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Quantum Mechanics
Photons, matter waves, uncertainty, wavefunctions and quantum phenomena. 71 topics, from beginner to advanced. Open the interactive version with labs and quizzes
Quantum mechanics describes matter and light at the smallest scales, where energy comes in discrete packets and particles behave like waves. This area traces how classical physics failed, through Planck's quantum hypothesis and the photoelectric effect, to de Broglie waves, the uncertainty principle, the Schrodinger equation and wavefunctions. It shows why atoms are stable and why many everyday technologies depend on quantum effects.
How the topics connect
Quantum mechanics replaces the classical picture of definite paths with wavefunctions and probabilities. It starts with photons and electron waves, then uses the Schrödinger equation, uncertainty and tunnelling to explain atoms, lasers and modern technology. Its ideas are the foundation for the rest of modern physics.
Real-world uses
Explain how electrons jump between discrete energy levels to emit light of a precise colour, the basis of lasers and light-emitting diodes.
Understand how photons with enough energy release electrons in a semiconductor, the photoelectric effect applied to electricity generation.
Use the wave nature of electrons, with wavelengths far shorter than visible light, to image structures too small for optical microscopes.
Relate quantum tunnelling and band structure to how transistors switch and how flash memory stores data.
Light sources depend on quantised energy levels and stimulated emission, which were first understood with quantum theory.
Transistors, flash memory and quantum computers rely on electron wave behaviour, tunnelling and superposition.
Failure of classical physics
Blackbody, photoelectric effect and atomic spectra defeated classical theory.
By 1900 classical theory failed on blackbody radiation (the ultraviolet catastrophe), the photoelectric effect, atomic stability and line spectra, and the heat capacity of solids at low temperature. These failures pointed to energy being quantised.
Classical theory predicted infinite energy in the ultraviolet from a hot body.
Classical physics remains valid as a limit of quantum theory.
Planck's quantum hypothesis
Energy is emitted in discrete packets.
E = hfPlanck proposed that oscillators in a cavity exchange energy only in discrete amounts E = hf, which reproduced the blackbody curve exactly. He didn't initially believe light itself was quantised.
A 500 nm photon has E = 2.48 eV.
Higher frequency means a larger energy quantum.
Planck's constant
Fundamental constant of quantum physics.
h = 6.626×10⁻³⁴ J·sh = 6.626×10⁻³⁴ J·s sets the scale of quantum effects; it's tiny, which is why quantum behaviour is hidden in daily life. ħ = h/2π.
A 1 kg ball at 1 m/s has de Broglie wavelength 6.6×10⁻³⁴ m.
If h were large, everyday life would look quantum.
Photoelectric effect
Light ejects electrons from a metal only above a threshold frequency.
photoLight of sufficiently high frequency ejects electrons from a metal immediately; below a threshold frequency no electrons come out regardless of intensity. Classical waves couldn't explain this.
UV ejects electrons from zinc; bright red light doesn't.
Intensity changes the number of electrons, not their maximum energy.
Einstein's photoelectric equation
Kinetic energy equals photon energy minus work function.
K_max = hf − φEach photon gives all its energy hf to one electron: K_max = hf − φ, where φ is the work function. This confirmed light's particle nature.
Sodium (φ = 2.3 eV) with 400 nm light (3.1 eV): K = 0.8 eV.
Electrons ejected from deeper levels have less than K_max.
Threshold frequency
Minimum frequency to eject electrons.
f₀ = φ/hBelow f₀ = φ/h no electrons are emitted, even with intense light, because single photons lack enough energy.
φ = 2.3 eV: f₀ = 5.6×10¹⁴ Hz (green-yellow).
Intensity doesn't matter below threshold.
Stopping potential
Voltage that stops the fastest photoelectrons.
eV₀ = hf − φThe reverse voltage needed to stop the fastest photoelectrons: eV₀ = K_max = hf − φ. A graph of V₀ vs f has slope h/e, letting you measure h.
f = 8×10¹⁴ Hz, φ = 2.3 eV: V₀ = 1.0 V.
Independent of intensity.
