The Need for Quantization

Chem 3240 · Lecture 1.2

Davit Potoyan

What is the nature of light?

• Light as a traveling electromagnetic wave
• Perpendicular electric and magnetic components
• Needs no medium, travels in vacuum at speed c
• This wave picture is not the whole story

The electromagnetic spectrum

• Visible light is a narrow band
• High frequency carries more energy (X-rays, gamma)
• Low frequency carries less energy (microwave, radio)
• Clear link between an object’s temperature and its radiation

Frequency, wavelength, speed of light

\lambda \nu = c \qquad c = 3 \cdot 10^8 \, \text{m/s}

• Key question: how does the energy E of light relate to its frequency \nu?

How a light bulb started quantum mechanics

• 1887: Siemens and Helmholtz found the PTR in Berlin, the first national standards lab, to serve German industry
• Its customer: the electric lighting business. How much light does a glowing filament give per watt, and at what temperature?
• 1890s: Lummer, Pringsheim, Rubens, Kurlbaum measure black body spectra with unmatched precision, deep into the infrared
• October 1900: Wien’s formula fails at long wavelengths; Rubens tells Planck on a Sunday; Planck has the right formula by evening
• December 1900: the only derivation that works needs E = h\nu. Planck: “an act of desperation”
• The full story, with the people and instruments (video)

The black body

• Idealized model: absorbs and emits every wavelength
• In thermal equilibrium at temperature T
• Emitted spectrum set only by T
• Heating up: intensity rises, peak shifts to shorter \lambda, color goes red to white to blue

The recipe for a radiation spectrum

\rho_{\nu}(T)\, d\nu = \underbrace{\langle E \rangle}_{\text{energy per mode}} \times \underbrace{dN_{\nu}}_{\text{number of modes}}

• A mode is one standing wave that fits in the box, like one harmonic of a guitar string; each mode has its own frequency
• Energy at frequency \nu = average energy of one mode \times number of modes at that frequency
• Average mode energy \langle E\rangle is thermodynamics: what the waves are, how much energy each holds. This is where classical and quantum physics part ways
• Number of modes dN_\nu is geometry: how many standing waves fit in a box. Same classically and quantum

Average mode energy, the classical way

• Heated atoms vibrate like springs, radiating at frequency \nu
• Equipartition: every oscillator gets the same energy, whatever its frequency \langle E\rangle = k_BT
• The same k_BT for a slow, long wave and a fast, short one. Hold that thought

Number of modes: counting the waves in a box

• Only whole half-waves fit: \lambda_n = 2L/n
• Shorter waves fit more easily: N \sim L/\lambda \sim \nu in 1D
• In 3D the wave must fit along each direction, so N \sim \nu^3
• A thin slice of frequency holds dN_{\nu} = \frac{8\pi}{c^3}\,\nu^2\, d\nu

The ultraviolet catastrophe

• Energy per mode \times modes, the Rayleigh-Jeans law: \rho(\nu) = k_B T \cdot \frac{8\pi}{c^3}\nu^2
• It shoots to infinity at high \nu: a light bulb could destroy the universe
• The mode count is solid geometry, so the average energy must be wrong

Live: heat it up

Planck vs Rayleigh-Jeans (classical), plotted per unit frequency; shaded band is visible light, dotted line is the peak

viewof Tbb = Inputs.range([1000, 10000], {step: 100, value: 5800, label: "temperature T (K)"})

{
  const h = 6.626e-34, c = 2.998e8, kB = 1.381e-23;
  const nus = d3.range(0.02, 30, 0.05);   // in units of 1e14 Hz
  const planck = nus.map(n => { const nu = n * 1e14; return {x: n, y: 8 * Math.PI * h * Math.pow(nu, 3) / Math.pow(c, 3) / Math.expm1(h * nu / (kB * Tbb))}; });
  const rj = nus.map(n => { const nu = n * 1e14; return {x: n, y: 8 * Math.PI * Math.pow(nu, 2) * kB * Tbb / Math.pow(c, 3)}; });
  const ymax = 1.6 * d3.max(planck, d => d.y);
  const nuPeak = 5.879e10 * Tbb / 1e14;   // Wien's law in frequency form
  return Plot.plot({
    width: 1000, height: 360,
    x: {label: "frequency (10^14 Hz)", domain: [0, 30]},
    y: {label: "energy density per unit frequency", domain: [0, ymax], ticks: 0},
    marks: [
      Plot.rect([{x1: 4.0, x2: 7.9, y1: 0, y2: ymax}], {x1: "x1", x2: "x2", y1: "y1", y2: "y2", fill: "gold", fillOpacity: 0.15}),
      Plot.ruleX([nuPeak], {stroke: "#999", strokeDasharray: "2,3"}),
      Plot.line(rj, {x: "x", y: "y", stroke: "#7fb3d5", strokeWidth: 2, strokeDasharray: "6,4", clip: true}),
      Plot.line(planck, {x: "x", y: "y", stroke: "#C8102E", strokeWidth: 2.5})
    ]
  });
}

md`Peak at **${(5.879e10 * Tbb / 1e14).toFixed(1)} x 10^14 Hz**: the object ${Tbb < 3500 ? "glows dull red" : Tbb < 6500 ? "looks yellow-white, like the Sun" : "looks blue-white"}. The classical curve never comes down.`

Planck’s trick: quantization

In 1900 Planck postulated that an oscillator can hold only discrete amounts of energy:

\boxed{E = n\,h\nu}, \qquad n = 0, 1, 2, \ldots

• Planck’s constant: h = 6.63 \cdot 10^{-34} \, \text{J} \cdot \text{s}
• Atoms absorb and emit radiation in quanta, multiples of h\nu
• h is tiny, so quantization is invisible at the macro scale (classical limit h \to 0)

Planck’s law: same recipe, new average energy

Quantized oscillators give a frequency-dependent average energy (Boltzmann-weighted sum over n): \langle E \rangle = \frac{h\nu}{e^{\frac{h\nu}{ kT}} - 1} \qquad \Big(\to k_BT \text{ for } h\nu \ll k_BT\Big)

Energy per mode \times modes, with the mode count untouched:

\rho_{\nu}(T) = \frac{8\pi \nu^2}{c^3} \cdot \frac{h\nu}{e^{\frac{h\nu}{kT}} - 1}

• The exponential beats the \nu^2: the curve comes down, the total is finite (\propto T^4)

Wien’s displacement law

• Peak wavelength is inversely proportional to temperature:

\lambda_{max} = \frac{b}{T}

• b = 2.898 \cdot 10^{-3} \, m \cdot K
• Connects an object’s temperature to its color
• Hotter object: shorter \lambda_{max} (bluer)

Takeaway

A spectrum is modes times energy per mode. Classical equipartition gives every mode k_BT and the ultraviolet catastrophe; Planck’s E = nh\nu starves the high-frequency modes, cures the catastrophe, and starts quantum mechanics.