But all that comforting certitude was to be turned upside down with the scientific revelations and revolutions of the early 20th Century. They resulted in the Quantum Paradigm Shift which gave us the models we use today for the atomic scale world.
Confidence in the impending closure of Physics started to unravel not due to the successes of understanding in the realms of Light and Matter but the failures in understanding their interaction!
Wien 1893
Black Body Displacement
Law
(In "Black Bodies ,"
light gets lost as
it bounces from
surface to surface,
thoroughly
equilibrating
before coming
out again.)
max = c2/5T
shifting to shorter wavelengths (higher frequencies) as the
temperature rises. c2=1.44 cm K, and we'll see it again
when we study spectroscopy where it will be given as 1.44 K/cm-1
since cm-1, wavenumbers, are the spectroscopist's
unit of measure.max = c2/5TWien's discovery gave us the possibility of measuring temperatures remotely via photometry, T = c2 / 5
max.
Exercise # 11.14 encourages you to determine the temperature of the
photosphere of the sun given it's spectral peak at 480 nm in
wavelength. (That's green light to which our eyes are most sensitive.
We're less sensitive to red >480 nm
and very much less sensitive to blue < 480 nm.
The short wavelengths of visible light are most efficiently
scattered by the atmosphere; so the sky is blue. That means blue
light comes from all directions, washing out shadows cast by
objects in direct sunlight. As predators, we depended upon shadows
to unmask camoflaged prey; so we've evolved insensitivity to blue,
and shadows appear darker to us than they really are.)
Stefan- Boltzmann 1879
Total
Radiation
Law
M = 
T 4
where T 4
=5.67×10-8 W m-2 K-4
and W=Watts=Js-1, the SI unit of power.And interesting application of this expression is in the Heat Balance of the Earth. Light from the sun has a flux of about 2.00±0.04 cal cm-2 min-1 at the distance of Earth's orbit 'round the Sun. But what comes in (insolation) must go out (reradiation) or we'd accumulate energy until we burnt up!
HOMEWORK
Use the Stefan-Boltzmann equation to estimate Earth's reradiation temperature.
You'll need to know that 1 cal = 4.184 J and that although Earth absorbs sunlight over an area equal to its shadow (
R2),
it reradiates it over its entire surface (4R2).
Oddly enough, you don't need to know R, but it's 6378 km.Your result will be chilling until you realize that its not Earth's warm surface which is doing the lion's share of the reradiation. (What is protecting us from that somewhat low temperature?)
Rayleigh-
Jeans
1900
Black Body
Distribution
Law
max
but the entire dependence of Black Body radiation upon wavelength.What's that mean?
Just as the matter of the radiating body was presumed to be oscillators in thermodynamic equilibrium at temperature T, so to were the imagined light oscillators to be in equilibrium. That led to an energy density in the range
to +d of
d
= 8kT/4 d
and it's that = 8kT/4 d
4
in the denominator which screamed ERROR since it implies
that energies in the UV and beyond were unlimited! The
Ultraviolet Catastrophe.It's easy to understand why this happens. The Equipartition Theorem demands that every oscillator in equilibrium retains an average energy of kT (or 3kT if you permit oscillation in 3-d). Take a 1-d violin string. It's fundamental frequency,
0
has kT. So does it's overtone at 20, and
it's second overtone at 30 etc.
as high as you want to go. But there is a natural upper
limit, since each overtone halves the wavelength of its predecessor.
That can continue until the violin string is no longer continuous;
you eventually hit the atomic/molecular limit. No overtone can
split atoms! Thus, it is the discrete, atomic nature of matter
which prevents an Ultraviolet Catastrophe in the Black Body.Why not posit some quantization in Light as well to see if that solved Rayleigh's problem in some similar way?
Planck
1900
Black Body
Distribution
Law
Max Planck thought so, and did so, not by presuming a smallest wavelength
possible but rather by assuming a smallest unit of energy carried by the
electromagnetic field in equilibrium with the Black Body. That unit was
taken to be proportional to the light's frequency,
=c/,
and the constant of proportionality came to be called Planck's Constant,
h. Thus
E = h
When Planck used this quantization for light's energy, those short
wavelengths which decimated Rayleigh couldn't accumulate the enormous minimum
energy necessary to constitute even a single "quantum!" They couldn't be
"turned on" by the thermal energies available. And the radiation distribution
over wavelength became
d = 8hc-5 [exp(hc/kT) - 1]-1 d
Now while that -5
appears an even worse explosive pole as 
0
than did -4,
the rapidly enlarging exponential in that same regime more than compensates for it
since exponential growth beats any polynomial growth no matter the polynomial's
(constant) exponent!
