Quantum Theory

Syllabus reference

Unit 4, Topic 2 — 16 hours (including practicals)

Black-body radiation

A black body is an idealised object that absorbs all incident electromagnetic radiation and re-emits it based only on its temperature.

Wien's displacement law — the peak wavelength of emission is inversely proportional to the temperature:

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

where \(b = 2.898 \times 10^{-3}\) m K.

The ultraviolet catastrophe: classical physics predicted that a black body would radiate infinite energy at short wavelengths. This was resolved by Planck's hypothesis — energy is emitted in discrete packets (quanta).

\[ E = hf = \frac{hc}{\lambda} \]

where \(h = 6.626 \times 10^{-34}\) J s (Planck's constant).

Photoelectric effect

When light shines on a metal surface, electrons may be ejected. Classical wave theory could not explain the observations. Einstein explained it using the photon model:

Key formula

\[ E_k = hf - \phi \]

where:

\(E_k\) = maximum kinetic energy of ejected photoelectrons (J)
\(hf\) = energy of the incident photon (J)
\(\phi\) = work function of the metal (J)

Key observations:

Below the threshold frequency (\(f_0 = \frac{\phi}{h}\)), no electrons are emitted regardless of intensity
Increasing intensity increases the number of photoelectrons, not their energy
Increasing frequency increases the maximum kinetic energy of photoelectrons
Emission is instantaneous — no time delay

Worked example: Photoelectric effect

Question: Light of frequency \(8.0 \times 10^{14}\) Hz strikes a metal with work function \(3.0 \times 10^{-19}\) J. Find the maximum kinetic energy of the ejected electrons.

Solution:

[ E_k = hf - \phi = (6.626 \times 10^{-34})(8.0 \times 10^{14}) - 3.0 \times 10^{-19} ] [ E_k = 5.30 \times 10^{-19} - 3.0 \times 10^{-19} = 2.3 \times 10^{-19} \text{ J} ]

Bohr model

Bohr proposed that electrons orbit the nucleus in discrete energy levels (stationary states). Key features:

Electrons can only exist in specific orbits with quantised energies
Electrons can transition between levels by absorbing or emitting a photon of specific energy
The energy of the emitted/absorbed photon equals the difference between energy levels:

\[ E_{photon} = E_{higher} - E_{lower} = hf = \frac{hc}{\lambda} \]

Emission spectra — electrons drop to lower energy levels, emitting photons of specific wavelengths (bright lines on a dark background).

Absorption spectra — atoms absorb photons of specific wavelengths, exciting electrons to higher levels (dark lines on a continuous spectrum).

Wave–particle duality

de Broglie proposed that all matter has wave-like properties. The wavelength of a particle:

Key formula

\[ \lambda = \frac{h}{mv} = \frac{h}{p} \]

For everyday objects, the wavelength is negligibly small. For electrons and other subatomic particles, the wavelength is significant and can be observed (e.g. electron diffraction).

Heisenberg uncertainty principle

It is impossible to simultaneously know both the exact position and exact momentum of a particle:

\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]

This is a fundamental limit of nature, not a limitation of measurement technology.

Simulations and videos

PhET Simulations:

Crash Course Physics:

External resources: