Quantum Mechanics
Quantum mechanics is the branch of physics that governs phenomena at atomic and subatomic scales. Developed between 1900 and 1930 by Planck, Einstein, Bohr, de Broglie, Heisenberg, Schrödinger, Born, Dirac and others, it replaces deterministic trajectories with probability amplitudes and underlies the chemistry of atoms, the behaviour of solids, and the design of lasers, transistors and quantum computers.
A complex-valued function ψ(x, t) whose squared modulus |ψ|² gives the probability density of finding a particle at position x at time t. Wavefunctions must be normalised (∫|ψ|² dx = 1) and satisfy the Schrödinger equation.
The Schrödinger equation
In 1926 Erwin Schrödinger proposed an equation for the wavefunction:
iℏ ∂ψ/∂t = Ĥ ψ (time-dependent)
For stationary states, Ĥψ = Eψ — the time-independent form. The Hamiltonian operator for a non-relativistic particle is
Ĥ = −(ℏ²/2m) ∇² + V(r)
Solutions yield quantised energy levels — discrete eigenvalues — explaining atomic spectra and the stability of matter. ℏ = h/(2π) = 1.055 × 10⁻³⁴ J·s is the reduced Planck constant.
Operators and observables
In quantum mechanics every physical observable corresponds to a Hermitian operator:
| Observable | Operator |
|---|---|
| Position x | x̂ = x (multiplication) |
| Momentum p | p̂ = −iℏ ∂/∂x |
| Energy E | Ĥ |
| Angular momentum L_z | L̂_z = −iℏ ∂/∂φ |
The result of a measurement is always an eigenvalue of the operator; after measurement, the system collapses into the corresponding eigenstate. The expectation value of an observable in state ψ is ⟨A⟩ = ∫ψ* Â ψ dx.
Wave-particle duality
Quantum objects exhibit both wave and particle aspects depending on the experiment performed. Classic demonstrations:
- Electron double-slit experiment: even when sent one at a time, electrons accumulate to form an interference pattern.
- Photon Compton scattering: light scatters off electrons like a particle with momentum p = h/λ.
- Matter-wave diffraction of neutrons, atoms and even large molecules confirms de Broglie's hypothesis.
- Quantisation: energies, angular momenta and spins take only discrete values in bound systems.
- Superposition: a quantum state can be a linear combination of basis states until measurement.
- Entanglement: two particles can share a joint state that cannot be factorised — Einstein's "spooky action at a distance."
- Tunnelling: particles can pass through classically forbidden barriers; basis of α-decay, STM and Josephson junctions.
The uncertainty principle
Werner Heisenberg (1927) showed that certain pairs of observables cannot be simultaneously measured with arbitrary precision:
Δx · Δp ≥ ℏ/2
A similar relation holds for energy and time: ΔE · Δt ≥ ℏ/2. This is not a limitation of apparatus but a fundamental property of nature: position and momentum operators do not commute, [x̂, p̂] = iℏ.
Solvable model problems
Particle in a one-dimensional box
For a particle confined between x = 0 and x = L with infinite walls:
ψ_n(x) = √(2/L) sin(nπx/L), E_n = n²π²ℏ²/(2mL²), n = 1, 2, 3, …
Energy is quantised and the lowest state has non-zero (zero-point) energy.
Quantum harmonic oscillator
For V(x) = ½mω²x²:
E_n = (n + ½)ℏω, n = 0, 1, 2, …
The ground state has energy ½ℏω — again non-zero — reflecting the uncertainty principle.
Hydrogen atom
Solving the Schrödinger equation in a Coulomb potential yields energy levels E_n = −13.6 eV / n², matching Bohr's result. Three quantum numbers emerge:
- Principal n = 1, 2, 3, … (energy)
- Orbital angular momentum l = 0, 1, …, n−1 (shape: s, p, d, f)
- Magnetic m = −l, …, +l (orientation)
Adding electron spin s = ½ gives a fourth quantum number m_s = ±½. The Pauli exclusion principle forbids two electrons in an atom from sharing all four quantum numbers, explaining the periodic table.
Most exam questions in this area test (a) the form of the Schrödinger equation, (b) the uncertainty relation, and (c) the quantum-number rules. Memorise: l < n, |m| ≤ l, m_s = ±½, and Pauli's exclusion principle.
Interpretations and modern developments
- Copenhagen interpretation (Bohr, Heisenberg): the wavefunction encodes probabilities; measurement causes collapse.
- Many-worlds interpretation (Everett): all measurement outcomes occur in branching universes.
- Bohmian (pilot-wave) interpretation: particles have definite trajectories guided by the wavefunction.
Bell's theorem (1964) and subsequent experiments by Aspect (1982) and others ruled out local hidden-variable theories, confirming the strange but true quantum predictions for entangled systems. This non-locality is the resource behind quantum cryptography and quantum computing, fields that translate quantum strangeness into useful technology.