PREVIEW: We'll see that same exponential later in connection with the thermal
population of molecular vibration. Simple algebraic rearrangement of the [square
bracketted] denominator above gives
exp(-h/kT)/[1 - exp(-h/kT)]
which will be the probability of a molecular vibration being excited to its first
quantum state. But we're getting ahead of ourselves because in 1900, Planck only
had to postulate light energy quantization not yet matter energy
quantization. The latter leap was made by Einstein.
Planck's Black Body Distribution Law was a wild success. It fit experiment so well
that it's postulate of irreducible units of light energy (now called photons) was
treated with respect. It also reproduced both the Stefan-Boltzmann and Wein Laws
as might be imagined since they were just the integral and differentials of that
distribution, respectively.
HOMEWORK
For homework, differentiate Planck's distribution with respect to wavelength and
set it to zero (to get the maximum), making the simplifying
assumption that max< hc/kT
so that 1 can be ignored relative to an enormous exp(hc/maxkT),
and thus evaluate max.
You'll get Wien's Law that way, and that assumption will be tantamount to saying that
e5>>1 which is quite true; check it out. (Atkins is blowing steam
about needing to assume high temperatures.)
Einstein
1905
Temperature
Dependence
of
Heat Capacities
The advantages of quantizing energy in matter became apparent even before
the Bohr atom. The Equipartion Theorem (gets around, doesn't it?) required that 3-d
vibrational oscillators in a solid (crystal, say) would each contribute 3kT to the
average energy, making the molar heat capacity 3kNAv=3R. But, experimentally,
it wasn't that until temperatures became very, very high.
Einstein borrowed Planck's trick and applied to the molecular oscillation of the
solid; if it too could only acquire energy in units of h0,
it's natural frequency (times Planck's constant), Einstein showed that
CV = 3R { exp(h0/2kT) / [exp(h0/kT) - 1] }2
which fell from 3R as T dropped from infinity (or 1/T rose from zero, take your pick).
Peter Debye, realizing that solids resonate with a range of frequencies, not just
0,
included that range in his derivation of CV and produced an
expression with an even closer fit to experiment!
So, the stage was set. Quantization of energies in both light and matter seemed to
be universal solvents for the grime of intransigent 19th Century physics failures.
And it only required liberating oneself from the belief that energy should
be continuous as demanded by the precepts of Newtonian (classical) physics.
If only it were that simple.
Einstein
1905
Photoelectric
Effect
One more cherished presumption had yet to fall. That was that light must follow
wave physics as matter follows particulate physics. Belief in that would lead one
to expect light to wash over its target which would accumulate its energy continuously,
But that can't be since we just required discontinuous energy for light! The
new logic would suggest that quanta of light energy would get deposited on the target
in its complete bundle PING as if the light were a colliding...particle!
And this was what Einstein demonstrated by explaining the Photoelectric Effect
in which light ejects electrons from low ionization materials like alkali metals.
Since it takes a minimum energy (called the work function,
) just to wrench the
electron away from the atom, and if the only source of that energy is an incoming
"particle" of light, only wavelengths below a critical maximum, =hc/max
would do the trick. You could pound all day with longer wavelengths, and all you'd
succeed in doing is heat up the metal since that critical energy minimum wouldn't be
in the needed place all at once! That was in perfect accord with experiment.
