Module 04 — Atomic & Quantum Physics

Duration: 2–3 weeks | Prereq: Modules 00–03

This module is where classical physics runs out of road. The atomic world does not behave like a miniature version of the everyday world — it operates on fundamentally different rules. Quantum mechanics is the framework that describes it, and it is simultaneously the most precisely tested theory in the history of science and the most philosophically unsettling.

Managing expectations: Quantum mechanics is famously hard to visualise. You can't quite picture it because it doesn't work the way everyday objects work. The goal of this module is to understand what happens and why the equations work, not to have a comfortable mental image. Feynman's advice: "If you think you understand quantum mechanics, you don't understand quantum mechanics."

1. The Structure of the Atom

Historical models

Dalton (1803): atoms are indivisible solid spheres.

Thomson (1897): discovered the electron. "Plum pudding" model — electrons embedded in a diffuse positive charge cloud.

Rutherford (1911): gold foil experiment. Fired alpha particles at thin gold foil. Most went straight through. Some bounced back at large angles. Conclusion: atoms are mostly empty space, with a tiny dense positive nucleus.

Bohr (1913): electrons orbit the nucleus in fixed circular orbits (energy levels). Can't orbit at any radius — only specific allowed orbits. This explained why atoms emit discrete spectral lines.

Modern quantum model: electrons don't follow definite orbits. They exist in probabilistic "orbitals" — regions of space where they're likely to be found. The picture is inherently fuzzy.

Nuclear composition

Protons: positive charge (+e), mass ≈ 1.67 × 10⁻²⁷ kg Neutrons: no charge, mass ≈ 1.67 × 10⁻²⁷ kg Electrons: negative charge (−e), mass ≈ 9.11 × 10⁻³¹ kg (about 1/1836 of a proton)

Atomic number (Z): number of protons — defines the element. Mass number (A): total number of protons + neutrons. Isotopes: same element (same Z), different number of neutrons.

Notation: ᴬ_Z X (e.g., ¹²₆C = carbon-12, with 6 protons and 6 neutrons)

2. Radioactivity

Some nuclei are unstable and spontaneously emit radiation to become more stable. This is radioactive decay — a quantum process that can't be predicted for any individual nucleus (only statistically for large numbers).

Types of radiation

Type	Particle	Charge	Mass	Range in air	Stopped by
Alpha (α)	⁴He nucleus (2p + 2n)	+2e	4 u	~5 cm	Paper, skin
Beta-minus (β⁻)	Electron	−e	~1/1836 u	~1 m	Few mm aluminium
Beta-plus (β⁺)	Positron (antielectron)	+e	~1/1836 u	~1 m	Few mm aluminium
Gamma (γ)	High-energy photon	0	0	~100s of metres	Several cm lead

Decay equations

Alpha decay: nucleus loses 2 protons, 2 neutrons.

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

Beta-minus decay: neutron becomes proton + electron + antineutrino.

¹⁴₆C → ¹⁴₇N + ⁰₋₁e + v̄_e

In any decay equation: total mass number and total atomic number are conserved.

Half-life and exponential decay

Radioactive decay is random and spontaneous for each nucleus. For a large sample, the rate of decay is proportional to the number of nuclei present:

N(t) = N₀ × (½)^(t/T½)

Or equivalently: N(t) = N₀ e^(−λt)

Where:

N₀ = initial number of nuclei
T½ = half-life (time for half the nuclei to decay)
λ = decay constant = ln(2) / T½

Exponential decay means the quantity keeps halving over equal time intervals — never quite reaching zero, but decreasing rapidly. This shape appears all over physics (capacitor discharge, Newton's law of cooling, light absorption).

Applications: Carbon-14 dating (T½ ≈ 5730 years — useful for organic materials up to ~50,000 years old), medical imaging (short half-life isotopes for safety), nuclear power, smoke detectors (americium-241 alpha source).

3. The Photoelectric Effect — Where Quantum Begins

In 1905, Einstein explained an experiment that classical physics could not. Shine UV light on a metal surface and electrons are ejected. The finding that broke classical physics:

Below a threshold frequency, no electrons are ejected — no matter how intense the light.
Above the threshold, electrons are ejected immediately, even with very dim light.
Increasing intensity increases the number of electrons ejected but not their maximum kinetic energy.
Increasing frequency increases the maximum kinetic energy of ejected electrons.

