Waves, Light, the Quanta and the Atom
Review Including Material from Chapters 26 & 27
The behavior of EM waves—light waves—can be characterized in many ways by
describing their wave fronts, that is, the lines traced out by their peaks.
Wave fronts move out from oscillating sources in expanding patterns of
concentric circles. (Exceptions to this general pattern are beams from lasers or
optical lens systems.) Each point on a wave front acts as a new source of wave
fronts (Huygen’s Principle)—which helps us understand how wave behave as
they encircle obstacles or expand through tiny apertures.
EM waves are transverse, and thus have two possible polarizations (e.g. the
oscillating E-fields can be oriented horizontally or vertically).
As wave fronts hit a higher-index transparent surface, their propagation speed
slows, giving rise to refraction. They can also reflect off the surface. The fraction
of light that transmits verses reflects depends on the wave polarization.
If EM waves pass through one or more small holes, they spread out in fan
shapes (diffract), superimposing with themselves in constructive and destructive
interference (diffraction patterns). Three dimensional images (holograms) can be
captured as interference patterns and later reproduced using laser light.
EM radiation is quantized: its energy is packed in units called photons. The
amount of energy is proportional to the photon’s frequency: E = hf, where h is
Planck’s constant (6.6 * 10-34 joules sec). Photons have dual natures: in some
circumstances they behave as waves (e.g. when traveling, refracting &
diffracting), and in other circumstances they behave like particles (e.g. when
being emitted or absorbed by atoms).
Particles, like electrons, have dual natures: in some circumstances they behave
as waves (e.g. when traveling through vacuum, refracting & diffracting), and in
other circumstances they behave like particles (e.g. when interacting with atoms).
When behaving as a wave, the electron’s wavelength () is inversely proportional
to its momentum (p): electron = h/p, where p = mv.
Heisenberg’s Uncertainty Principle is a feature of Quantum Mechanics: there is a
limit to how well we can know both the position and momentum of a particle or
photon, that is, x p > ħ, where ħ = h/2, and  denotes an interval of
uncertainty. Similarly, there is a limit to how well we can not know both the time
interval (pulse interval) and energy of a particle or photon: t E > ħ.
The photoelectric effect demonstrates the particle nature of light: electrons are
confined to a metal’s surface by attractive electric forces. A certain level of
energy, the work function, W0, is required to eject an electron from the surface. If
a photon with an energy quanta greater than W0 is incident on the metal (that is,
hf ≥ W0), an electron can absorb the quanta, giving it enough kinetic energy to
break free from the “potential well” of the metal’s surface. (This is similar to a
rocket having enough KE to escape Earth’s gravity.) If a photon’s frequency is
too low, the electron may still absorb it, but it will be insufficient to kick it free of
the metal surface. The number of ejected electrons is proportional to the intensity
of the light (i.e. the number of photons in the light beam).
A double slit experiment demonstrates the wave nature of both photons and
particles. Low intensity beams of photons or electrons form characteristic wave
interference patterns when passing through two closely placed slits. If one slit is
covered, the pattern readjusts to a single-slit wave diffraction pattern. Somehow,
each “particle” spreads out in space so that it interferes with itself just as water
wave does.
The wave-like behavior of matter is mathematically described by Schrödinger’s
wave equation: (-ħ 2/2m 2 + U) = iħ /t, where  is a particle’s wave
function, 2 extracts the spatial rate of change of the rate of change of the wave
function, U is the potential energy of the particle, i is the imaginary unit number,
and /t is the temporal rate of change of the wave function. Like most wave
equations in physics,  is cast as a function of complex numbers (i.e. having
both real and imaginary parts). However, the value of the square of the wave
function, | |2, is always a function of real numbers, and for a given (x,y,z)
location, it’s value equals the probability that the described particle can be found
at that point of space. For example, | |2 for an electron orbiting a single proton
describes the shape of the “electron cloud” of the hydrogen atom.
It is a character of nature that as electrons orbit an atomic nucleus, the electrons
are moving in a wave-like manner surrounding the nucleus, and stable orbits
correspond to electron standing waves—that is, every stable orbit corresponds to
an integer number of electron wavelengths. The lowest electron orbit
corresponds to 1 electron wavelength wrapped around the nucleus, the 2 nd orbit
corresponds to 2 electron wavelengths wrapped around the nucleus, and so on.
All of these are solutions to Schrödinger’s | |2 equation. Higher level orbits can
take unusual shapes (figure-8’s, 4-leaf clovers, doorknobs…) because the wraparound occurs in 3 dimensions instead of one.
These integer-oriented electron orbits correspond nicely with an earlier, simpler
model of electron orbits developed by Niels Bohr. His model is planetary—the
electrons follow circular orbits, except the electron’s orbits can only have discreet
energy levels. When an electron absorbs energy (e.g. a photon), the atom is
excited, the electron jumps up 1 or more orbital levels. When the atom relaxes to
its ground state, the electron jumps back to a lower energy orbit, emitting a
photon in the process. Because the possible energy levels are discreet, only
certain wavelengths of photons can be absorbed or emitted—the atomic
spectrum of a given element. This means each type of atom has a unique pattern
of absorption and emission lines, determined by the number of protons in the
nucleus, and the configuration of underlying lower-level electrons that shield
outer electrons from the protons’ pull.
The diameters of various elements are remarkable similar, varying only by a
factor of seven. Heavy elements can have 100 or more electrons, but their inner
electron orbits are pulled correspondingly tighter by the large number of protons
in the nuclei.
