VCE Physics Unit 4 light and matter overview: photons, matter waves and special relativity

What this guide is for

Unit 4 carries half the Year 12 content and roughly half the exam marks. The conceptual transitions in this unit are bigger than anywhere else in the course: light is a wave, then a particle, then both; matter is a particle, then a wave too; time and length are absolute, then relative. This guide gives the big-picture view of all the Unit 4 content so the dot-point detail has somewhere to attach.

The wave model of light

Newton thought light was a stream of corpuscles. Huygens and later Young thought it was a wave. Young's double-slit experiment in 1801 decided the debate, at least for a century.

Young's double-slit. Coherent monochromatic light through two slits forms an interference pattern of bright (constructive) and dark (destructive) fringes on a screen. The fringe spacing is

with $\lambda$ the wavelength, $L$ the screen distance and $d$ the slit separation. Measuring $\Delta x$ at known $L$ and $d$ gives $\lambda$. Visible light wavelengths span 400-700 nm.

Polarisation. Light is a transverse electromagnetic wave with the electric field oscillating perpendicular to propagation. A polariser transmits the component aligned with its axis. Malus's law: if light of intensity $I_0$ passes through one polariser and then a second at angle $\theta$ to the first,

For unpolarised input, the first polariser removes half the intensity.

Refraction. Light bends when crossing a boundary between media of different refractive index. Snell's law: $n_1 \sin\theta_1 = n_2 \sin\theta_2$. Dispersion arises because $n$ depends on wavelength, so different colours bend by different amounts.

The photon model

By 1900 the wave model could not explain three observations:

• Black-body radiation (Planck, 1900). Energy quanta $E = hf$ needed to fit the spectrum.
• The photoelectric effect (Einstein, 1905). Threshold frequency, no intensity dependence on kinetic energy.
• Atomic line spectra (Bohr, 1913). Discrete energy levels.

Photoelectric effect. Light of frequency $f$ on a metal ejects electrons only if $f > f_0$, where $hf_0 = \phi$ is the work function. Maximum kinetic energy of ejected electrons:

Intensity controls the number of electrons ejected, not their kinetic energy. Plotted as $E_{k,\max}$ versus $f$, the result is a straight line of gradient $h$ and $x$-intercept $f_0$. Millikan's 1916 measurement gave $h = 6.6 \times 10^{-34}$ J s, matching Planck.

Worked example. Light of wavelength 400 nm shines on a metal with $\phi = 2.0$ eV. Find $E_{k,\max}$ of the ejected electrons.

$f = c/\lambda = 3.0 \times 10^8 / 4.0 \times 10^{-7} = 7.5 \times 10^{14}$ Hz.

$hf = 4.14 \times 10^{-15} \times 7.5 \times 10^{14} = 3.1$ eV.

$E_{k,\max} = 3.1 - 2.0 = 1.1$ eV.

Atomic spectra. Each atom has discrete bound-state energies. A transition from level $E_i$ to $E_f$ ($E_i > E_f$) emits a photon of frequency

The hydrogen Balmer series lies in the visible. Each element's line spectrum is unique, which underpins spectroscopy.

Matter waves

If light has particle character, perhaps particles have wave character. De Broglie proposed

For an electron accelerated through potential $V$, $eV = p^2/(2m_e)$, so $\lambda = h/\sqrt{2m_e eV}$. A convenient shortcut: $\lambda \approx 1.226/\sqrt{V}$ nm for $V$ in volts.

Davisson-Germer. Electrons fired at a nickel crystal diffracted, producing a Bragg-style pattern with $\lambda$ matching the de Broglie prediction. Electron diffraction is now used in electron microscopes, which exploit short electron wavelengths (sub-nanometre) to resolve detail far below the optical diffraction limit.

Wave-particle duality. Light and matter both exhibit wave behaviour (interference, diffraction) and particle behaviour (photoelectric effect, electron tracks in detectors). Which behaviour is observed depends on the experiment, not on the underlying entity. The behaviour cannot be reduced to either pure wave or pure particle.

Special relativity

Einstein's 1905 postulates:

1. The laws of physics are the same in all inertial reference frames.
2. The speed of light in vacuum is the same in all inertial frames, regardless of the motion of source or observer.

Two startling consequences:

Time dilation. A clock moving at speed $v$ relative to an observer ticks slowly by a factor

The proper time $t_0$ is measured in the frame in which the two events occur at the same place.

Length contraction. An object of proper length $L_0$ moving at speed $v$ relative to an observer is measured to have length

The proper length $L_0$ is measured in the rest frame of the object.

Worked example. A muon has a proper lifetime $t_0 = 2.2 \mu$s. If it moves at $v = 0.99c$ relative to Earth, what lifetime does an Earth observer measure?

$\gamma = 1/\sqrt{1 - 0.99^2} = 1/\sqrt{0.0199} = 7.09$.

$t = \gamma t_0 = 7.09 \times 2.2 = 15.6\ \mu$s.

This is why cosmic-ray muons created in the upper atmosphere reach the Earth's surface: at rest they would decay within a few hundred metres, but time dilation gives them many kilometres of travel.

Mass-energy equivalence. $E = mc^2$ relates rest energy to mass. Energy released in nuclear reactions corresponds to a measurable mass deficit. Total relativistic energy is $E = \gamma m c^2$, of which $mc^2$ is rest energy and the remainder is kinetic.

Cross-links to dot points

Unit 4 dot points covered by this overview:

• Wave model of light and interference.
• Refraction and dispersion of light.
• Polarisation and Malus's law.
• Photoelectric effect and photons.
• Atomic energy levels and emission spectra.
• Electromagnetic spectrum and EM waves.
• Matter waves and de Broglie wavelength.
• Wave-particle duality.
• Practical investigation design and uncertainty.

For numerical practice see the worked-problems guide. For exam structure and scaling see the Units 3 and 4 exam structure guide.

Why this content matters in physics history

Unit 4 is a tour of the conceptual revolution between 1900 and 1925. The classical picture (Newtonian mechanics, Maxwell's electromagnetism, Galilean relativity) failed for three regimes: the very small (atomic), the very fast (relativistic) and the very cold (black-body). Quantum mechanics and special relativity rescued the physics. Today both are routine: relativity is in every GPS satellite, quantum mechanics in every transistor.

In one sentence

VCE Physics Unit 4 covers the wave model of light (interference $\Delta x = \lambda L/d$, polarisation $I = I_0\cos^2\theta$, refraction by Snell's law), the photon model (photoelectric effect with $E_{k,\max} = hf - \phi$, atomic line spectra from transitions $hf = E_i - E_f$), matter waves with de Broglie wavelength $\lambda = h/p$, wave-particle duality, and Einstein's special relativity (time dilation $t = \gamma t_0$, length contraction $L = L_0/\gamma$, mass-energy $E = mc^2$); together these dot points tell the early-20th-century revolution that gave physics its modern foundations.