83 terms from classical mechanics, electrodynamics, quantum mechanics and quantum field theory.
A · B · C · D · E · F · G · H · I · K · L · M · N · O · P · Q · R · S · U · V · W
A
Action
The time integral of the Lagrangian along a path. The principle of least action says the path a system actually takes makes the action stationary. The Euler-Lagrange equation follows from this condition. (book)
Angular momentum
The quantity L = r × p for a point mass. It is conserved when the system is symmetric under rotations, which is an instance of Noether's theorem. In quantum mechanics its components do not commute. (book)
Annihilation operator
An operator that lowers the number of quanta in a state by one. In the harmonic oscillator it lowers the energy by one step; in quantum field theory it removes a particle with a given momentum. (book)
Anomaly
A classical symmetry that does not survive quantization. Anomalies in gauge symmetries must cancel for a theory to be consistent, which constrains the particle content of the Standard Model. (book)
B
Bohr radius
The most probable distance between the electron and the proton in the ground state of hydrogen, about 0.529 × 10⁻¹⁰ m. (book)
Bound state
A state in which a particle is confined by a potential and has a discrete energy, such as the electron in a hydrogen atom. (book)
Bra-ket notation
Dirac's notation for quantum states. A ket |ψ⟩ is a vector in Hilbert space, a bra ⟨φ| is a linear map from kets to numbers, and ⟨φ|ψ⟩ is their inner product. (book)
C
Canonical commutation relation
The relation [x, p] = iħ between the position and momentum operators. It is the starting point of canonical quantization and implies the Heisenberg uncertainty relation. (book)
Canonical quantization
The procedure that turns a classical theory into a quantum one by replacing Poisson brackets with commutators of operators. Applied to fields, it gives quantum field theory. (book)
Canonical transformation
A change of coordinates in phase space that preserves the form of Hamilton's equations. Generating functions produce them, and the Hamilton-Jacobi equation is built on them. (book)
Charge conservation
The total electric charge in a closed system does not change. Locally it is expressed by the continuity equation, which follows from Maxwell's equations. (book)
Configuration space
The space of all possible positions of a system. A system with N degrees of freedom has an N-dimensional configuration space. The Lagrangian formalism works here. (book)
Conservative force
A force whose work between two points does not depend on the path. It can be written as minus the gradient of a potential energy. (book)
Continuity equation
The equation ∂ρ/∂t + ∇·J = 0. It states that a density can only change by flowing in or out. It holds for electric charge and for probability in quantum mechanics. (book)
Coulomb's law
The electric force between two point charges is proportional to the product of the charges and inversely proportional to the square of their distance. (book)
Covariant derivative
A derivative that includes the gauge field so that the result transforms like the field itself under gauge transformations. Writing the Lagrangian with it couples matter to gauge fields. (book)
Creation operator
An operator that raises the number of quanta by one. In quantum field theory, acting with it on the vacuum creates a one-particle state. (book)
Cross section
A measure of how likely a scattering process is, with units of area. Quantum field theory predicts it from the squared scattering amplitude. (book)
D
Degrees of freedom
The number of independent coordinates needed to describe the configuration of a system. A free particle in three dimensions has three. (book)
Delta distribution
The object δ(x) that is zero everywhere except at x = 0 and integrates to one. It describes point charges and the normalization of position and momentum eigenstates. (book)
Dirac equation
The relativistic wave equation for spin-1/2 particles such as the electron. It predicts antimatter and the electron's magnetic moment. (book)
Displacement current
The term ε₀ ∂E/∂t that Maxwell added to Ampère's law. Without it the equations would violate charge conservation and would not predict electromagnetic waves. (book)
E
Eigenstate
A state that an operator maps to a multiple of itself. Measuring the observable on an eigenstate always gives the corresponding eigenvalue. (book)
Electric field
The force per unit charge that a test charge would feel at a point. Charges and changing magnetic fields produce it. (book)
Electromagnetic field tensor
The antisymmetric tensor F that holds the electric and magnetic fields as components. It makes the Lorentz covariance of Maxwell's equations explicit. (book)
Electromagnetic potential
The scalar potential φ and vector potential A, combined in the four-potential. The fields follow from their derivatives, and gauge transformations change the potentials without changing the fields. (book)
Electromagnetic wave
A solution of Maxwell's equations in which electric and magnetic fields oscillate perpendicular to each other and to the direction of travel, moving at the speed of light. (book)
Energy conservation
Total energy is constant when the laws of a system do not change with time. Noether's theorem connects the two. (book)
Euler-Lagrange equation
The equation of motion in the Lagrangian formalism: d/dt (∂L/∂q̇) − ∂L/∂q = 0. It follows from the principle of least action. (book)
Expectation value
The average result of measuring an observable on many identically prepared systems, written ⟨A⟩ = ⟨ψ|A|ψ⟩. (book)
F
Faraday's law
A changing magnetic flux through a loop induces an electric field around it. It is one of Maxwell's equations. (book)
Feynman diagram
A picture that stands for a term in the perturbative expansion of a scattering amplitude. Lines are particles, vertices are interactions, and fixed rules turn each diagram into a formula. (book)
Feynman propagator
The amplitude for a particle to travel between two spacetime points. It appears as the internal lines of Feynman diagrams. (book)
G
Gauge symmetry
A redundancy in the description of a theory: field transformations that change the variables without changing anything physical. Requiring it locally produces gauge fields such as the photon. (book)
Gauss's law
The electric flux through a closed surface equals the enclosed charge divided by ε₀. It is one of Maxwell's equations. (book)
Generalized coordinates
Any set of coordinates that fixes the configuration of a system, chosen for convenience, such as angles for a pendulum. The Lagrangian formalism works with them directly. (book)
H
Hamilton's equations
The equations of motion in the Hamiltonian formalism: q̇ = ∂H/∂p and ṗ = −∂H/∂q. They are first order in time and act on phase space. (book)
Hamilton-Jacobi equation
A partial differential equation for a generating function that makes the equations of motion trivial. It links classical mechanics to the Schrödinger equation. (book)
Hamiltonian
The function H(q, p) that generates time evolution. For most systems it equals the total energy. In quantum mechanics it becomes the operator in the Schrödinger equation. (book)
Harmonic oscillator
A system with a restoring force proportional to displacement. Its quantum version has equally spaced energy levels and is the model for free quantum fields. (book)
Heisenberg picture
The description of quantum mechanics in which operators change with time and states stay fixed. It gives the same predictions as the Schrödinger picture. (book)
Hermitian operator
An operator equal to its own adjoint. Its eigenvalues are real, so observables are represented by Hermitian operators. (book)
Hilbert space
The vector space of quantum states, with an inner product that gives probabilities. Observables act on it as Hermitian operators. (book)
Hydrogen atom
The bound state of one proton and one electron. Its energy levels follow from solving the Schrödinger equation with the Coulomb potential. (book)
I
Inertial frame
A frame of reference in which a free body moves in a straight line at constant speed. Newton's laws hold in their simple form there. (book)
K
Klein-Gordon equation
The relativistic wave equation for spin-0 fields. It follows from the energy-momentum relation E² = p²c² + m²c⁴. (book)
L
Lagrange multiplier
An extra variable used to find extrema subject to constraints. In mechanics it gives the constraint forces. (book)
Lagrangian
The function L = T − V, kinetic minus potential energy, for a mechanical system. Its time integral is the action. Field theories use a Lagrangian density. (book)
Legendre transform
The operation that turns the Lagrangian L(q, q̇) into the Hamiltonian H(q, p) by trading velocities for momenta. (book)
Liouville's theorem
The volume of a region of phase space stays constant as the system evolves under Hamilton's equations. It is the basis of statistical mechanics. (book)
Lorentz force
The force F = q(E + v × B) on a charge moving through electric and magnetic fields. (book)
Lorentz transformation
The coordinate change between inertial frames in special relativity. Maxwell's equations keep their form under it. (book)
M
Magnetic field
The field that exerts a force on moving charges perpendicular to their velocity. Currents and changing electric fields produce it. (book)
Maxwell's equations
The four equations for electric and magnetic fields: Gauss's law, Gauss's law for magnetism, Faraday's law and the Ampère-Maxwell law. With the Lorentz force law they make up classical electrodynamics. (book)
