Nuclear and Quantum Physics

Q: Why does increasing light intensity not increase photoelectron kinetic energy?

Photon energy $E=hf$ depends only on frequency. Increasing intensity increases the number of photons arriving per second, not the energy of each individual photon, so KE_max stays fixed by frequency alone.

Q: Why does energy in a particle-in-a-box scale with $n^2$ instead of $n$?

The energy formula $E_n = n^2h^2/8mL^2$ comes directly from the boundary conditions of confining a wave in a box of width L, which forces energy to scale as the square of the quantum number $n$, not linearly.

Photons, matter waves, radioactive decay and HL quantum rules — the exam-ready essentials.

Split illustration of a photon hitting a metal surface on one side and an exponential radioactive decay curve on the other

Subject

Physics

Curriculum

IB Diploma Programme

Grade

Topic

Nuclear and Quantum Physics

Reading

8 min

Difficulty

Advanced

Quick facts

Difficulty

★★★★☆

Exam weight

~14h SL / ~28h HL (~10% of HL course)

Prerequisites

Atomic structure, energy, basic mechanics

You'll learn

Photon model, matter waves, decay law, HL quantum rules

Revision time

45–60 min

Nuclear and quantum physics is one of the most graph-heavy, formula-dense corners of IB DP Physics — and it shows up guaranteed in Paper 1 and Paper 2 every single session. The topic hinges on a handful of signature ideas: light and matter behaving as both particles and waves, unstable nuclei decaying according to a precise exponential law, and (at HL) the strange rules of wavefunctions, uncertainty and confined-particle energy levels. Examiners love testing whether you can read meaning off a graph's gradient or intercept rather than just plugging numbers into an equation. This teaser walks through the five ideas that appear most often — the photoelectric effect, de Broglie wavelength, radioactive decay, alpha/beta/gamma transformations and HL quantum mechanics — with the exact traps students fall into. For the full worked examples, formula derivations and complete practice sets, head to the full revision notes.

What you’ll be able to do

✓Explain why the photoelectric effect proves light behaves as photons

✓Apply Einstein's photoelectric equation to find KE_max, f_0 or φ

✓Interpret gradient and intercepts of a KE_max vs f graph

✓Calculate the de Broglie wavelength of matter and link it to diffraction evidence

✓Apply N = N₀e^(-λt) and A = λN to decay and half-life problems

✓Write balanced nuclear equations for alpha, beta-minus, beta-plus and gamma decay

✓Apply the Heisenberg uncertainty principle (HL)

✓Find energy levels for a particle confined in a 1D infinite well (HL)

The Photoelectric Effect: Light Behaving as Photons

Shine light on a metal and electrons only get ejected instantly once the frequency clears a sharp threshold, $f_{0}$ — no matter how intense a lower-frequency beam is, nothing happens. This only makes sense if light arrives as discrete photons of energy $E = h f$ , where raising intensity increases photon number but never photon energy. Einstein's equation $E_{k, ma x} = h f - ϕ$ turns this into a straight-line graph: gradient $h$ , y-intercept $- ϕ$ , x-intercept $f_{0}$ .

Graph of maximum kinetic energy of photoelectrons against light frequency showing gradient h and intercepts

Exam tip

Calculate the photon energy once and reuse the exact value across a question — recomputing it separately for each metal or threshold introduces rounding drift that costs marks.

Common mistake

Confusing the incident frequency with the threshold frequency $f_{0}$ — $f_{0}$ is always lower than the incident frequency whenever photoemission actually happens.

Mini summary

Photon energy depends only on frequency; intensity changes photon rate, never photon energy.

de Broglie Waves: Matter Behaving Like a Wave

If light can behave as a particle, de Broglie argued matter must be able to behave as a wave too, with wavelength $λ = h / p$ . Electron and neutron diffraction experiments confirm this directly — particles produce fringe and diffraction patterns exactly like light does, and Bragg's law $nλ = 2 d sin θ$ describes diffraction off crystal planes.

