The crisis of classical physics

Start with something slightly counter-intuitive: physicists in 1900 did not feel anything was missing.

The textbooks divided the world up like this:

• Mechanics handles how things move. Newton’s three laws plus gravitation get you from cannonballs to Neptune.
• Electromagnetism handles electricity, magnetism and light. Maxwell’s four equations unified them and predicted electromagnetic waves into the bargain — which Hertz duly produced in 1887.
• Thermodynamics and statistical mechanics handle heat, energy and entropy. Boltzmann reduced macroscopic temperature to molecules banging about.

Between them they covered essentially every experiment that could then be done. Students were occasionally advised against reading physics on the grounds that all that remained was measuring the constants to another decimal place.

This chapter is about how the second cloud turned into a downpour. To make the road ahead easier, let us first lay out the three most damaging contradictions — not solving them, just being precise about where they get stuck.

Dead end one: heat something up and the formula returns infinity

Anything with a temperature glows. Iron at 600 °C is dull red, at 1200 °C orange-white; the Sun’s surface at 5800 K is white. Physicists idealised this as a black body: a cavity that only absorbs and never reflects, observed through a small hole.

The question is plain: how is the radiated energy distributed across frequencies?

The measured curve is of course nothing like this. It rises, peaks at some frequency, then falls, and the total energy is finite.

Note the character of this failure: the theory is not slightly off, it returns infinity. That kind of error cannot be repaired with a correction factor.

Dead end two: cool it down and the heat capacity “vanishes”

The second dead end uses the same equipartition theorem in a different setting.

Each atom in a solid vibrates about its lattice site, which we can treat as three springs. Each oscillator has kinetic and potential energy, equipartition hands it $k_BT$, and a mole of solid therefore has

$$C_V=3N_Ak_B=3R\approx 25\ \text{J}/(\text{mol}\cdot\text{K})\tag{1.1.3}$$

This is the Dulong–Petit law, and at room temperature it is startlingly accurate for most metals.

How can equipartition fail? Its derivation uses only the most basic assumptions of statistical mechanics. Either statistical mechanics is wrong, or the premise that “every degree of freedom can take up energy continuously” is wrong.

Dead end three: a classical atom lasts under a ten-billionth of a second

In 1911 Rutherford fired alpha particles at gold foil and found that the positive charge and nearly all the mass sit in a tiny nucleus, with the electrons outside. So what are the electrons doing?

A stationary electron would simply be pulled straight into the nucleus by the Coulomb force. So it must orbit — like a planet round the Sun.

Estimate: how long does a classical atom livebasic~5 minexpand

The trouble is that the electron is charged, and classical electrodynamics has an ironclad rule: an accelerating charge radiates. Circular motion is accelerated motion (the direction keeps changing), so the electron must radiate continuously, lose energy continuously, and spiral inward continuously.

The radiated power is given by the Larmor formula:

$$P=\frac{e^2a^2}{6\pi\varepsilon_0c^3}\tag{1.1.4}$$

Take the hydrogen scale $r\approx0.5\ \text{Å}=5\times10^{-11}\ \text{m}$. The Coulomb force supplies the centripetal acceleration:

$$a=\frac{e^2}{4\pi\varepsilon_0 m r^2}\approx 9\times10^{22}\ \text{m}/\text{s}^2\tag{1.1.5}$$

The electron’s total energy (kinetic plus potential) is

$$E=-\frac{e^2}{8\pi\varepsilon_0 r}\approx-2.2\times10^{-18}\ \text{J}\tag{1.1.6}$$

Putting these in gives a radiated power of about $5\times10^{-8}\ \text{W}$. Dividing the total energy by the power gives an order-of-magnitude lifetime:

$$\tau\sim\frac{|E|}{P}\sim\frac{2.2\times10^{-18}}{5\times10^{-8}}\approx4\times10^{-11}\ \text{s}\tag{1.1.7}$$

About 40 picoseconds. Doing the integral properly (the power grows as the orbit shrinks) gives $1.6\times10^{-11}$ s, the same order.

In other words: according to classical electrodynamics, the block of wood on your desk should have collapsed into a heap of nuclei within the first 0.00000000002 seconds after the Big Bang.

The classical model cannot account for spectra either. A continuously shrinking orbit means a continuously changing frequency, so atoms should emit a continuous spectrum. What is actually measured is a set of sharp lines, positioned precisely enough to serve as an element’s fingerprint.

∎Key formulas

• Rayleigh–Jeans law

  $$u(\nu,T)=\frac{8\pi\nu^2}{c^3}k_BT$$

  Flawless derivation, divergent total energy — the ultraviolet catastrophe

• Dulong–Petit law

  $$C_V=3R\approx25\ \text{J}/(\text{mol}\cdot\text{K})$$

  Accurate at room temperature, useless in the cold

• Larmor radiated power

  $$P=\frac{e^2a^2}{6\pi\varepsilon_0c^3}$$

  Gives a classical atom under 10⁻¹¹ s to live

?Self-check3 questions

1. 1.

   What is the crucial point about the ultraviolet catastrophe?

   • The classical derivation contains no error, yet the total radiated energy diverges — a failure no correction factor can repair
   • Rayleigh and Jeans made a mistake counting the modes
   • The equipment of the day could not measure the ultraviolet accurately
   • A black body is an idealisation that does not exist

   Submit
2. 2.

   The vanishing of specific heat at low temperature directly challenges which of these?

   • Equipartition: every degree of freedom gets an average energy k_BT
   • Conservation of energy
   • The second law of thermodynamics
   • Coulomb's law

   Submit
3. 3.

   Classical electrodynamics predicts an orbiting electron falls into the nucleus within about 10⁻¹¹ s. That prediction rests on:

   • An accelerating charge necessarily radiates electromagnetic waves
   • The electron mass being very small
   • The Coulomb force being far stronger than gravity
   • The nucleus carrying positive charge

   Submit

What comes next

Of the three dead ends, the first was prised open first. At the end of 1900, in order to make a formula match the measured curve, Planck made an assumption he himself was unhappy with.

The next section looks at what he actually did, and why he spent more than a decade refusing to believe it.