Non-relativistic quantum mechanics: Schrödinger equation, Born rule, Heisenberg uncertainty, Pauli exclusion, entanglement, wave-particle duality. Interpretation-neutral where possible.
quantum
Schrödinger equation
Fundamental equation of non-relativistic quantum mechanics governing time-evolution of a wavefunction ψ under Hamiltonian H.
Born rule
The probability of measuring a quantum system in state |ψ⟩ to be in eigenstate |a⟩ equals |⟨a|ψ⟩|².
Heisenberg uncertainty principle
Conjugate observables cannot simultaneously be known to arbitrary precision; e.g. σ_x · σ_p ≥ ℏ/2.
Pauli exclusion principle
Two identical fermions cannot occupy the same quantum state simultaneously. Consequence of antisymmetry of the joint wavefunction.
Quantum superposition
A quantum system can exist as a linear combination (superposition) of basis states prior to measurement. Applies to amplitudes, not…
Quantum entanglement
A multi-particle quantum state that cannot be factored into product states; measurement outcomes on subsystems are correlated in ways that…
Wave-particle duality
Quantum entities (photons, electrons, etc.) exhibit both wavelike (interference) and particle-like (localized detection) behavior depending…
Energy quantization
Bound quantum systems admit only discrete energy eigenvalues. Originally observed in black-body radiation and the hydrogen spectrum.
Hilbert-space formulation of QM
Pure states = rays in a complex separable Hilbert space 𝓗; observables = self-adjoint operators; measurement = spectral projection. …
Canonical commutation relation
[x̂_i, p̂_j] = iℏ δ_{ij}. Cornerstone of quantisation; underlies Heisenberg uncertainty and the ℏ → 0 classical limit (Poisson-bracket…
Heisenberg picture
Equivalent formulation of QM in which operators evolve by A(t) = e^{iHt/ℏ} A e^{−iHt/ℏ} and states are time-independent; Heisenberg eq: iℏ…
Interaction picture
Intermediate picture in which states evolve under H_int only; basis of time-dependent perturbation theory and Dyson series.
Density matrix ρ̂
ρ̂ = Σ p_i |ψ_i⟩⟨ψ_i|: unifies pure and mixed states. Expectation ⟨A⟩ = Tr(ρ̂ A), evolution iℏ ∂_t ρ̂ = [H, ρ̂] (von Neumann).
Feynman path integral
⟨x_f, t_f|x_i, t_i⟩ = ∫ 𝒟x(t) e^{iS[x]/ℏ}. Equivalent to canonical quantisation; fundamental basis of QFT and lattice gauge theory.
Quantum harmonic oscillator
Exactly-solvable quantum system with H = p²/2m + ½ mω²x². Spectrum E_n = ℏω(n + ½); ladder operators a, a† manipulate levels. Building…
Quantum angular momentum
L_i operators satisfy [L_i, L_j] = iℏ ε_{ijk} L_k; eigenvalues L² = ℏ²ℓ(ℓ+1), L_z = ℏm with ℓ ∈ {0, ½, 1, …} and m ∈ {−ℓ, …, ℓ}.
Quantum spin
Intrinsic angular momentum S with S² = ℏ²s(s+1), s ∈ {0, ½, 1, …}. Electron carries s = ½; Pauli matrices realise it. Not a classical…
Quantum perturbation theory
Systematic expansion of eigenenergies/states in a small parameter (Rayleigh–Schrödinger, Brillouin–Wigner). Time-dependent version leads…
Bell inequality
Inequality satisfied by any local hidden-variable theory but violated by quantum mechanics. Experimentally confirmed violations (Aspect…
Quantum decoherence
Loss of quantum coherence through entanglement with an environment: density matrix's off-diagonal elements decay, yielding an effective…
Stone–von Neumann theorem
CCR [x,p]=iℏ has essentially unique irreducible unitary representation on L²(ℝ) up to unitary equivalence. Fails for infinitely many DOF…
Wigner's theorem
Every symmetry of a quantum system is represented by a unitary or antiunitary operator on Hilbert space, unique up to phase. Foundation of…
Gleason's theorem
In dim ≥ 3 Hilbert space, every probability measure on projections comes from a density operator ρ via Tr(ρP). Derives Born rule from…
Kochen–Specker theorem
No non-contextual hidden-variable theory can reproduce QM predictions in dim ≥ 3; incompatible projective measurements cannot all have…
No-cloning theorem
No unitary can copy arbitrary unknown quantum state: U|ψ⟩|0⟩ = |ψ⟩|ψ⟩ fails by linearity. Foundation of quantum cryptography.
