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Let $|\psi ⟩,|\varphi ⟩$ be normalized independent quantum states spanning a two-dimensional space, and define a linear operator byWriting $a=⟨\psi |\varphi ⟩$, this operator is Hermitian exactly when $ga$ is real. To see this, the diagonal matrix elements are $ga$ and $(ga{)}^{\ast }$; their reality is necessary. The off-diagonal Hermiticity identity holds automatically, so reality is also sufficient on the spanning space.

For $g\ne 0$, set $r=ga/|g|\in \mathbb{R}$. Independence implies $|a|<1$, hence $|r|<1$. The normalized energy eigenstates areSubstitution verifies the eigenvalue equations. Expanding their inner products gives the displayed denominators and zero overlap between the two eigenstates. For $g=0$, $H$ is zero on this space; every orthonormal basis is an eigenbasis with eigenvalue zero.

This example shows why a coefficient matrix written in a nonorthogonal basis need not look Hermitian under an ordinary conjugate transpose. The physical inner product, including the basis overlaps, determines whether the represented linear operator is Hermitian.

A potential barrier is a region of higher potential energy separating accessible regions of a physical system. A classical particle cannot pass through a barrier higher than its total energy along its trajectory, but a quantum particle can have nonzero transmission probability through a finite barrier by quantum tunnelling.

In a one-dimensional region with constant $V_0>E$, the Schrödinger equation gives exponential solutions with tunnelling decay constant $\kappa =\sqrt{2m(V_0-E)}/\hslash$. The wavefunction can therefore penetrate a finite barrier and match an outgoing wave on the other side. The rectangular potential barrier is an exactly solvable model; smooth barriers lead naturally to a semiclassical approximation. Even for $E>V_0$, quantum reflection can occur where the potential changes.

The Planck constant $h$ connects the frequency $\nu$ of an electromagnetic field mode with a single photon's energy, $E=h\nu$. Its physical dimension is action, or energy multiplied by time. It sets the scale of quantum mechanics: the de Broglie wavelength of a particle is $\lambda =h/p$, and discrete energy levels often have spacings involving $h$.

In the International System of Units, $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$ exactly. Fixing this value defines the kilogram through already defined units of length and time; it does not remove experimental uncertainty from a practical mass measurement.^[ref] The reduced Planck constant $\hslash =h/(2\pi )$ is convenient when angular frequency replaces ordinary frequency.

A quantum state describes a preparation through its predicted measurement in quantum mechanics probabilities. In the Hilbert space formulation, a pure state is a quantum state ray: a normalized vector representative $|\psi ⟩$ satisfies $⟨\psi |\psi ⟩=1$, and representatives differing by a global phase describe the same quantum state. ^[ref]

Both pure states and mixed states are represented by a density operator on the chosen complex Hilbert space:Thus $\rho$ is Hermitian, positive semidefinite and of unit trace; in infinite dimension it must also be a trace-class operator. For a normalized pure state, $\rho =|\psi ⟩⟨\psi |$ is a rank-one orthogonal projection. A mixed state has no single such vector representative. Its density matrix is the matrix of $\rho$ in a chosen orthonormal basis. ^[ref] For a discrete POVM with effects $E_a\ge 0$ and ${\sum }_a{E}_a=I$, the Born rule readsThese constraints and Born rule probabilities describe arbitrary finite-dimensional preparations. ^[ref]

Randomly choosing normalized pure states with probabilities $w_j$ and forgetting the label givesThis quantum state ensemble averages probabilities; a quantum superposition adds probability amplitudes. Several ensembles can give the same density operator. Quantum entanglement can make a reduced density matrix mixed while the joint quantum state is pure. ^[ref]

The preparation, the choice of basis, and the measurement are separate ingredients. A wavefunction $\psi (x)=⟨x|\psi ⟩$ is the position representation of a pure state; a general mixed state requires a density operator. For a closed quantum system, a unitary operator evolves it as $\rho \mapsto U\rho U^{†}$, preserving its eigenvalues and purity of a density operator. ^[ref] The Bloch sphere and Bloch ball give the qubit geometry; quantum state tomography connects repeated preparations to an estimated density matrix; projective measurement treats the conditional state after an outcome.