$Diagram of electrons diffracting through two slits producing an interference pattern on a screen$

Exam tip

Sanity-check fringe spacing answers: with tiny matter wavelengths, the calculated fringe spacing should come out far smaller than the slit separation used.

Common mistake

Using $cos θ$ instead of $sin θ$ in Bragg's law — the glancing angle is measured from the crystal plane, not the normal.

Radioactive Decay: Random Nuclei, Exponential Population

You can never predict when one specific unstable nucleus will decay — it's spontaneous and random, unaffected by temperature or chemical state. But with billions of identical nuclei, the population obeys a precise exponential law $N = N_{0} e^{- λ t}$ , because the decay constant $λ$ (probability of decay per unit time) is fixed for a given isotope. Half-life links directly to $λ$ via $T_{1/2} = ln 2/ λ$ .

Exponential decay curve showing number of undecayed nuclei halving at each successive half-life

Exam tip

Always check that the units of $λ$ match the units of $t$ before substituting — mixing $s^{- 1}$ with years is a silent error that gives no obvious red flag.

Common mistake

Assuming the same number of nuclei decay in each half-life, or that everything is gone after two half-lives — each half-life halves whatever is currently present: $N = N_{0} / 2^{n}$ .

Alpha, Beta and Gamma Decay

Alpha decay ejects a helium nucleus, dropping mass number $A$ by 4 and atomic number $Z$ by 2. Beta-minus decay converts a neutron into a proton, electron and antineutrino, raising $Z$ by 1 with $A$ unchanged; beta-plus does the reverse, lowering $Z$ by 1. Gamma decay changes neither $A$ nor $Z$ — the nucleus just sheds excess energy as a photon.

Four nuclear decay diagrams showing changes in mass number and atomic number for alpha, beta-minus, beta-plus and gamma decay

Decay type	Change in A	Change in Z	What's emitted
Alpha (α)	-4	-2	Helium nucleus
Beta-minus (β⁻)	0	+1	Electron + antineutrino
Beta-plus (β⁺)	0	-1	Positron + neutrino
Gamma (γ)	0	0	High-energy photon

Common mistake

Forgetting that gamma decay changes the energy state of the nucleus but leaves both A and Z completely unchanged.

HL Extension: Uncertainty, Wavefunctions and Confined Particles

At HL, the uncertainty principle $Δ x Δ p \geq h /4 π$ sets an absolute limit on knowing position and momentum simultaneously. A particle confined in a 1D infinite well has quantised energies $E_{n} = n^{2} h^{2} /8 m L^{2}$ — energy scales with $n^{2}$ , not $n$ . Probability density comes from the Born interpretation $P (x) = ∣ ψ (x) ∣^{2}$ , and for superposed states you must add the wavefunctions (with sign) before squaring.

Energy level diagram for a particle in a 1D infinite potential well showing energies scaling as n squared

Exam tip

For a state like $∣ ψ ⟩ = \frac{1}{2} ∣0 ⟩ + \frac{1}{2} ∣1 ⟩$ , the probability of measuring $∣1 ⟩$ is the coefficient squared (0.5), not the coefficient itself.

Common mistake

Computing $E_{3} = 3 \times E_{1}$ instead of $E_{3} = 9 \times E_{1}$ — energy in a box scales with $n^{2}$ , the single most common error on this formula.

Quick formula sheet

E = h f

Photon energy in terms of frequency. — Bigger frequency, bigger energy — intensity never enters.

E = \frac{h c}{λ}

Photon energy in terms of wavelength.

E_{k, ma x} = h f - ϕ

Einstein's photoelectric equation; gradient

h

, y-intercept

- ϕ

, x-intercept

f_{0}

e V_{s} = E_{k, ma x}

Stopping potential relation for the fastest photoelectrons.

λ = \frac{h}{p} = \frac{h}{m v}

de Broglie wavelength of matter.

nλ = 2 d sin θ

Bragg diffraction condition off crystal planes (glancing angle

θ

s = \frac{λ D}{d}

Fringe spacing for double-slit interference of light or matter waves.