Photon momentum
Photons carry momentum.
p = h/λA photon has p = h/λ = E/c, even though it is massless. Light exerts pressure and can push atoms (laser cooling).
A 500 nm photon: p = 1.3×10⁻²⁷ kg·m/s.
Momentum arises from energy, not mass.
Compton effect
X-ray wavelength increases after scattering by electrons.
Δλ = (h/mc)(1−cosθ)X-rays scattering off electrons emerge with longer wavelength: Δλ = (h/mₑc)(1 − cosθ). It can only be explained by photon–electron collisions that conserve energy and momentum.
At θ = 90° the shift is 2.43 pm.
Visible light shifts are too small to notice.
Pair production
A photon converts to an electron–positron pair.
A photon with energy > 2mₑc² = 1.022 MeV near a nucleus can convert to an electron–positron pair; the nucleus absorbs the extra momentum.
Demonstrates E = mc² and antimatter.
Can't happen in empty space (momentum conservation).
de Broglie hypothesis
Matter has wave properties.
λ = h/pAll matter has an associated wavelength λ = h/p. Electrons accelerated through 100 V have λ = 0.12 nm, similar to atomic spacings.
A 1 g bullet at 500 m/s: 1.3×10⁻³⁴ m, unobservable.
It's a probability wave, not a physical vibration.
Davisson–Germer experiment
Electron diffraction confirmed matter waves.
Electrons scattered from a nickel crystal showed diffraction peaks at angles matching λ = h/p, confirming matter waves in 1927.
54 eV electrons gave a peak at 50°.
Followed by diffraction of atoms and molecules.
Wave–particle duality
Quantum objects show both wave and particle behaviours.
Quantum objects show wave behaviour (interference) in propagation and particle behaviour (localised detection) on measurement. Neither classical picture is complete.
Single photons build up an interference pattern one dot at a time.
Not a switch between two states.
Double slit with electrons
Single electrons build up an interference pattern.
Fire electrons one at a time through two slits: each lands as a single dot, yet over time an interference pattern appears. If you detect which slit, the pattern disappears.
Done with molecules of C₆₀ and larger.
Requires the path to be unknowable.
Heisenberg uncertainty principle
Position and momentum cannot both be known precisely.
ΔxΔp ≥ ħ/2The position and momentum cannot both be sharply defined: ΔxΔp ≥ ħ/2. It's a fundamental limit from the wave nature of matter.
An electron confined to 0.1 nm has Δp ≥ 5×10⁻²⁵ kg·m/s, speed spread about 10⁶ m/s.
Not due to measurement disturbance alone.
Energy–time uncertainty
Short-lived states have uncertain energy.
ΔEΔt ≥ ħ/2ΔEΔt ≥ ħ/2: a state that lives only for Δt has an energy spread. Explains natural line widths and virtual particles.
An excited state with a 10 ns lifetime has ΔE ≈ 3×10⁻⁸ eV.
Time isn't an operator in the same way as position.
Wavefunction
Complex function whose squared magnitude is probability density.
P = ∣ψ∣²The state of a quantum system is described by a complex wavefunction ψ(x, t); |ψ|² gives the probability density of finding the particle at x. Phase information gives interference.
Hydrogen 1s ψ ∝ e^{−r/a₀}.
ψ itself isn't directly observable.
Born rule
Probability from the wavefunction.
The probability of finding a particle in a region is ∫|ψ|²dx; for general measurement, probability of eigenvalue a is |⟨a|ψ⟩|².
|ψ|² = 0.2 at x means a 20% density per unit length.
Probabilities are fundamental, not due to ignorance, in standard QM.
Normalisation
Total probability is one.
∫∣ψ∣²dx = 1Probabilities must add to one: ∫|ψ|²dx = 1. Multiply ψ by a constant to normalise it.
ψ = Ae^{−x²/2} needs A = π^{−1/4}.
Non-normalisable states like plane waves need special treatment.
Schrödinger equation (time-dependent)
Governs the evolution of the wavefunction.
iħ∂ψ/∂t = Ĥψiħ ∂ψ/∂t = Ĥψ determines how the wavefunction evolves. It's linear, so superpositions evolve independently.
A free particle's ψ spreads over time.
Evolution is deterministic until measurement.
Schrödinger equation (time-independent)
Eigenvalue equation for stationary states.
Ĥψ = EψĤψ = Eψ, an eigenvalue equation: allowed energies are eigenvalues and the stationary states are eigenfunctions. Boundary conditions quantise energy.
For a box, ψ must vanish at the walls.
Stationary means probability density doesn't change.
Operators and observables
Measurable quantities are represented by Hermitian operators.
Each observable (position, momentum, energy) is a Hermitian operator; measured values are its eigenvalues. Momentum p̂ = −iħ ∂/∂x.
Energy operator Ĥ = p̂²/2m + V.
Measuring gives one eigenvalue, never the average.
Commutators
Non-commuting operators imply uncertainty.
[x,p] = iħ[A, B] = AB − BA; if non-zero the observables can't be sharply defined simultaneously. [x̂, p̂] = iħ leads to the uncertainty principle.
Spin components don't commute.
Commuting observables share eigenstates.
Particle in a box
Energy levels are quantised.
E_n = n²h²/8mL²A particle confined to a region of width L has energy levels E_n = n²h²/(8mL²), with nodes in ψ. The lowest energy is non-zero.
Electron in 1 nm box: E₁ = 0.38 eV.
Energy grows as n²; spacing grows with n.
Free particle
Plane waves with continuous energy.
With V = 0 the solutions are plane waves e^{ikx} with E = ħ²k²/2m, continuous energy. Localised wave packets spread with time.
A moving electron beam.
Plane waves aren't normalisable.
Finite potential well
Wavefunctions leak into walls.
ψ penetrates into the classically forbidden walls, giving lower energies than the infinite well and a finite number of bound states.
Quantum wells in semiconductor lasers.
Always at least one bound state in 1D.
Quantum tunnelling
Particles cross barriers they classically could not.
T ≈ e^{−2κa}A particle can cross a potential barrier higher than its energy, with probability T ≈ e^{−2κa}, κ = √(2m(V − E))/ħ. It decreases rapidly with width and mass.
Electrons tunnel across 1 nm gaps in electronics.
Not a violation of energy conservation.
Tunnelling applications
Scanning tunnelling microscopes and nuclear fusion in stars.
Scanning tunnelling microscopes (current ∝ e^{−2κd} for atomic resolution), tunnel diodes, flash memory and fusion in the Sun.
STM sees individual atoms.
Current is extremely sensitive to the gap.
Alpha decay as tunnelling
Alpha particles escape the nucleus through the Coulomb barrier.
Alpha particles escape the nucleus by tunnelling through the Coulomb barrier (Gamow, 1928), explaining the huge range of half-lives (10⁻⁷ s to 10¹⁷ years) from small energy differences.
Uranium-238: 4.5 billion years.
Geiger–Nuttall law.
Quantum harmonic oscillator
Equally spaced levels with zero point energy.
E_n = (n+½)ħωEnergy levels E_n = (n + ½)ħω are equally spaced; the wavefunctions are Hermite polynomials times a Gaussian. It models vibrations of molecules, phonons and fields.
Photons are quanta of an EM oscillator.
Spacing is ħω.
Zero point energy
Minimum energy not zero.
E₀ = ½ħωEven at absolute zero a confined particle retains energy E₀ = ½ħω (oscillator) because Δx and Δp can't both vanish. Helium remains liquid at zero temperature at ordinary pressure.
It causes the Casimir force.
Can't be extracted as usable energy.
Hydrogen atom in quantum mechanics
Energy levels from the Schrödinger equation.
E_n = −13.6 eV/n²Solving the Schrödinger equation with a Coulomb potential gives E_n = −13.6 eV/n² and orbitals labelled by n, l, m. Fine structure and hyperfine splitting arise from relativistic and spin effects.
The 21 cm hydrogen line maps galaxies.
Degeneracy n² (without spin).
Quantum numbers
n, l, m, s specify electron states.
n (principal), l (orbital, 0 to n−1), m_l (−l to +l) and m_s (±½) specify an electron's state in an atom. They determine energy, shape, orientation and spin.
n = 3 has l = 0, 1, 2 subshells: 9 orbitals.
