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

\[\lambda \nu = c\]

  • Speed of light: \(c = 3 \cdot 10^8 \, m/s\) (fundamental constant)
  • Frequency \(\nu\): cycles per second, \(1\,Hz = 1\,s^{-1}\)
  • Wavelength \(\lambda\): distance between successive peaks
  • Key question: how does energy \(E\) relate to frequency \(\nu\)?

The black body

  • Idealized model: absorbs and emits every wavelength
  • In thermal equilibrium at temperature \(T\)
  • Emitted spectrum set only by \(T\)
  • Heating up: (1) intensity rises, (2) peak shifts to higher \(\nu\), (3) color red to yellow to blue

The classical picture

  • Heated atoms vibrate like springs, radiating at frequency \(\nu\)
  • More short-wavelength modes fit in a box than long ones: \[dN_{\nu} = \frac{8\pi}{c^3} \cdot \nu^2 d\nu\]
  • Equipartition: each oscillator gets the same energy \[\langle E\rangle = k_BT\]

The ultraviolet catastrophe

  • Classical radiation distribution: \[\rho({\nu}) = k_B T \cdot \frac{8\pi}{c^3}\nu^2\]
  • Shoots to infinity at high \(\nu\)
  • Total radiation would be infinite: a light bulb could destroy the universe
  • Classical physics fails

Planck’s trick: quantization

In 1900 Planck postulated that only discrete energy values are allowed:

\[\boxed{E= h\nu}\]

  • 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 radiation law

Quantized oscillators give a frequency-dependent average energy: \[\langle E \rangle = \frac{h\nu}{e^{\frac{h\nu}{ kT}} - 1}\]

This yields a distribution that vanishes at high frequency: \[\rho_{\nu}(T) = \frac{8\pi \nu^2}{c^3} \cdot \frac{1}{e^{\frac{h\nu}{kT}} - 1}\]

  • Integrating over all \(\nu\) gives a finite total: \(\int^{\infty}_0 \rho_{\nu}(T)d\nu = \sigma T^4\)
  • No more catastrophe!

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

Energy is quantized: by postulating \(E = h\nu\), Planck cured the ultraviolet catastrophe and gave birth to quantum mechanics.