Classical wave theory predicted: intensity should determine energy, not frequency. More intense = more energetic electrons. This was wrong.

Einstein's explanation: light comes in discrete packets — photons — each carrying energy:

E = hf (energy of a photon = Planck's constant × frequency)

h = 6.63 × 10⁻³⁴ J s (Planck's constant — one of the most fundamental constants in nature)

A single photon interacts with a single electron. If the photon has enough energy to liberate the electron from the metal (overcoming the work function φ), the electron escapes. Any leftover energy becomes kinetic energy:

Eₖ(max) = hf − φ

Threshold frequency: f₀ = φ/h (minimum frequency to eject electrons)

Einstein won the 1921 Nobel Prize for this — not for relativity.

The profound implication: Light, which wave optics showed was a wave (interference patterns, diffraction), also behaves as a stream of particles. This is wave-particle duality — the central mystery of quantum mechanics.

4. Energy Levels and Spectra

Electrons in atoms occupy discrete energy levels — they can't have just any energy. When an electron moves between levels, it emits or absorbs a photon with energy exactly equal to the gap between levels.

E_photon = hf = E_upper − E_lower

Emission spectra

Heat a gas and electrons jump to higher energy levels. As they fall back down, they emit photons at specific frequencies. Each element has a unique set of energy levels and therefore a unique spectrum — a spectroscopic fingerprint.

This is how we know the composition of distant stars — we split their light into a spectrum and read the fingerprint.

Absorption spectra

Shine white light through a cool gas. Electrons absorb photons of exactly the right frequencies to jump up. The spectrum shows dark lines at those frequencies — the same lines that appear bright in the emission spectrum of the same gas.

Energy levels in hydrogen

For hydrogen (simplest case), energy levels are:

E_n = −13.6 / n² eV

Where n = 1 (ground state), 2, 3, ... and eV is electron-volts.

1 eV = 1.6 × 10⁻¹⁹ J (the energy one electron gains through 1 volt of potential difference)

The ground state energy is −13.6 eV. The ionisation energy (to remove the electron completely) is +13.6 eV.

5. Wave-Particle Duality

The double-slit experiment with electrons reveals the heart of quantum weirdness.

Fire electrons one at a time through a double slit at a detector screen. Over time, an interference pattern builds up — even though each electron goes through individually. Each electron seems to interfere with itself.

If you try to detect which slit the electron went through, the interference pattern disappears.

The electron behaves as a wave while unobserved, and as a particle when detected. This isn't a measurement limitation — it's how nature works.

De Broglie wavelength: all particles have an associated wavelength:

λ = h / p = h / mv

Where p is momentum. For an electron moving at significant speed, λ is on the order of atomic spacings — which is why electron diffraction can probe atomic structure.

For everyday objects (e.g., a cricket ball at 40 m/s, mass 0.16 kg): λ = 6.63×10⁻³⁴ / (0.16 × 40) ≈ 10⁻³⁴ m — far smaller than any observable scale. This is why quantum effects don't appear at human scales.

6. The Heisenberg Uncertainty Principle

You cannot simultaneously know both the position and momentum of a particle with arbitrary precision:

Δx · Δp ≥ h / 4π

This is not a measurement limitation — it's a fundamental feature of reality. The more precisely you know position, the less precisely you can know momentum, and vice versa. There's a similar relation for energy and time:

ΔE · Δt ≥ h / 4π

Implication: An electron confined to a small space (like inside a nucleus) must have a large uncertainty in momentum — meaning it's moving very fast. This "zero-point energy" means electrons can never truly be at rest.

Counterintuitive implication: The energy-time relation is sometimes used to motivate the idea of "virtual particles" briefly borrowing energy, and the Casimir effect (a measurable force between nearby metal plates in a vacuum) is real. However, the "energy borrowing" language is a simplified picture from quantum field theory, not a direct consequence of the uncertainty principle as stated here. Treat it as an evocative analogy rather than a precise statement.