Each state holds one electron (Pauli).
Atomic orbitals
Shapes s, p, d, f define probability clouds.
Orbitals are regions of high electron probability: s spherical, p dumbbell, d cloverleaf. They're not paths. Their shapes explain chemical bonding.
2p has a node at the nucleus.
Orbitals ≠ orbits.
Electron spin
Intrinsic angular momentum of ½ħ.
Spin is intrinsic angular momentum of ½ħ with two projections (up or down). It gives electrons a magnetic moment and is essential for the periodic table and magnetism.
Not a literal spinning ball.
Spin-½ needs a 720° rotation to return.
Stern–Gerlach experiment
Beam splits by spin orientation.
Silver atoms passing through a non-uniform magnetic field split into two beams, not a continuous smear, showing spin quantisation.
1922 experiment.
Sequential measurements show non-commuting components.
Pauli exclusion principle
No two fermions share the same quantum state.
No two identical fermions can occupy the same quantum state. It builds atomic shells and prevents the collapse of white dwarfs and neutron stars.
Helium's two electrons have opposite spins.
Only for identical fermions.
Fermions and bosons
Half-integer vs integer spin particles.
Fermions (electrons, quarks, spin ½) obey exclusion and antisymmetric wavefunctions; bosons (photons, gluons, Higgs, spin 0 or 1) can occupy the same state.
Lasers (photons) and superfluid helium (bosons).
Composite particles are bosons if they contain an even number of fermions.
Identical particles
Symmetry under exchange affects statistics.
Swapping identical particles can only change ψ by a sign (+ bosons, − fermions). It produces exchange forces and quantum statistics.
Two electrons in the same spin state must avoid each other.
Not an actual force.
Angular momentum in QM
Quantised magnitude and projection.
L = √(l(l+1))ħThe magnitude L = √(l(l+1))ħ and one component L_z = mħ are quantised. The other components remain uncertain.
l = 1: L = 1.41ħ, L_z ∈ {−ħ, 0, ħ}.
L never equals lħ exactly.
Addition of angular momenta
Combining spins and orbits.
Combining spins and orbital momenta gives total J ranging from |l − s| to l + s. Controls the multiplet structure of spectra.
Two spin-½ particles: J = 0 (singlet) or 1 (triplet).
Use Clebsch–Gordan coefficients.
Spin–orbit coupling
Interaction creating fine structure.
The electron's magnetic moment interacts with the magnetic field produced by its orbital motion, splitting energy levels (fine structure). Larger for heavy atoms.
Sodium D lines at 589.0 and 589.6 nm.
Relativistic effect.
Zeeman effect
Spectral lines split in a magnetic field.
An external magnetic field splits atomic levels by ΔE = m_j g_J μ_B B, making spectral lines split into several components.
Used to measure magnetic fields on the Sun.
Anomalous Zeeman effect involves spin.
Stark effect
Spectral lines split in an electric field.
An electric field shifts and splits atomic energy levels; linear in E for hydrogen, quadratic otherwise.
Used in molecular spectroscopy.
Smaller than the Zeeman splitting for typical fields.
Perturbation theory
Approximate solutions for small changes to solvable systems.
Approximate the energy and states of a system with small extra interaction: E ≈ E⁰ + ⟨ψ⁰|H'|ψ⁰⟩. It's the main tool for real atoms and molecules.
Helium atom treated as hydrogen-like plus electron repulsion.
Fails for strong perturbations.
Variational principle
Upper bound on ground-state energy.
The expectation value of H in any trial state is ≥ the true ground-state energy; adjusting parameters to minimise it gives a good approximation.
Helium ground state: −77.5 eV from a simple trial function vs −79.0 measured.
Always an upper bound.
Dirac notation
Bras and kets compactly represent quantum states.
⟨ψ∣φ⟩States are kets |ψ⟩, duals are bras ⟨φ|, and inner products ⟨φ|ψ⟩ are amplitudes. Operators act as Â|ψ⟩. It makes abstract state space reasoning compact.
⟨x|ψ⟩ = ψ(x).
|ψ⟩ is not a function of x, but a vector.
