Simple Nature - Light and Matter

(Martin Jones) #1
librium position, so ∆x= 0. But this would violate the Heisenberg
uncertainty principle and so is impossible.
The inevitable motion that is present even in the ground state is
known as zero-point motion, and its energy is the zero-point energy.
Relativity tells us thatE =mc^2 , so the zero-point energy of par-
ticles is equivalent to a certain amount of mass. In fact, nearly all
the mass of ordinary matter arises from the zero-point energy of the
quarks inside the neutrons and protons. Another interesting appli-
cation is to spontaneous nuclear fission, which is the basis for nuclear
energy, providing the kick-off for a chain reaction. Spontaneous fis-
sion requires that a nucleus become more and more elongated until
it breaks apart into two pieces. The very elongated shapes have a
high potential energy, so that spontaneous fission requires quantum-
mechanical tunneling. If it were not for the zero-point vibrational
energy associated with this motion, the tunneling probability for
uranium and plutonium isotopes would be extremely small. These
isotopes would decay only by alpha emission, and nuclear reactors
and bombs would not work.

f/Each panel of the figure shows
a standing wave on a sphere, with
the convention that gray is zero,
white is a positive real number,
and black is a negative real num-
ber. (These could instead have
been drawn as traveling waves,
but then we would have needed
to represent complex numbers
using color, as in figure c on
p. 922.) Only 2 is a solution of the
Figure f shows visually the reason for the correction of^2 to Schrodinger equation. ̈ (+ 1). Each of these standing waves has|z|= 16, wherezis the
vertical axis. But only f/2 is a solution of the Schr ̈odinger equation
for a state of definite. To be a solution of the Schr ̈odinger equa- tion, such a wave must have the same kinetic energy everywhere. Each of these three has the same kinetic energy associated with its wavelength in the “east-west,” or azimuthal, direction. Wave f/1 is not a solution, because near the equator, it has an extremely short wavelength in the “north-south,” or longitudinal, direction, and this gives it a greater kinetic energy near the equator than elsewhere. The opposite problem occurs in f/3, where the wave is constant in the longitudinal direction; at the poles, the wavefunction varies in- finitely rapidly, and therefore the kinetic energy blows up to infinity there. The only valid solution is f/2, which has a Goldilocks-style just-right wavelength in the longitudinal direction. The kinetic en- ergy associated with this wavelength is the difference between the semiclassical^2 and the correct quantum mechanical(+ 1).
A different example that is particularly easy to reason about is


Section 14.2 Rotation and vibration 963
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