Quantum teleportation
Alice+Bob share Bell pair; Alice measures joint state |ψ⟩_C⊗|Φ⁺⟩_AB in Bell basis; Bob applies Pauli correction from 2 classical bits.
Environment-induced decoherence
Entanglement with environment diagonalizes reduced density matrix in pointer basis; selects classical-like states without collapse…
Berry geometric phase
Adiabatic cyclic evolution picks up phase γ = i∮⟨n|∇_R|n⟩·dR; observable in Aharonov–Bohm, topological insulators, molecular dynamics.
Aharonov–Bohm effect
Charged particle acquires phase e∮A·dℓ/ℏ around solenoid with B=0 outside; interference shift shows gauge potential is physical in QM.
Feynman path integral (QM)
⟨x_f,t_f|x_i,t_i⟩ = ∫𝒟x e^(iS[x]/ℏ); semiclassical stationary-phase gives Hamilton principal function; basis for QFT.
WKB semiclassical approximation
ψ ≈ A(x) exp((i/ℏ)∫p dx); connection formulas at turning points; Bohr–Sommerfeld quantization ∮p dq = (n+½)h.
Adiabatic theorem
System starting in eigenstate |n(0)⟩ remains in |n(t)⟩ under slow H(t) change (gap condition); dynamical + geometric phases.
Sudden approximation
Very fast H change leaves state unchanged; transition amplitudes given by overlaps ⟨n_new|m_old⟩. Beta decay, shake-up in photoemission.
Fermi's golden rule
Γ_{i→f} = (2π/ℏ)|⟨f|V|i⟩|² ρ(E_f); first-order time-dependent PT; applies to decay rates, absorption cross sections.
Landau–Zener formula
Diabatic passage probability P = exp(-2π|H_12|²/(ℏv|Δα|)) at avoided crossing; governs nonadiabatic transitions.
Dyson series & time-ordering
U(t,t₀) = T exp(-i∫H_I(t')dt'/ℏ); Dyson equation for interacting Green functions; basis of scattering perturbation.
Lippmann–Schwinger equation
|ψ⁺⟩ = |φ⟩ + (E-H_0+iε)⁻¹ V|ψ⁺⟩; T-matrix and Born series; analytic scattering theory.
Partial wave analysis
f(θ) = (1/k)Σ(2ℓ+1)e^(iδ_ℓ)sin δ_ℓ P_ℓ(cos θ); low-energy s-wave scattering length; resonances as poles in δ_ℓ.
Levinson's theorem
δ_ℓ(0) - δ_ℓ(∞) = n_ℓ π counts bound states with angular momentum ℓ. Connects scattering to spectrum.
Quantum Zeno effect
Frequent projective measurements suppress evolution: decay probability ~ t² for short t makes repeated measurement freeze state.
Rabi oscillation
Two-level system on-resonance driven at Ω_R = |⟨e|d·E|g⟩|/ℏ oscillates between levels; basis of coherent control, qubit gates.
Ramsey interferometry
Two π/2 pulses separated by free evolution give fringes in detuning; basis of atomic clocks reaching 10⁻¹⁸ fractional uncertainty.
Quantum error correction (Shor/Steane codes)
Encode 1 logical qubit into 9/7 physical qubits; syndrome measurement detects bit/phase flips without disturbing logical state. Threshold…
Shor's quantum factoring via modular exponentiation and period finding (quantum)
Quantum application of L0 multiplication and Peano successor. Shor's 1994/1997 algorithm factors an integer N in polynomial time on a…
Grover's quantum search: ⌊π/4·√N⌋ oracle-query optimum (quantum)
Quantum application of L0 Peano successor and natural numbers. Grover's 1996 unstructured-search algorithm finds a marked element among N…
von Neumann type-III factor: projection-lattice cardinality and Connes classification (quantum)
Quantum application of L0 ZFC power-set and Cantor's theorem. Murray and von Neumann's 1936 factor classification (types I_n, I_∞, II_1,…
Haag's theorem: continuum-many unitarily inequivalent CCR representations (quantum)
Quantum application of L0 Zorn's lemma and well-ordering-theorem. Haag's theorem (1955) asserts that the interaction picture of quantum…
Birkhoff–von Neumann quantum logic: orthomodular non-distributive projection lattice (quantum)
Quantum application of L0 distributive-laws and L0 commutative-laws. Birkhoff–von Neumann (1936) observed that the lattice L(𝓗) of closed…
Piron–Solèr theorem: Hilbert-space QM reconstructed from propositional-lattice axioms (quantum)
Quantum application of L0 law-of-non-contradiction and L0 soundness. Piron's 1964 representation theorem (strengthened by Solèr 1995)…