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

VCEPhysicsStudy guide19 min readUpdated 21 Jun 2026

Quick answer

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.

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What this guide is for
The wave model of light
The photon model
Matter waves
Special relativity
Cross-links to dot points
Why this content matters in physics history
Check your knowledge

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

\Delta x = \frac{\lambda L}{d},

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.

Figure 1. Young's double-slit experiment: panel (a) gives the geometry with six rays from each slit reaching the screen; panel (b) is the true cos-squared intensity envelope I(y) sampled at 121 points and plotted as a polyline alongside the screen. Fringe spacing Δx rises with wavelength and screen distance, and falls with slit separation.

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,

I = I_0 \cos^2\theta.

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:

E_{k,\max} = hf - \phi.

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.

Figure 2. Photoelectric stopping-voltage analogue plot: maximum kinetic energy E_k versus frequency is a straight line of slope h above the threshold f₀; below the threshold no electrons are emitted. Millikan's 1916 measurement matched Planck's value to within 0.5 percent.

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

f = \frac{E_i - E_f}{h}.

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

Figure 3. Bohr hydrogen energy levels: panel (a) plots the levels at y computed from E_n = −13.6/n² eV. Three Lyman transitions to n = 1 emit ultraviolet photons (accent); three Balmer transitions to n = 2 give the visible Hα, Hβ and Hγ lines used to measure stellar hydrogen.

Matter waves

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

\lambda = \frac{h}{p}.

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:

The laws of physics are the same in all inertial reference frames.
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

t = \frac{t_0}{\sqrt{1 - v^2/c^2}}.

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

L = L_0 \sqrt{1 - v^2/c^2}.

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.

Check your knowledge

A focused set on Unit 4 light, matter and special relativity in the VCAA Section A and B style. Attempt under exam conditions before checking the solutions block. Use the data sheet for $h$ , $c$ , $m_e$ , $e$ .

State two pieces of experimental evidence in support of the photon model of light. (2 marks)
(a, 2) Calculate the energy of a single photon of yellow sodium light, $\lambda = 589 \ \text{nm}$ , in both joules and electronvolts. (b, 2) Calculate the de Broglie wavelength of an electron moving at $1.0 \times 10^{6} \ \text{m s}^{-1}$ . (4 marks)
In a photoelectric experiment using a clean potassium surface (work function $\phi = 2.30 \ \text{eV}$ ), monochromatic light of wavelength 400 nm is incident on the metal. (a) Calculate the photon energy in eV. (b) Calculate the maximum kinetic energy of the emitted photoelectrons. (c) Calculate the stopping voltage. (d) State and justify what happens to the photocurrent if the intensity (not the wavelength) is doubled. (7 marks)
(a, 3) In a Young's double-slit experiment with slits 0.20 mm apart, the second-order maximum for red laser light is observed at 6.5 mm from the central maximum on a screen 1.00 m away. Calculate the wavelength. (b, 2) Compare in one sentence how the pattern would change if electrons of the same de Broglie wavelength replaced the light. (5 marks)
The hydrogen atom emits a photon when an electron drops from $n = 3$ to $n = 2$ . The energies of the levels are $E_n = -13.6 / n^2 \ \text{eV}$ . (a) Calculate the photon energy and wavelength. (b) Identify the visible-spectrum colour and name the spectral series. (c) Sketch in words what an emission spectrum looks like compared to an absorption spectrum for the same atom. (6 marks)
A muon is created in the upper atmosphere at an altitude of 15.0 km and travels toward Earth at $v = 0.9995c$ . The muon's proper lifetime is $2.2 \ \mu\text{s}$ . (a) Calculate the Lorentz factor $\gamma$ . (b) From the Earth's frame, calculate the muon's lifetime and the distance it travels before decay. (c) From the muon's frame, calculate the contracted altitude. (d) Determine whether the muon reaches the surface, and explain why both frames agree on this observable outcome. (7 marks)
(a, 2) State Einstein's two postulates of special relativity. (b, 4) A 1.00 kg fuel sample is fully converted to energy in a hypothetical engine. Calculate (i) the energy released in joules and (ii) the equivalent number of household-year electricity uses, taking 8,000 kWh per household per year. (6 marks)
A practical investigation measures the diffraction pattern of a HeNe laser ( $\lambda = 632.8 \ \text{nm}$ ) through a single slit of unknown width. The first minimum lies at 4.20 mm from the central maximum on a screen 1.50 m from the slit. (a) Calculate the slit width. (b) Estimate the percent uncertainty if the screen distance is known to $\pm 0.01$ m and the fringe position to $\pm 0.10$ mm. (c) Suggest one modification that would reduce the dominant source of uncertainty. (6 marks)

