|Scientists in the 19th Century thought Physics to be "done."
had formulated an excellent framework for discussion of
Chemical Thermodynamics from concepts of Free Energy.
had tied those to molecular energies via Statistical
Mechanics. Maxwell had permitted a detailed understanding of
electromagnetic radiation via his four laws. And J.J. Thomson had
even begun the systematic deconstruction of the atom with his
Cathode Ray Tube
which studied electrons in flight; Thomson had
by the turn of the century even postulated an (inaccurate) model
of the atom (as a "plum pudding" with negative electrons and
positive protons mixed as raisins in a dough).|
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!
|Light in equilibrium with heated bodies gives rise to a
continuous spectrum with a maximum in its wavelength distribution.
Wien found that maximum to be temperature dependent:|
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.
Stefan- Boltzmann 1879
|Similarly, the integrated intensity of such "Black Body" light
per unit area of emitter was found by Stefan and Boltzmann to
increase as the 4th power of the temperature|
where =5.67×10-8 W m-2 K-4
and W=Watts=Js-1, the SI unit of power.
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?)
Lord Rayleigh and James Jeans used the Equipartition Theorem
to determine not just 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
and it's that 4
in the denominator which screamed ERROR since it implies
that energies in the UV and beyond were unlimited! The
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
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.)
|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
which fell from 3R as T dropped from infinity (or 1/T rose from zero, take your pick).
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
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'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
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
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
|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)
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.
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).