So photoelectric ejection would be like the old Texas phrase, "One riot? One Ranger!",
reconstituted as "One photon? One electron!" The clincher was Energy Conservation;
the fate of photonic energy above the minimum would be to eject the electron
with increasing kinetic energy. That to was verifiable according to the simple
½meve2 = hlight -
and the transformation of light into an entity which travels via wave rules but is
created and destroyed via particle rules was complete. All that was needed to finish
the Revolution was to do the same for matter.
deBroglie
1925
Momentum
and
Wavelength
Light's wave properties were legendary. Even in the Photoelectric Effect, where
its collision with the alkali metal was as a particle, called a photon, those
wave properties were de rigeur. The experimentalist had to produce a
monochromatic light to test Einstein's theory, and in those days before lasers,
monochromatic light came out of a diffractometer, so named because it
diffracted light into its component "colors" (i.e., wavelengths) by passing it over
a diffraction grating whose regular striations interacted with light's regularly-
spaced waves to send different wavelengths in different directions. Without wave
behavior, the photon's particulate nature couldn't have been tested! Bit of irony
there.
So it only seemed fair for light which had borrowed matter's particulate nature
to loan matter its own wave nature. Indeed, Louis deBroglie postulated that both
light and matter had wavelengths which depended in a simple way upon their
momenta. Ah, yes; light has momentum, and it has spin too, but we'll get to that
later. Einstein had even used light's momentum in his famous E=mc2
equation which refers only to objects at rest. His generalization for objects with
momentum, p, was actually
E2 = m02c4 + p2c2
and since light has no rest mass, m0, that leaves only E=pc for light.
But E=h=hc/
for light as well. It's the same E; so that means that light's momentum is
p = h /
And deBroglie concluded that "what's good for the goose must be good for the gander,"
and predicted that matter would exhibit wavelike properties dependent upon its
momentum, p, in exactly the same way
= h / p
Davisson-
Germer
1925
Electron
Diffraction
The challenge was taken up by Davisson and Germer. Since X-rays diffracted nicely
off the regular spacings, d, of crystalline planes (the essence of X-ray diffraction
as a tool for determining molecular shapes), they would accelerate electrons through
a potential drop until their kinetic energy, K=½mv2=p2/2m,
dictated a momentum yielding a deBroglie wavelength near d.
As the X-rays would be expected to diffract into angle, 
,
according to (see Atkins, Chapter 21, page 726)
= (2d/n) sin
they should look for electrons at those angles as well.
And there they were. The electrons had interacted with the crystal planes
of the target just as if they were following wave mechanics at deBroglie's wavelength!
Thus, light was a wave which collided like a particle. And matter was a particle which
travelled like a wave! All of this was true on the atomic scale over which the deBroglie
wavelengths were comparable to atomic spacings; wave interferences would then be
inescapable. But if wavelengths are very short, relative to the confines of their
environment, interference virtually disappears. We know this to be so from everyday
experience: light passing through a doorway casts an undistorted shadow of
the door, not some oddly fringed wave interference pattern! The doorway is orders of
magnitude larger then light's wavelength, and classical path behavior results.
So to, when matter waves shrink to dimensions tiny compared to their environmental
constraints, classical path behavior should re-emerge! Tiny in this context means
sub-atomic or, better still, sub-nuclear, 10-12 m.
HOMEWORK
Show this to be true even for a molecule of albumin, MW about 60,000 daltons (a
dalton is an amu), with the momentum, p, associated with
only trivial thermal energy, kT, at room temperature (300 K).
=c/,
and the constant of proportionality came to be called Planck's Constant,
h. Thus
E = h
When Planck used this quantization for light's energy, those short
wavelengths which decimated Rayleigh couldn't accumulate the enormous minimum
energy necessary to constitute even a single "quantum!" They couldn't be
"turned on" by the thermal energies available. And the radiation distribution
over wavelength became
d = 8hc-5 [exp(hc/kT) - 1]-1 d
Now while that = 8hc-5 [exp(hc/kT) - 1]-1 d
-5
appears an even worse explosive pole as 0
than did -4,
the rapidly enlarging exponential in that same regime more than compensates for it
since exponential growth beats any polynomial growth no matter the polynomial's
(constant) exponent!PREVIEW: We'll see that same exponential later in connection with the thermal population of molecular vibration. Simple algebraic rearrangement of the [square bracketted] denominator above gives
/kT)/[1 - exp(-h/kT)]
which will be the probability of a molecular vibration being excited to its first quantum state. But we're getting ahead of ourselves because in 1900, Planck only had to postulate light energy quantization not yet matter energy quantization. The latter leap was made by Einstein.