Measurement problem
The open question of how and why a measurement produces one definite outcome when the Schrödinger equation alone evolves states smoothly. Interpretations of quantum mechanics answer it differently. (book)
N
Newton's second law
F = dp/dt, or F = ma for constant mass. It is the equation of motion in the Newtonian formulation. (book)
Noether's theorem
Every continuous symmetry of the action gives a conserved quantity: time translation gives energy, space translation gives momentum, rotation gives angular momentum. (book)
O
Observable
A physical quantity that can be measured. In quantum mechanics it is a Hermitian operator, and the possible results are its eigenvalues. (book)
P
Path integral
Feynman's formulation of quantum mechanics: the amplitude to go between two points is a sum over all paths, each weighted by e^(iS/ħ). It is the standard tool for quantum field theory. (book)
Pauli exclusion principle
No two identical fermions can occupy the same quantum state. It explains the structure of the periodic table. (book)
Phase space
The space of all positions and momenta of a system. A point in it fixes the state completely, and Hamilton's equations move that point. (book)
Poisson bracket
The bracket {f, g} of two phase space functions. It gives their time evolution and becomes the commutator divided by iħ under quantization. (book)
Poynting vector
The vector S = E × B / μ₀. It gives the energy flow per unit area per unit time of an electromagnetic field. (book)
Probability amplitude
A complex number whose squared magnitude is a probability. Quantum mechanics adds amplitudes, not probabilities, which produces interference. (book)
Q
Quantum field
An operator-valued function of spacetime. Particles are its quanta, and every electron, for example, is an excitation of the same electron field. (book)
Quantum tunneling
A particle passing through a potential barrier higher than its energy, with a probability that falls exponentially with the barrier width. (book)
R
Renormalization
The procedure that absorbs the infinities of loop diagrams into a redefinition of masses and couplings, which then depend on the energy scale of the measurement. (book)
Running coupling
A coupling constant whose effective value changes with the energy scale. The electromagnetic coupling grows at higher energies; the strong coupling shrinks. (book)
S
S-matrix
The operator that maps incoming particle states to outgoing ones in a scattering process. Its matrix elements are the scattering amplitudes. (book)
Scattering amplitude
The probability amplitude for a set of incoming particles to turn into a set of outgoing particles. Its squared magnitude gives cross sections and decay rates. (book)
Schrödinger equation
The equation iħ ∂ψ/∂t = Hψ for the time evolution of a quantum state. Its time-independent form gives the allowed energies. (book)
Schrödinger picture
The description of quantum mechanics in which states change with time and operators stay fixed. (book)
Spin
Intrinsic angular momentum of a particle, not caused by motion through space. Electrons have spin 1/2, photons spin 1. (book)
Spinor
An object that changes sign under a full 360° rotation. Spinor fields describe spin-1/2 particles in the Dirac equation. (book)
Spontaneous symmetry breaking
A situation where the laws are symmetric but the ground state is not. In the Standard Model it gives mass to the W and Z bosons through the Higgs field. (book)
Standard Model
The quantum field theory of the known elementary particles and of the electromagnetic, weak and strong interactions. Gravity is not part of it. (book)
Superposition
A quantum state that is a sum of other states. The system does not hold one of the components until measured; measurement probabilities come from the coefficients. (book)
U
Uncertainty principle
The relation Δx Δp ≥ ħ/2: a state cannot have both a sharp position and a sharp momentum. It follows from the canonical commutation relation. (book)
V
Vacuum
The lowest-energy state of a quantum field theory, with no particles present. Fields still fluctuate in it. (book)
Variational principle
A law stated as the requirement that some quantity is stationary, such as the principle of least action in mechanics. (book)
Vector potential
The field A whose curl is the magnetic field. Together with the scalar potential it forms the electromagnetic four-potential. (book)
W
Wave function
The function ψ(x) giving the probability amplitude to find a particle at position x. Its squared magnitude is the probability density. (book)
Work
The energy transferred by a force along a path, the line integral of F · dx. The work-energy theorem equates it with the change in kinetic energy. (book)