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The idea of matter waves was soon tested after de Broglie wave with well-defined shapes and frequencies. A wave is
published his theory. Experiments with a beam of light pass- not localized like a particle and is spread out in space. The
ing by the edge of a sharp-edged obstacle produced interfer- quantum mechanics model is, therefore, a series of orbitlike
ence patterns. This was part of the evidence for the wave nature smears, or fuzzy statistical representations, of where the elec-
of light, since such results could only be explained by waves, tron might be found.
not particles. When similar experiments were performed with
a beam of electrons, identical wave property behaviors were
observed. This and many related experiments showed without THE QUANTUM MECHANICS MODEL
doubt that electrons have both wave properties and particle
The quantum mechanics model is a highly mathematical treat-
properties. And, as was the case with light waves, measurements
ment of the mechanics of matter waves. In addition, the wave
of the electron interference patterns provided a means to mea-
properties are considered as three-dimensional problems,
sure the wavelength of electron waves.
and three quantum numbers are needed to describe the fuzzy
Recall that waves confined on a fixed string establish reso-
electron cloud. The mathematical details will not be presented
nant modes of vibration called standing waves (see chapter 5).
here. The following is a qualitative description of the main ideas
Only certain fundamental frequencies and harmonics can exist
in the quantum mechanics model. It will describe the results of
on a string, and the combination of the fundamental and over-
the math ematics and will provide a mental visualization of what
tones gives the stringed instrument its particular quality. The
it all means.
same result of resonant modes of vibrations is observed in any
First, understand that the quantum mechanical theory is
situation where waves are confined to a fixed space. Charac-
not an extension or refinement of the Bohr model. The Bohr
teristic standing wave patterns depend on the wavelength and
model considered electrons as particles in circular orbits that
wave velocity for waves formed on strings, in enclosed columns
could be only certain distances from the nucleus. The quantum
of air, or for any kind of wave in a confined space. Electrons
mechanical model, on the other hand, considers the electron as
are confined to the space near a nucleus, and electrons have
a wave and considers the energy of its harmonics, or modes, of
wave properties, so an electron in an atom must be a confined
standing waves. In the Bohr model, the location of an electron
wave. Does an electron form a characteristic wave pattern?
was certain—in an orbit. In the quantum mechanical model, the
This was the question being asked in about 1925 when Heisen-
electron is a spread-out wave.
berg, Schrödinger, Dirac, and others applied the wave nature
Quantum mechanics describes the energy state of an elec-
of the electron to develop a new model of the atom based on
tron wave with four quantum numbers:
the mechanics of electron waves. The new theory is now called
wave mechanics, or quantum mechanics. 1. Distance from the Nucleus. Th e principal quantum
number describes the main energy level of an electron
in terms of its most probable distance from the nucleus.
WAVE MECHANICS The lowest energy state possible is closest to the nucleus

Erwin Schrödinger, an Austrian physicist, treated the atom and is assigned the principal quantum number of 1
as a three-dimensional system of waves to derive what is now (n = 1). Higher states are assigned progressively higher
called the Schrödinger equation. Instead of the simple circular positive whole numbers of n = 2, n = 3, n = 4, and so on.
planetary orbits of the Bohr model, solving the Schrödinger Electrons with higher principal quantum numbers have
equation results in a description of three-dimensional shapes higher energies and are located farther from the nucleus.
of the patterns that develop when electron waves are con- 2. Energy Sublevel. Th e angular momentum quantum number
fined by a nucleus. Schrödinger first considered the hydrogen defines energy sublevels within the main energy levels.

atom, calculating the states of vibration that would be pos- Each sublevel is identified with a letter. Th e first four of

sible for an electron wave confined by a nucleus. He found these letters, in order of increasing energy, are s, p, d, and f.
that the frequency of these vibrations, when multiplied by The letter s represents the lowest sublevel, and the letter f

Planck’s constant, matched exactly, to the last decimal point, represents the highest sublevel. A principal quantum
the observed energies of the quantum states of the hydrogen number and a letter indicating the angular momentum
atom (E = hf ). The conclusion is that the wave nature of the quantum number are combined to identify the main energy
electron is the important property to consider for a successful state and energy sublevel of an electron. For an electron
model of the atom. in the lowest main energy level, n = 1 and in the lowest
The quantum mechanics theory of the atom proved to be sublevel, s, the number and letter are 1s (read as “one-s”).

very successful; it confirmed all the known experimental facts Thus, 1s indicates an electron that is as close to the nucleus
and predicted new discoveries. The theory does have some of as possible in the lowest energy sublevel possible.

the same quantum ideas as the Bohr model; for example, an There are limits to how many sublevels can occupy
electron emits a photon when jumping from a higher state to a each of the main energy levels. Basically, the lowest main
lower one. The Bohr model, however, considered the particle energy level can have only the lowest sublevel, and another
nature of an electron moving in a circular orbit with a defi- sublevel is added as you move up through the main energy
nitely assigned position at a given time. Quantum mechan- levels. Thus, the lowest main energy level, n = 1, can have

ics considers the wave nature, with the electron as a confined only the s sublevel. Th e n = 2 main energy level can have s
8-11 CHAPTER 8 Atoms and Periodic Properties 213