Failures of Classical Physics

Chm 3312 Lecture for 13 Jan 1997


Scientists in the 19th Century thought Physics to be "done." Gibbs had formulated an excellent framework for discussion of Chemical Thermodynamics from concepts of Free Energy. Boltzmann 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!

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.)
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:

 lambda 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.

Wien's discovery gave us the possibility of measuring temperatures remotely via photometry, T = c2 / 5 lambda 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  lambda >480 nm and very much less sensitive to blue  lambda < 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

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

M =  sigma T 4

where  sigma =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 ( pi R2), it reradiates it over its entire surface (4 pi R2). 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

Lord Rayleigh and James Jeans used the Equipartition Theorem to determine not just  lambda 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
 lambda to  lambda +d lambda of

d epsilon = 8 pi kT/ lambda 4 d lambda

and it's that  lambda 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,  nu 0 has kT. So does it's overtone at 2 nu 0, and it's second overtone at 3 nu 0 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,  nu =c/ lambda , and the constant of proportionality came to be called Planck's Constant, h. Thus

E = h  nu

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 epsilon = 8 pi hc lambda -5 [exp(hc/ lambda kT) - 1]-1 d lambda

Now while that  lambda -5 appears an even worse explosive pole as  lambda  arrow right 0 than did  lambda -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 nu /kT)/[1 - exp(-h nu /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  lambda max< hc/kT so that 1 can be ignored relative to an enormous exp(hc/ lambda maxkT), and thus evaluate  lambda 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 h nu 0, it's natural frequency (times Planck's constant), Einstein showed that

CV = 3R { exp(h nu 0/2kT) / [exp(h nu 0/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  nu 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,  phi ) 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,  phi =hc/ lambda 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 = h nu light -  phi

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 nu =hc/ lambda for light as well. It's the same E; so that means that light's momentum is

p = h /  lambda

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

 lambda = 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,  Theta , according to (see Atkins, Chapter 21, page 726)
 lambda = (2d/n) sin Theta
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).


Last modified 27 January 1997. Chris Parr