7. Nuclear Physics

Binding energy and mass defect

A nucleus weighs slightly less than the sum of its constituent protons and neutrons. This "missing" mass is the mass defect, and by E = mc², it represents the energy released when the nucleus formed — the binding energy.

E = mc² (energy = mass × speed of light squared)

c = 3 × 10⁸ m/s, so even tiny mass differences represent enormous energies.

Binding energy per nucleon peaks at nickel-62 (Ni-62), with iron-56 (Fe-56) a close second. In practice, Fe-56 is often cited because it is the dominant end-product of stellar nucleosynthesis — iron is what accumulates in stellar cores. Elements lighter than iron release energy by fusion (combining nuclei); elements heavier than iron release energy by fission (splitting nuclei).

This is why stars burn: they fuse hydrogen to helium, releasing energy. And why they explode at the end of their life: elements heavier than iron require energy to make — stellar physics runs out of "fuel" at iron.

Nuclear fission

A heavy nucleus (e.g., U-235) absorbs a neutron and splits into two medium-sized nuclei plus more neutrons and gamma radiation. The released neutrons can trigger more fissions — a chain reaction.

Nuclear power: controlled chain reaction — the neutrons are moderated (slowed) and the reaction rate is controlled.

Nuclear weapons: uncontrolled chain reaction.

Nuclear fusion

Two light nuclei combine to form a heavier one, releasing energy. Requires extremely high temperatures (millions of degrees) to overcome electrostatic repulsion.

The Sun: fuses ~600 million tonnes of hydrogen per second. The energy from this powers all life on Earth.

Fusion power (ITER, JET): the goal of controlled fusion research. More energy-dense and cleaner than fission. Still in development.

8. Self-Check Questions

A photon has wavelength 420 nm. What is its energy in Joules?
Uranium-238 undergoes alpha decay. Write the decay equation.
An element has a half-life of 8 days. What fraction remains after 32 days?
An electron is accelerated through 50 V. What is its de Broglie wavelength? (electron mass = 9.11 × 10⁻³¹ kg)
Sodium has a work function of 2.3 eV. What is the minimum frequency of light needed to eject electrons?

Answers:

E = hc/λ = (6.63×10⁻³⁴ × 3×10⁸) / 420×10⁻⁹ = 1.989×10⁻²⁵ / 4.2×10⁻⁷ = 4.73 × 10⁻¹⁹ J
²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He
32 days = 4 half-lives → (½)⁴ = 1/16
KE = eV = 1.6×10⁻¹⁹ × 50 = 8×10⁻¹⁸ J. p = √(2mKE) = √(2 × 9.11×10⁻³¹ × 8×10⁻¹⁸) = √(1.46×10⁻⁴⁷) = 3.82×10⁻²⁴ kg m/s. λ = h/p = 6.63×10⁻³⁴ / 3.82×10⁻²⁴ = 1.73 × 10⁻¹⁰ m ≈ 0.17 nm
φ = 2.3 eV = 2.3 × 1.6×10⁻¹⁹ = 3.68×10⁻¹⁹ J. f = φ/h = 3.68×10⁻¹⁹ / 6.63×10⁻³⁴ = 5.55 × 10¹⁴ Hz

Go Deeper

You've now touched the edge of one of the deepest areas in science:

Quantum field theory (QFT) — the full framework combining quantum mechanics with special relativity. This is the theory of electrons, photons, and their interactions (QED). Not approachable without university-level maths, but the conceptual landscape is beautifully described in Feynman's QED.
Interpretations of quantum mechanics — what does it mean? The Copenhagen interpretation, many-worlds, pilot wave theory, relational quantum mechanics... physicists still disagree. This is genuinely open philosophy of physics.
Quantum computing — uses superposition and entanglement to process information in fundamentally new ways. An active research area with practical implications.
Standard Model of particle physics — the full classification of fundamental particles (quarks, leptons, bosons). The most precisely tested theory in science. PBS Space Time on YouTube covers this with remarkable depth at a non-technical level.
Sixty Symbols: their episodes on the photoelectric effect, Heisenberg, and the double-slit experiment are outstanding.

Next: Module 05 — Cosmology & Space