Superposition
A system can be in a combination of states.
A quantum state can be a linear combination of other states: |ψ⟩ = a|0⟩ + b|1⟩, with probabilities |a|² and |b|². Interference arises from the relative phase.
A spin pointing along x is a superposition of up and down.
Not the same as a classical mixture.
Measurement and collapse
Observation yields one eigenvalue with Born probability.
Measuring an observable yields one eigenvalue with probability given by the Born rule, and the state then becomes the corresponding eigenstate. How and why the collapse occurs is the measurement problem.
Measuring spin up/down on a superposition gives 50/50.
Repeated measurement gives the same result.
Schrödinger's cat
A thought experiment about superposition at large scales.
Schrödinger's 1935 thought experiment: a cat linked to a radioactive atom would be in a superposition of alive and dead, highlighting the puzzle of applying QM to macroscopic objects.
Experiments realise "cats" with superconducting circuits and large molecules.
Decoherence explains why we never see it.
Entanglement
Correlations between particles that cannot be explained classically.
Two or more particles share a joint state that can't be written as separate states; measuring one instantly fixes correlations in the other, however far apart.
Two photons with correlated polarisations.
Can't be used to send signals faster than light.
Bell's inequalities
Experiments rule out local hidden variables.
Bell showed that local hidden-variable theories satisfy certain inequalities, while quantum mechanics violates them; experiments (Aspect 1982, 2015 loophole-free tests) confirm the violation.
Nobel Prize 2022.
Rules out local realism.
EPR paradox
Einstein, Podolsky and Rosen questioned quantum completeness.
Einstein, Podolsky and Rosen (1935) argued that QM must be incomplete because entangled particles seem to have definite properties without being measured. Bell's theorem later tested this.
Spin singlet states.
The paradox dissolves if reality is nonlocal.
Decoherence
Interaction with environment destroys superposition.
Interaction with the environment rapidly destroys phase relations between components of a superposition, making a system look classical. It explains the disappearance of quantum interference for large objects.
A dust grain decoheres in 10⁻³¹ s in air.
Doesn't solve the measurement problem by itself.
Interpretations of QM
Copenhagen, many-worlds and others.
Copenhagen (collapse), many-worlds (no collapse; branching), de Broglie–Bohm (pilot wave), and others make the same predictions but differ in the picture of reality.
Experiments can't yet distinguish them.
Choice is partly philosophical.
Quantum computing: qubits
Quantum bits can be 0 and 1 simultaneously.
A qubit is a two-state quantum system holding α|0⟩ + β|1⟩. n qubits represent 2ⁿ amplitudes at once. Interference algorithms (Shor, Grover) can beat classical ones.
300 qubits could encode more numbers than atoms in the universe.
Reading out gives only one outcome.
Quantum gates
Unitary operations on qubits.
Reversible unitary operations: Hadamard creates superposition, CNOT entangles two qubits, phase gates rotate relative phase. Circuits of gates implement algorithms.
H|0⟩ = (|0⟩ + |1⟩)/√2.
Gates must preserve probabilities.
Quantum cryptography
Uses quantum states for secure keys.
Quantum key distribution (BB84) uses photon polarisations; any eavesdropping disturbs the states and is detected. It's secure by physics rather than computation.
Used in satellite links.
Needs authenticated classical channel.
Quantum teleportation
Transfers a quantum state using entanglement.
An unknown qubit state can be transferred to a distant location using an entangled pair plus two classical bits; the original is destroyed. No matter is transported.
Demonstrated over 1400 km via satellite.
Cannot exceed light speed since classical bits are needed.
Lasers
Stimulated emission produces coherent light.
Light Amplification by Stimulated Emission of Radiation: photons stimulate excited atoms to emit identical photons, building a coherent beam in a resonant cavity. Needs a gain medium and pumping.
He–Ne at 632.8 nm.
Not a source of 'energy from nothing'.
Population inversion
More atoms in excited than ground state, required for lasing.
A laser requires more atoms in the excited state than in the lower state so stimulated emission beats absorption. Achieved with three- or four-level systems and external pumping.
