The Breakdown of Classical Physics

At the turn of the 20th century there was a widely held belief that most of physics was known or understood and little remained to be discovered. Witness the following statements:

The Munich physics professor Philipp von Jolly advised Max Planck against going into physics, saying, "in this field, almost everything is already discovered, and all that remains is to fill a few holes."(Lightman, Alan P. (2005). The discoveries: great breakthroughs in twentieth-century science, including the original papers. Toronto: Alfred A. Knopf Canada. p. 8.)

Albert A. Michelson (who along with Edward Morley made the first precise to six significant figures measurements on the speed of light) remarked in the introduction to the Department of Physics section of the University of Chicago Catalog in 1901-1902:

“While it is never safe to affirm that the future of Physical Science has no marvels in store even more astonishing than those of the past, it seems probable that most of the grand underlying principles have been firmly established and that further advances are to be sought chiefly in the rigorous application of these principles to all the phenomena which come under our notice.” It is here that the science of measurement shows its importance – where quantitative results are more to be desired than qualitative work. An eminent physicist has remarked that the future truths of Physical Science are to be looked for the in the sixth place of decimals.” (The Annual Registrar, The University of Chicago, p. 270, 1901-1902. For more background on this quote see Robert Lagemann, American Journal of Physics, Volume 27, Issue 3, pp. 182-184 (1959).

Of course, there were a few things that remained to be tidied up:

• the ultraviolet catastrophe;• heat capacities at low temperature;• the mechanism of the propagation of light;• the photoelectric effect;• the line spectra of atoms;• the interference patterns of diffracted electrons …

As often happens the explanations of these still to be explained details led to a revolution in physics and our understanding of the world!

Blackbody Radiation and the Ultraviolet Catastrophe

Blackbody radiation is the radiation that is emitted as a result of a material’s temperature. An ideal blackbody consists of a hot metal sphere with blackened inside walls containing a small hole that allows for the absorption and emission of radiation (the figure below is reproduced from Wikimedia Commons):

Radiation entering this cavity is essentially completely absorbed (indicated by the decreasing width of the green line in the above figure). Radiation within the cavity is at thermal equilibrium with the metal cavity and when emitted is characteristic of the temperature of the blackbody.

The classical analysis of the radiation emitted by a blackbody viewed the metal as consisting of oscillating dipoles that both absorbed and emitted radiation characteristic of the temperature of the dipoles, i.e., of the metal. In this view the energy density of the radiation emitted by the blackbody is given by:

ρ (ν,T) dν = ( 8 π kB T ν2 / c3 ) dν

Here the energy density, ρ (ν,T), has units of energy per unit volume per unit frequency interval, ν is the frequency of the emitted radiation, and kB is Boltzmann’s constant, i.e., the gas constant on a per molecule basis:

kB = R / No = 8.314 J / (mol K) / 6.022 1023 molecules / mol

= 1.381 10-23 J / (molecule K) = 1.381 10-23 J / K

Does the expression for the energy density have the units suggested?

The problem with this view was that the classical equation predicted that the energy density would increase as the square of the radiation frequency leading to catastrophic energy density for ultraviolet energies, something that was not observed experimentally.

Max Planck solved this problem by postulating that the energy of the oscillators was proportional to the frequency of the radiation and not the square of the amplitude, as was classically thought. The relationship between radiation energy and frequency has become known as Planck’s law:

E = h ν

The proportionality constant, h, is known as Planck’s constant:

h = 6.626 10-34 J sec

What is the energy in Joules of the light emitted by a red laser pointer of wavelength 650 nm?

Planck further postulated that the oscillator energies were quantized, i.e., that not all energies were possible:

En = n h ν

where n is a quantum number that can take on only integer values:

n = 1, 2, 3, 4, ….

These ideas led to an expression for the energy density of a blackbody in which the ultraviolet catastrophe is avoided and which provided very good agreement with experiment:

ρ (ν,T) dν = ( 8 π h ν3 / c3 ) dν / (e + h ν / kB T – 1)

Does this expression give the correct units for the energy density?

A plot of the energy density for both the classical case and for the case where the energies of the oscillators are quantized is shown below:

Where would red light of wavelength 650 nm fall on this diagram?

Where would blue light of wavelength 475 nm fall on this diagram?

Why would a material at 6000 K be described as white hot?

The correspondence principle says that as the values of the variables change from microscopic to macroscopic, expressions derived for and valid in the quantum mechanical realm, will move smoothly over into those expressions known to hold in the classical realm.

Show that in the limit as the energy of the blackbody oscillator, hν, becomes very small compared to the thermal energy, kBT, the quantum mechanical expression for the energy density:

ρ (ν,T) dν = ( 8 π h ν3 / c3 ) dν / (e + h ν / kB T – 1)

becomes equal to the classical expression:

ρ (ν,T) dν = ( 8 π kB T ν2 / c3 ) dν

The Photoelectric Effect

In the late 1890s Heinrich Hertz discovered the photoelectric effect in which light incident on certain metal surfaces could cause electrons to be ejected from that surface. The plot below for potassium shows that below a certain threshold frequency of the incident light no electrons are ejected and further shows above this threshold frequency the kinetic energy of the ejected electrons is proportional to the frequency of the incident radiation:

These results were at odds with classical physics which predicted:• that the kinetic energy of the ejected electrons should be some function of the intensity, i.e., proportional to the square of the amplitude of incident light wave and not proportional, as was observed, to the frequency of the incident light.• the effect should be observed for light of any frequency of sufficient intensity and should not exhibit a threshold frequency, as again was observed.