Δ x Δ p \geq \frac{h}{4 π}

Heisenberg's uncertainty principle (HL).

E_{n} = \frac{n ^{2} h ^{2}}{8 m L ^{2}}

Energy levels of a particle confined in a 1D infinite potential well (HL). — Energy scales with $n^2$, never $n$.

P (x) = ∣ ψ (x) ∣^{2}

Probability density from the Born interpretation (HL) — add wavefunctions before squaring.

A = λ N

Activity is proportional to the number of undecayed nuclei present.

N = N_{0} e^{- λ t}

Exponential decay of the number of undecayed nuclei.

A = A_{0} e^{- λ t}

Activity decays with the same exponential shape as N.

T_{1/2} = \frac{ln 2}{λ}

Relationship between half-life and decay constant.

Practice questions

Easy

State two observations from the photoelectric effect that cannot be explained by a wave model of light.
Write the nuclear equation for the alpha decay of a generic nucleus with mass number A and atomic number Z.
Define half-life and decay constant, and state the relationship between them.

Medium

Light of known frequency ejects photoelectrons with a given stopping potential. Use Einstein's equation to find the work function of the metal.
A sample's activity falls from an initial value to a quarter of that value. How many half-lives have passed, and what does this tell you about the amount of sample remaining?
Electrons of a given de Broglie wavelength pass through two slits of known separation onto a screen at a known distance. Find the fringe spacing.

Challenge

Two points on a KE_max vs f graph are given. Determine Planck's constant and the work function from the gradient and intercept, without extrapolating by eye.
A particle in a 1D infinite well has a known ground-state energy. Find the energy of the n=3 state and explain why it is not simply three times the ground state.
For a superposition state $ψ = ψ_{1} + ψ_{2}$ with given amplitudes, find the correct probability density and explain the mistake of summing $∣ ψ_{1} ∣^{2} + ∣ ψ_{2} ∣^{2}$ directly.

Frequently asked questions

What's the difference between the photoelectric effect and double-slit diffraction in terms of what they prove about light?+

The photoelectric effect (instant emission, sharp threshold frequency, intensity-independent KE_max) proves light behaves as photons. Double-slit diffraction (fringe patterns from constructive/destructive interference) proves light behaves as a wave. Both are true depending on the experiment.

Why does increasing light intensity not increase photoelectron kinetic energy?+

Photon energy $E = h f$ depends only on frequency. Increasing intensity increases the number of photons arriving per second, not the energy of each individual photon, so KE_max stays fixed by frequency alone.

How is radioactive decay both random and predictable?+

Any single nucleus decays at a completely random, unpredictable moment. But because the decay constant $λ$ is fixed for an isotope, a large population of nuclei follows the precise exponential law $N = N_{0} e^{- λ t}$ .

Why does energy in a particle-in-a-box scale with $n^2$ instead of $n$?+

The energy formula $E_{n} = n^{2} h^{2} /8 m L^{2}$ comes directly from the boundary conditions of confining a wave in a box of width L, which forces energy to scale as the square of the quantum number $n$ , not linearly.

Do I need to know the uncertainty principle and wavefunctions at SL?+

No — the wavefunction, uncertainty principle and particle-in-a-box content are HL-only extensions. SL students focus on the photoelectric effect, de Broglie wavelength and radioactive decay.

What's the most common mistake with alpha, beta and gamma decay equations?+

Forgetting that gamma decay leaves both mass number A and atomic number Z completely unchanged — it only releases a photon of excess energy, unlike alpha and beta decay which transform the nucleus.

Get the complete Nuclear and Quantum Physics notes

Full worked examples for photoelectric, de Broglie, decay and HL quantum questions Every formula, definition and graph explained step by step Original mock papers and exam-style questions with full solutions Covers both SL essentials and the complete HL extension

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Physics: Nuclear and Quantum Physics (SL)Fields: The IB DP Physics Guide to Flux, Induction & Charged Particle Motion Physics: The Particulate Nature of Matter Physics: Wave Behaviour