Planck's Black Body Distribution Law was a wild success. It fit experiment so well that it's postulate of irreducible units of light energy (now called photons) was treated with respect. It also reproduced both the Stefan-Boltzmann and Wein Laws as might be imagined since they were just the integral and differentials of that distribution, respectively.
HOMEWORK
For homework, differentiate Planck's distribution with respect to wavelength and
set it to zero (to get the maximum), making the simplifying
assumption that max< hc/kT
so that 1 can be ignored relative to an enormous exp(hc/maxkT),
and thus evaluate max.
You'll get Wien's Law that way, and that assumption will be tantamount to saying that
e5>>1 which is quite true; check it out. (Atkins is blowing steam
about needing to assume high temperatures.)
Einstein
1905
Temperature
Dependence
of
Heat Capacities
Einstein borrowed Planck's trick and applied to the molecular oscillation of the solid; if it too could only acquire energy in units of h
0,
it's natural frequency (times Planck's constant), Einstein showed that
CV = 3R { exp(h0/2kT) / [exp(h0/kT) - 1] }2
which fell from 3R as T dropped from infinity (or 1/T rose from zero, take your pick).0/2kT) / [exp(h0/kT) - 1] }2
Peter Debye, realizing that solids resonate with a range of frequencies, not just
0,
included that range in his derivation of CV and produced an
expression with an even closer fit to experiment!So, the stage was set. Quantization of energies in both light and matter seemed to be universal solvents for the grime of intransigent 19th Century physics failures. And it only required liberating oneself from the belief that energy should be continuous as demanded by the precepts of Newtonian (classical) physics.
If only it were that simple.
Einstein
1905
Photoelectric
Effect
And this was what Einstein demonstrated by explaining the Photoelectric Effect in which light ejects electrons from low ionization materials like alkali metals. Since it takes a minimum energy (called the work function,
) just to wrench the
electron away from the atom, and if the only source of that energy is an incoming
"particle" of light, only wavelengths below a critical maximum, =hc/max
would do the trick. You could pound all day with longer wavelengths, and all you'd
succeed in doing is heat up the metal since that critical energy minimum wouldn't be
in the needed place all at once! That was in perfect accord with experiment.So photoelectric ejection would be like the old Texas phrase, "One riot? One Ranger!", reconstituted as "One photon? One electron!" The clincher was Energy Conservation; the fate of photonic energy above the minimum would be to eject the electron with increasing kinetic energy. That to was verifiable according to the simple
½meve2 = hlight -
and the transformation of light into an entity which travels via wave rules but is
created and destroyed via particle rules was complete. All that was needed to finish
the Revolution was to do the same for matter.
light -
deBroglie
1925
Momentum
and
Wavelength
So it only seemed fair for light which had borrowed matter's particulate nature to loan matter its own wave nature. Indeed, Louis deBroglie postulated that both light and matter had wavelengths which depended in a simple way upon their momenta. Ah, yes; light has momentum, and it has spin too, but we'll get to that later. Einstein had even used light's momentum in his famous E=mc2 equation which refers only to objects at rest. His generalization for objects with momentum, p, was actually
=hc/
for light as well. It's the same E; so that means that light's momentum is
p = h /
And deBroglie concluded that "what's good for the goose must be good for the gander,"
and predicted that matter would exhibit wavelike properties dependent upon its
momentum, p, in exactly the same way
= h / p
= h / p
Davisson-
Germer
1925
Electron
Diffraction
As the X-rays would be expected to diffract into angle,
,
according to (see Atkins, Chapter 21, page 726) = (2d/n) sin
And there they were. The electrons had interacted with the crystal planes of the target just as if they were following wave mechanics at deBroglie's wavelength!
Thus, light was a wave which collided like a particle. And matter was a particle which travelled like a wave! All of this was true on the atomic scale over which the deBroglie wavelengths were comparable to atomic spacings; wave interferences would then be inescapable. But if wavelengths are very short, relative to the confines of their environment, interference virtually disappears. We know this to be so from everyday experience: light passing through a doorway casts an undistorted shadow of the door, not some oddly fringed wave interference pattern! The doorway is orders of magnitude larger then light's wavelength, and classical path behavior results.
So to, when matter waves shrink to dimensions tiny compared to their environmental constraints, classical path behavior should re-emerge! Tiny in this context means sub-atomic or, better still, sub-nuclear, 10-12 m.