavelengths, furthermore, fall into
definite sequences that depend upon the quantum number nfof the final energy
level of the electron (Fig. 4.16). Since ninfin each case, in order that there be
an excess of energy to be given off as a photon, the calculated formulas for the first
five series are
Lyman nf1:     n2, 3, 4,
1

n^2
1

12
E 1

ch
1


1

n^2 i
1

n^2 f
E 1

ch
1


Hydrogen
spectrum
1

n^2 i
1

n^2 f
E 1

h
EiEf

h
1

n^2 i
1

n^2 f
1

n^2 f
1

n^2 i
Quantization in the Atomic World
S
equences of energy levels are characteristic of all atoms, not just those of hydrogen. As in
the case of a particle in a box, the confinement of an electron to a region of space leads to
restrictions on its possible wave functions that in turn limit the possible energies to well-defined
values only. The existence of atomic energy levels is a further example of the quantization, or
graininess, of physical quantities on a microscopic scale.
In the world of our daily lives, matter, electric charge, energy, and so forth appear to be con-
tinuous. In the world of the atom, in contrast, matter is composed of elementary particles that
have definite rest masses, charge always comes in multiples of eore, electromagnetic waves
of frequency appear as streams of photons each with the energy h, and stable systems of par-
ticles, such as atoms, can possess only certain energies. As we shall find, other quantities in na-
ture are also quantized, and this quantization enters into every aspect of how electrons, protons,
and neutrons interact to endow the matter around us (and of which we consist) with its famil-
iar properties.
bei48482_ch04.qxd 1/29/02 4:50 PM Page 136