Ruby laser pumped by a flash lamp.
Impossible in a two-level system in equilibrium.
Einstein coefficients
Rates of absorption, spontaneous and stimulated emission.
Einstein defined rates of absorption B₁₂, stimulated emission B₂₁ = B₁₂ and spontaneous emission A₂₁ and derived Planck's law from them, introducing stimulated emission.
A/B = 8πhν³/c³.
Spontaneous emission dominates at high frequencies.
Blackbody and Planck's law
Quantisation resolves ultraviolet catastrophe.
blackbodyPlanck's law B(λ,T) = (2hc²/λ⁵)/(e^{hc/λkT} − 1) describes the spectrum, peaking at λ_max = b/T. Quantisation removes the ultraviolet catastrophe.
The CMB follows it to 1 part in 10⁵.
Integrates to the Stefan–Boltzmann law.
Quantum statistics of light
Photons obey Bose–Einstein statistics.
Photons are bosons, so thermal light has Bose–Einstein statistics: bunching of detection events. Laser light has Poisson statistics, and single photons show antibunching.
Hanbury Brown–Twiss experiment.
Distinguishes classical and nonclassical light.
Quantum field theory intro
Particles are excitations of fields.
Combines quantum mechanics and special relativity: particles are excitations of fields that fill space. Allows particle creation and annihilation.
The electron is a ripple in the electron field.
Calculations use perturbation (Feynman diagrams).
Feynman path integral
Sum over all histories gives amplitudes.
The amplitude to go from A to B is the sum over all paths of e^{iS/ħ}. Classical paths arise where the action is stationary and phases add.
Describes the double slit as all paths through both slits.
Equivalent to Schrödinger's equation.
Quantum electrodynamics
The theory of light and electrons.
QED describes the interaction of light and charged particles by exchanging photons. It predicts the electron's magnetic moment to 12 digits, the most accurate theory in science.
Lamb shift in hydrogen.
Needs renormalisation to remove infinities.
Casimir effect
Attractive force between plates from vacuum fluctuations.
Two uncharged metal plates close together attract because they restrict vacuum electromagnetic modes between them: F/A = −π²ħc/(240d⁴).
At 100 nm: about 1 atm pressure.
Evidence for vacuum fluctuations.
Quantum Hall effect
Hall resistance quantised in 2D electron gases.
In a 2D electron gas at low temperature and high field the Hall resistance is quantised: R_H = h/(νe²), with plateaus independent of material details. Used as a resistance standard.
25,812.807 Ω for ν = 1.
Fractional versions reveal exotic quasi-particles.
Superfluidity
Frictionless flow of helium-4 at low temperature.
Helium-4 below 2.17 K flows without viscosity, climbs walls and has quantised vortices; explained as a partial Bose–Einstein condensate.
Fountain effect.
Related to but not the same as superconductivity.
Superconductivity and BCS theory
Cooper pairs carry current without resistance.
Bardeen–Cooper–Schrieffer theory: electrons form Cooper pairs through lattice vibrations, condense into a coherent state with an energy gap, and flow without resistance. High-temperature superconductors are not fully understood.
Aluminium's T_c = 1.2 K.
Pairs are bosons, not individual electrons.
Frequently asked questions
What is a photon?
A photon is a packet of electromagnetic energy with energy E = hf, where h is Planck's constant and f is the frequency. Light behaves as a stream of photons when it interacts with matter, as in the photoelectric effect.
What does the uncertainty principle mean?
The more precisely you know a particle's position, the less precisely you can know its momentum, and the reverse. It is a fundamental limit on what can be known at the same time, not just a measurement problem.
Why does quantum mechanics seem so strange?
Its predictions are very precise and have been confirmed many times, but they describe things, such as superposition and wave-particle duality, that have no everyday analogue. The mathematics is the reliable guide here, not intuition.
Does quantum mechanics mean nothing is certain?
It means individual outcomes are probabilistic, but the probabilities are precisely calculable, and they match experiments extremely well.
Why don't we see quantum effects in everyday life?
Quantum effects are tiny for large objects because their wavelengths are very small and interactions with the environment quickly destroy superpositions.
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