In 1905 Albert Einstein, at age 26, published four seminal papers:

On a heuristic viewpoint concerning the production and transformation of light, Annalen der Physik 17 (1905), 132-148. This paper on the photoelectric effect was the work for which Einstein was awarded the Nobel prize in 1921.

On the movement of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat, Annalen der Physik,, 17 (1905), 549-560. This paper explained Brownian motion, in which microscopically visible small particles suspended in a fluid undergo apparent random motion as a result of collisions with neighboring not visible molecular species. This link provides an animation of the Brownian motion concept (courtesy of Wikepedia). This video shows the Brownian motion of fat 0.5 – 3.0 um fat droplets suspended in milk.

On the electrodynamics of moving bodies, Annalen der Physik 17 (1905), 891-921. This paper constitutes the basis of Einstein’s special theory of relativity.

Does the inertia of a body depend upon its energy content?, Annalen der Physik, 18 (1905), 639-641. It was in this paper that Einstein demonstrated the equivalence of energy and mass in his famous equation, E = mc2.

(These links were originally provided by http://lorentz.phl.jhu.edu/AnnusMirabilis/ )

These papers along with the work of Max Planck and others turned classical physics on its head and laid the foundation for our current physical view of the world.

In Einstein’s paper on the photo electric effect he took the view that electromagnetic radiation could be viewed as a particle. A particle of electromagnetic radiation or light is called a photon and has an energy given by Planck’s law:

E = h ν

He then argued that some of the energy of the light incident on the metal surface would be used to overcome the energy that bound the electron to the metal (the work function of the metal) with any excess energy appearing as kinetic energy in the ejected electron:

Eincident photon = work function + kinetic energy of ejected electron

or:h ν = h νo + KE

Is this equation consistent with the plot of the photoelectric effect shown for K on an earlier slide?

The work function and electron kinetic energy are usually expressed in electron volts. An electron volt is the energy that is imparted to an electron, with a charge of 1.602 10-19 Coulombs (C), when it accelerated through potential of volt:

1 eV (electron volt) = (1.602 10-19 C) (1 V) (1 J / 1 C V)

= 1.602 10-19 J

kineticenergy of the

ejectedwavelength electron

(nm) (eV)========== ==========

666 0.00638 0.00612 0.00588 0.00566 0.00545 0.00526 0.06508 0.14491 0.22476 0.31461 0.39447 0.47434 0.55422 0.64411 0.72

Use the data given below for the photoelectric effect for K to determine the work function for K in eV and the value of Planck’s constant:

Wave Particle Duality

Just as a moving particle of finite mass can have associated with it a momentum, a particle of light or photon, should also have an associated momentum. An expression for the momentum of a photon can be derived by equating Einstein’s and Planck’s expressions for the energy of the photon:

E = m c2 = h ν = h c / λ

p = m c = h / λ

Calculate the momentum associated with a photon of red light of from a He-Ne laser pointer of wavelength of 653 nm.

Analogously Louis de Broglie argued that a particle, e.g., an electron, should have an associated wavelength (the de Broglie wavelength):

λ = h / (m v)

Calculate the de Broglie wavelength in nm of an electron moving at 2% of the speed of light? X-rays have wavelengths ranging from 0.010 nm to 10 nm. How does the de Broglie wavelength of an electron compare with that of X-rays?

The wave nature of electrons was first demonstrated by Davisson and Germer who observed that electrons scattered off a nickel surface gave rise to variations in intensity whose angular dependence agreed with what would be expected if the electrons were viewed as waves. The picture is a link to their original paper with comments.

shows that both x-rays and electrons exhibit the constructive and destructive interference patterns expected of waves diffracting from aluminum foil.

This link from Hitachi Corporation provides a video of electron interference.

Figure 1.8 below taken from Physical Chemistry A Molecular Approach, McQuarrie, D.A. and Simon, J.D., University Science Books, Sausalito (1997), p. 17:

Note that substances scatter or diffract light most strongly when the wavelength of the radiation is of the same order of magnitude as the distance between the scattering centers or the spacing in the diffracting surface.

Why when you are holding a CD in sunlight do you see the colors of the rainbow on the surface of the CD?

Take Aways

Boltzmann’s constant, kB, is just the gas constant, R, expressed on a per molecule basis:

kB = R / No = 8.314 J / (mol K) / 6.022 1023 molecules / mol

= 1.381 10-23 J / (molecule K) = 1.381 10-23 J / K

Often the per molecule is left off the units of kB.

The energy of a photon or particle of light is given by Planck’s law:

E = h ν = h c / λ

where h, is Planck’s constant:

h = 6.626 10-34 J sec

An electron volt is the energy that is imparted to an electron, with a charge of 1.602 10-19 Coulombs (C), when it accelerated through potential of volt:

1 eV (electron volt) = (1.602 10-19 C) (1 V) (1 J / 1 C V)

= 1.602 10-19 J

The de Broglie wavelength of a particle of mass, m, moving with velocity, v, is given by:

λ = h / (m v)

Note that substances scatter or diffract light most strongly when the wavelength of the radiation is of the same order of magnitude as the distance between the scattering centers or the spacing in the diffracting surface.