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11 Biomolecular Structure and Interactions
molecules. For example, the s-p ‘hybrid orbitals’ describe electron waves
participating in chemical bonds in terms of simpler hydrogenic waves. By the
superposition principle, expansion of a valence electron wave into hydrogenic
waves is always possible. Whether only a few terms in the expansion dominate
a given molecular bond depends on the full atomic interactions, and how the
electrons behave once they settle into the lowest energy states among those
available stationary molecular waves.
4. The evolution of a particle wave (between observations) is determined by a
linear diffusive-like differential equation called a quantum wave equation. This
means that if an electron is left unobserved, its wave (as described by the theory)
‘diffuses’ until it reaches spatial boundaries. With non-absorbing boundaries
confining the electron in some finite region, the wave evolves into a standing
wave. No further change in the probability distribution for these standing waves
occurs.
For electrons making transitions between bound atomic or molecular
orbitals, the time to reach a steady state is usually within nanoseconds, but
some transitions may take hours (such as within long-lifetime phosphors).
5. An electron initially confined by thin walls with finite confining forces will
not be completely confined, but rather the wave function for the unobserved
electron will have an exponential tail within the walls, and a small oscillating
part in the region outside the wall. This means there is a finite probability
of finding the electron outside the confining region, even though classically,
the electron would not have had enough energy to overcome the forces of the
walls. This effect, called ‘quantum tunneling’, has not only been observed, but
is common in the small world. Radioactive substances would not decay if not
for quantum tunneling, 2 nor would electrons be able to conduct through copper
wires twisted together with an oxide layer between them. The hydrogen atoms
in the ammonia molecule tunnel back and forth past the nitrogen at a microwave
frequency. Quantum tunneling of protons can also occur between base pairs in
your DNA, possibly leading to a gene mutation.
6. Quantum theory and Relativity predict that a stable localizable particle carries
a fixed mass and a quantized spin. The magnitude of the spin of a particle must
be
√
s(s + 1) ¯
h, where s is a quantum number taking one of the possible values
0, 1/2, 1, 3/2, 2, · · · . For integer spin, the particle is called a boson. For halfodd-integer spin, the particle is called a fermion. In 1939, Pauli and Fierz proved
that within relativistic quantum theory, any number of integer-spin particles can
be placed in a single quantum state, but no more than one half-odd-integerspin particle can be placed in a single quantum state. Photons are bosons (with
s = 1) and electrons are fermions (with s = 1/2). Light from a LASER will
be found with lots of photons in one quantum state. We cannot construct an
2 George Gamow, in 1928, was the first to apply quantum tunneling to explain radioactivity. In
1948, while at the George Washington University, he and his student Ralph Alpher, used quantum
theory and General Relativity to devise the hot ‘big bang’ theory of the Universe.
11 Biomolecular Structure and Interactions
molecules. For example, the s-p ‘hybrid orbitals’ describe electron waves
participating in chemical bonds in terms of simpler hydrogenic waves. By the
superposition principle, expansion of a valence electron wave into hydrogenic
waves is always possible. Whether only a few terms in the expansion dominate
a given molecular bond depends on the full atomic interactions, and how the
electrons behave once they settle into the lowest energy states among those
available stationary molecular waves.
4. The evolution of a particle wave (between observations) is determined by a
linear diffusive-like differential equation called a quantum wave equation. This
means that if an electron is left unobserved, its wave (as described by the theory)
‘diffuses’ until it reaches spatial boundaries. With non-absorbing boundaries
confining the electron in some finite region, the wave evolves into a standing
wave. No further change in the probability distribution for these standing waves
occurs.
For electrons making transitions between bound atomic or molecular
orbitals, the time to reach a steady state is usually within nanoseconds, but
some transitions may take hours (such as within long-lifetime phosphors).
5. An electron initially confined by thin walls with finite confining forces will
not be completely confined, but rather the wave function for the unobserved
electron will have an exponential tail within the walls, and a small oscillating
part in the region outside the wall. This means there is a finite probability
of finding the electron outside the confining region, even though classically,
the electron would not have had enough energy to overcome the forces of the
walls. This effect, called ‘quantum tunneling’, has not only been observed, but
is common in the small world. Radioactive substances would not decay if not
for quantum tunneling, 2 nor would electrons be able to conduct through copper
wires twisted together with an oxide layer between them. The hydrogen atoms
in the ammonia molecule tunnel back and forth past the nitrogen at a microwave
frequency. Quantum tunneling of protons can also occur between base pairs in
your DNA, possibly leading to a gene mutation.
6. Quantum theory and Relativity predict that a stable localizable particle carries
a fixed mass and a quantized spin. The magnitude of the spin of a particle must
be
√
s(s + 1) ¯
h, where s is a quantum number taking one of the possible values
0, 1/2, 1, 3/2, 2, · · · . For integer spin, the particle is called a boson. For halfodd-integer spin, the particle is called a fermion. In 1939, Pauli and Fierz proved
that within relativistic quantum theory, any number of integer-spin particles can
be placed in a single quantum state, but no more than one half-odd-integerspin particle can be placed in a single quantum state. Photons are bosons (with
s = 1) and electrons are fermions (with s = 1/2). Light from a LASER will
be found with lots of photons in one quantum state. We cannot construct an
2 George Gamow, in 1928, was the first to apply quantum tunneling to explain radioactivity. In
1948, while at the George Washington University, he and his student Ralph Alpher, used quantum
theory and General Relativity to devise the hot ‘big bang’ theory of the Universe.