11.2 Quantum Behavior of Matter
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space unaffected by who is looking, or how. Nature does not allow us this
luxury. If there is no instrument or system interacting with an electron, we
cannot properly think of the electron proceeding along any one path between
observations. With quantum theory, cartoons of electrons orbiting nuclei are
fundamentally wrong. Also, drawings of individual electrons spread out like a
classical wave are wrong.
2. In quantum theory, certain individual processes cannot be predicted. Rather,
only the probability of a given outcome can be predicted. For example, if an
atom is left unobserved for a second, then the position of one of its electrons
within one of the possible electron orbitals cannot be predicted. Rather, we can
predict the probability of finding the electron in a certain position. We first have
to measure that electron’s position many times for the same initially unobserved
state to get an average measured position. Then this average, or ‘mean position’,
can be compared with a prediction of quantum theory.
3. Each particle can be represented by a complex valued wave function, denoted
x). In space, the probability density of finding that particle is proportional
to the magnitude of the wave function squared, ∗ (t, x))(t, x). The wave
function is a solution to a wave equation linear in , which means that the
principle of linear superposition applies to such waves: If 1 (t, x) and 2 (t, x)
are two solutions, then so is (t, x) = 1 (t, x) + 2 (t, x). This principle of
superposition also means that any wave function can be ‘decomposed’ into a
linear combination of simpler waves which ‘span’ the possible waves in the
given space. 1
Quantum theory predicts that the probability of finding a particle somewhere
in space can show a pattern of constructive and destructive interference.
Interference is a consequence of the superposition principle. In quantum theory,
the interference of a particle wave with itself can occur. For example, a
sequence of electrons sent (unobserved) toward a double slit will arrive at points
on a phosphor screen beyond the slits, but the accumulation of points will have
a distribution which matches a classical interference pattern, with nodes and
antinodes. In quantum theory language, the electron wave which arrives at
the screen is a superposition of the waves which came through each slit. No
classical theory of particles is capable of showing such interference. To observe
this interference effect for the two-slit experiment, the electron wave must be
allowed to pass through both slits. If any instrument is used to determine which
slit the electron passes through, the interference pattern is lost.
In a localized bound system left undisturbed for a known relaxation time,
particles settle into ‘quantum orbitals’, i.e. stationary solutions to the appropriate wave equation.
The expression of atomic and molecular orbitals as a linear combination of
only a few of the dominant components in an expansion in terms of simpler
waves is often used in chemistry to describe valence electrons in atoms and
1 A Fourier series and its generalizations is an example of such a decomposition.
515
space unaffected by who is looking, or how. Nature does not allow us this
luxury. If there is no instrument or system interacting with an electron, we
cannot properly think of the electron proceeding along any one path between
observations. With quantum theory, cartoons of electrons orbiting nuclei are
fundamentally wrong. Also, drawings of individual electrons spread out like a
classical wave are wrong.
2. In quantum theory, certain individual processes cannot be predicted. Rather,
only the probability of a given outcome can be predicted. For example, if an
atom is left unobserved for a second, then the position of one of its electrons
within one of the possible electron orbitals cannot be predicted. Rather, we can
predict the probability of finding the electron in a certain position. We first have
to measure that electron’s position many times for the same initially unobserved
state to get an average measured position. Then this average, or ‘mean position’,
can be compared with a prediction of quantum theory.
3. Each particle can be represented by a complex valued wave function, denoted
x). In space, the probability density of finding that particle is proportional
to the magnitude of the wave function squared, ∗ (t, x))(t, x). The wave
function is a solution to a wave equation linear in , which means that the
principle of linear superposition applies to such waves: If 1 (t, x) and 2 (t, x)
are two solutions, then so is (t, x) = 1 (t, x) + 2 (t, x). This principle of
superposition also means that any wave function can be ‘decomposed’ into a
linear combination of simpler waves which ‘span’ the possible waves in the
given space. 1
Quantum theory predicts that the probability of finding a particle somewhere
in space can show a pattern of constructive and destructive interference.
Interference is a consequence of the superposition principle. In quantum theory,
the interference of a particle wave with itself can occur. For example, a
sequence of electrons sent (unobserved) toward a double slit will arrive at points
on a phosphor screen beyond the slits, but the accumulation of points will have
a distribution which matches a classical interference pattern, with nodes and
antinodes. In quantum theory language, the electron wave which arrives at
the screen is a superposition of the waves which came through each slit. No
classical theory of particles is capable of showing such interference. To observe
this interference effect for the two-slit experiment, the electron wave must be
allowed to pass through both slits. If any instrument is used to determine which
slit the electron passes through, the interference pattern is lost.
In a localized bound system left undisturbed for a known relaxation time,
particles settle into ‘quantum orbitals’, i.e. stationary solutions to the appropriate wave equation.
The expression of atomic and molecular orbitals as a linear combination of
only a few of the dominant components in an expansion in terms of simpler
waves is often used in chemistry to describe valence electrons in atoms and
1 A Fourier series and its generalizations is an example of such a decomposition.