Atom’s electrons can be excited to higher energy states in 3 basic ways: 1)
absorbing a photon, 2) being smacked into higher orbit by another high energy
electron, or 3) being jostled into a higher energy state by molecular impacts—
high temperature translational kinetic energy.
Crisp clean atomic spectra are observed in low-pressure gases. In denser
situations, innumerable different interactions of atoms cause each atom to have
slightly different electron wave patterns—each atom has slightly different orbital
energy levels—with the result that the composite materials’ absorption and
emission spectral lines spread wider (line broadening). In the extreme case
where thermal agitation causes atoms to emit light, the line broadening is so
extensive that the radiation wavelength profile forms a broad shape
(incandescence, or the black body or Planck distribution). Sunlight is an example
of high temperature incandescence—the high thermal energy from nuclear fusion
causes the gas in star’s surface to glow as a 6000K black body.
There are four main pathways for excited electrons to return to their ground
(relaxed) state: 1) fluorescence—accompanied by the quick emission of a
photon, 2) phosphorescence—accompanied by the delayed emission of a
photon, 3) radiationless relaxation—the electron relaxation involves the
redistribution of energy into molecular vibrations, and 4) stimulated emission—an
oncoming EM wave of the right frequency triggers the relaxation transition and
photon emission (the thing that makes lasers work).
Unlike normal light beams, laser light is coherent—its photon waves are all the
same frequency, in phase (the EM peaks are aligned), and moving in the same
direction.
Atomic electronic relaxation involves the emission of photons primarily in the
infrared, visible and ultraviolet wavelength ranges. Very high energy excitation,
involving the deepest electron levels of heavy elements, can cause very high
frequency photons to be emitted: X-rays.
The Atomic Nucleus, Fission and Fusion
Atomic nuclei are composed of protons and neutrons. (In turn, protons and
neutrons are composed of 3 quarks each.) The strong electrostatic repulsion of
protons in close proximity is overcome by the strong nuclear force, which acts
very powerfully but only over short distances (10-15m). The combination of two
protons and neutrons (“an alpha particle”) is particularly stable and selfcontained. Neutrons tend to stabilize a nucleus, and in heavy elements they
significantly outnumber protons. Within a given element, the number of neutrons
within the nucleus can vary (isotopes), and some combinations are more stable
than others. Unstable nuclear combinations will eventually break apart:
radioactive decay. The rate of breakage will vary from isotope to isotope. An
isotope’s half-life is the period of time over which half of a given amount of an
isotope will break down.
The shape of a nucleus varies: spherical, football-shaped & doorknob-shaped,
and is somewhat fluid. When a nucleus gets very large the mutual repulsion of
protons nearly overwhelms the binding strong nuclear force. Within the jostling of
neutrons and protons, there is a small probability that an alpha particle grouping
will break free and shoot out of the nucleus (alpha decay, alpha radiation),
accompanied by a burst of ultra-high energy radiation (a gamma ray).
Sometimes, through a process involving the weak nuclear force, a neutron will
spontaneously emit an electron and become a proton (beta decay, beta
radiation). Both of these radioactive decay paths change the element—it
transmutates into another element—determined by the final number of protons in
the nucleus. Of these three forms of radiation, alpha particles can be stopped by
a single layer of paper, beta particles (high energy electrons) can be stopped by
a layer of aluminum foil, while gamma rays can penetrate several centimeters of
lead.
Another form of radioactive decay involves the splitting of a nucleus into two
parts, releasing a substantial amount of energy: fission. This occurs naturally in
certain heavy isotopes like Uranium-235. In the process of breaking apart, two or
three free neutrons also shoot out. These can hit other unstable nuclei and
stimulate other atoms to break apart, releasing even more energy and free
neutrons. If many unstable nuclei are present—if a critical mass is exceeded—
the breakdown process will grow exponentially in a chain reaction, causing the
whole mass to explode: an atomic bomb.
On the other hand, by carefully controlling the nuclear breakdown rate, the fission
process can be harnessed to produce power: a nuclear powerplant.
The opposite process of fission is fusion: combining light nuclei to form heavier,
more stable nuclei. This also releases energy. (The most stable nucleus is iron.
Elements lighter than iron release energy during formation via fusion. Elements
heavier that iron release energy during formation via fission.)
Relativity
The postulates of Einstein’s Special Theory of Relativity are 1) within a given
frame of reference, all laws of nature are the same, 2) light (in a vacuum) always
has the same measured speed, regardless of the observing frame of reference,
and 3) events observed to be simultaneous in one frame of reference may not
appear simultaneous in another frame of reference.
The consequences of special relativity are, when we observe a frame of
reference moving relative to us: 1) time will appear slower in the other frame, 2)
lengths (along the direction of motion) will appear shorter, and 3) mass in the
other frame will appear to increase. The magnitude of these changes, , will not
appear noticeable until the speed of the other frame of reference is a significant
fraction of the speed of light {  = 1/sqrt(1 – v2/c2) }. Another, very important
consequence of special relativity is the realization that matter and energy are
different forms of the same thing, related via the formula E = mc2.
The key postulate of the General Theory of Relativity is the principle of
equivalence: an acceleration experienced in a frame of reference due to
increasing speed is indistinguishable from that caused by gravity. The
consequences are: 1) a light ray will bend in gravity (or alternatively, “space-time
gets curved”), 2) time will slow down in higher gravitational fields, and 3) moving
masses emit gravity waves.