Solutions

Q1: Two pieces of evidence: (i) the photoelectric effect (Einstein 1905): the existence of a threshold frequency below which no electrons are emitted, and the linear dependence of maximum photoelectron kinetic energy on frequency, are inexplicable on the wave model and demand quantised packets of energy $E = hf$ . (ii) The Compton effect (or, at VCE level, atomic line spectra): the discrete wavelengths emitted by excited atoms correspond to photons of specific energy carrying the energy difference between bound levels.
Q2: (a) $E = hc/\lambda = (6.63 \times 10^{-34})(3.00 \times 10^{8})/(589 \times 10^{-9}) = 3.38 \times 10^{-19} \ \text{J}$ . In eV: $3.38 \times 10^{-19} / 1.60 \times 10^{-19} = 2.11 \ \text{eV}$ . (b) $\lambda_{dB} = h/(mv) = 6.63 \times 10^{-34} / (9.11 \times 10^{-31} \times 1.0 \times 10^{6}) = 7.28 \times 10^{-10} \ \text{m}$ , about 0.73 nm.
Q3: (a) $E = hc / \lambda = (1240 \ \text{eV nm}) / 400 = 3.10 \ \text{eV}$ . (b) $KE_{max} = E_{photon} - \phi = 3.10 - 2.30 = 0.80 \ \text{eV}$ . (c) Stopping voltage $V_s = KE_{max} / e = 0.80 \ \text{V}$ . (d) Doubling intensity doubles the photon arrival rate (at fixed wavelength), so the photocurrent doubles. The maximum kinetic energy of each electron, and therefore $V_s$ , is unchanged because each electron still absorbs a single photon of the same energy.
Q4: (a) $d \sin\theta = m\lambda$ . For small angles $\sin\theta \approx y/L$ , so $\lambda = d y / (m L) = (2.0 \times 10^{-4} \times 6.5 \times 10^{-3}) / (2 \times 1.00) = 6.5 \times 10^{-7} \ \text{m} = 650 \ \text{nm}$ . (b) The pattern would be qualitatively identical, with electrons of the same de Broglie wavelength producing the same fringe spacing, demonstrating wave-particle duality.
Q5: (a) $E_3 - E_2 = -13.6/9 - (-13.6/4) = -1.51 + 3.40 = 1.89 \ \text{eV}$ . Wavelength $\lambda = hc / E = 1240 / 1.89 = 656 \ \text{nm}$ . (b) 656 nm is the deep red H $\alpha$ line of the Balmer series. (c) An emission spectrum is a series of bright coloured lines on a dark background, at the wavelengths $\lambda$ corresponding to allowed downward transitions. An absorption spectrum is a continuous (rainbow) spectrum from a hot blackbody seen through cool hydrogen, showing dark lines at exactly the same wavelengths where photons have been absorbed and re-emitted in all directions.
Q6: (a) $\gamma = 1 / \sqrt{1 - 0.9995^2} = 1 / \sqrt{0.00099975} = 31.63$ . (b) In the Earth's frame the muon's lifetime is dilated: $\tau = \gamma \tau_0 = 31.63 \times 2.2 \times 10^{-6} = 6.96 \times 10^{-5} \ \text{s}$ . Distance $= v \tau = 0.9995 \times 3.00 \times 10^{8} \times 6.96 \times 10^{-5} = 2.09 \times 10^{4} \ \text{m} = 20.9 \ \text{km}$ . (c) Muon's frame: altitude is contracted $L = L_0 / \gamma = 15.0 / 31.63 = 0.474 \ \text{km}$ . (d) The muon reaches the surface. In the Earth's frame it travels 20.9 km before decaying, which is more than the 15.0 km altitude (without time dilation it would travel only $v\tau_0 = 0.9995 \times 3.00 \times 10^{8} \times 2.2 \times 10^{-6} = 660 \ \text{m}$ ). In the muon's frame the contracted altitude of 474 m is crossed in $L/v = 474 / (0.9995 \times 3.00 \times 10^{8}) = 1.58 \times 10^{-6} \ \text{s}$ , which is less than its proper lifetime of $2.2 \ \mu\text{s}$ . Both frames predict the same observable outcome: time dilation (Earth's frame) and length contraction (muon's frame) are two descriptions of the same event, so the muon reaches the surface in both.
Q7: (a) (i) The laws of physics take the same form in all inertial reference frames. (ii) The speed of light in vacuum is the same constant $c$ in all inertial reference frames, independent of the motion of source or observer. (b) (i) $E = mc^2 = 1.00 \times (3.00 \times 10^{8})^2 = 9.00 \times 10^{16} \ \text{J}$ . (ii) $9.00 \times 10^{16} \ \text{J} / (3.6 \times 10^{6} \ \text{J kWh}^{-1}) = 2.50 \times 10^{10} \ \text{kWh}$ . Number of household-years $= 2.50 \times 10^{10} / 8000 = 3.13 \times 10^{6}$ , just over three million Australian household-years.
Q8: (a) For single-slit diffraction the first minimum is at $\sin\theta = \lambda / a$ . With small angles, $a = \lambda L / y = (632.8 \times 10^{-9})(1.50) / (4.20 \times 10^{-3}) = 2.26 \times 10^{-4} \ \text{m} = 0.226 \ \text{mm}$ . (b) Relative uncertainties: $L$ contributes $0.01/1.50 = 0.67$ percent, $y$ contributes $0.10/4.20 = 2.38$ percent. Total (sum in quadrature) $\sqrt{0.67^2 + 2.38^2} = 2.5$ percent. So $a = (0.226 \pm 0.006) \ \text{mm}$ . The fringe-position uncertainty dominates. (c) Use a longer screen distance, or use a CCD camera to localise the fringe position to micrometre precision; either would shrink the dominant uncertainty.