An energy eigenvalue is an eigenvalue of the Hamiltonian operator $H$: there is a nonzero vector $\psi$ in its operator domain $D(H)$ withFor a self-adjoint operator, $E$ is real. Normalizing $\psi$ produces an energy eigenstate; an ideal energy measurement then returns $E$ with probability one. The associated energy eigenspace includes all vectors with this same energy eigenvalue, including the zero vector, which represents no quantum state. Different energy eigenvalues have orthogonal energy eigenspaces. ^[ref].

The energy eigenvalues form the point spectrum, which can be only part of the spectrum. The term discrete spectrum more specifically means isolated eigenvalues of finite multiplicity. The usual fixed-nucleus hydrogen atom model has discrete negative bound state energy eigenvalues together with a continuum at nonnegative energy. ^[ref]. A quantum system need not have a ladder of discrete energy eigenvalues: a whole-line free particle has continuous spectrum. A plane wave used for a free particle is a generalized eigenfunction of the Hamiltonian operator, rather than a normalized vector in the physical Hilbert space. ^[ref].

For the infinite square well $0<x<L$, the Dirichlet boundary conditions giveSubstitution into $H{\varphi }_n=E_n{\varphi }_n$ and wavefunction normalization verify these statements directly. Here $E_1$ is the ground-state energy, even though its index is one; the labeling convention does not determine which quantum state is lowest. Equal populations of ${\varphi }_1$ and ${\varphi }_2$ have energy expectation value $(E_1+E_2)/2=5E_1/2$, but an energy measurement returns $E_1$ or $4E_1$, each with probability $1/2$. An average of allowed energy eigenvalues need not itself be an energy eigenvalue.

A ground state is a normalized energy eigenstate that attains the lowest possible energy of a quantum system. Precisely, let the Hamiltonian operator $H$ be a self-adjoint operator bounded below on a Hilbert space, and write $E_{\ast }=\inf \sigma (H)$ for the infimum of its spectrum. A ground state is a unit vector ${\psi }_0$ in the operator domain $D(H)$ satisfyingThus a ground state exists exactly when the spectral bottom $E_{\ast }$ is an energy eigenvalue. The lowest energy eigenvalue among some listed bound states is insufficient if other parts of the spectrum lie below it. The ground-state subspace collects all minimizing energy eigenstates; it can have ground-state degeneracy. The quantum variational principle characterizes $E_{\ast }$ as the infimum of normalized energy expectation values. ^[ref].

For the one-dimensional quantum harmonic oscillator, the ground state has energy $\hslash \omega /2$ above the minimum of the potential energy. This zero-point energy is compatible with being the lowest allowed energy: a localized wavefunction cannot simultaneously have zero kinetic energy. For a free particle on the whole line, the spectral bottom is zero but is unattained by any normalizable wavefunction; see ground-state energy. A lower bound therefore need not describe a physically preparable energy eigenstate.

The ground state supplies a reference for excitation costs and for the quantum variational principle. A ground state is also a stationary state under a time-independent Hamiltonian operator: its quantum phase changes, while its position probability density stays fixed. A nonzero ground-state energy does not imply motion along a classical trajectory, and ground-state degeneracy means there need not be a unique lowest quantum state.

A photon is one quantum of the electromagnetic field. In vacuum it has zero rest mass, energy $E=h\nu =\hslash \omega$, and momentum magnitude $p=E/c$. Its two physical polarization degrees of freedom are transverse to its propagation direction. A monochromatic field mode can contain zero, one or several photons; its photon number distinguishes these quantum states.

The photoelectric effect illustrates a photon's discrete energy exchange: increasing incident frequency increases the available electron kinetic energy, whereas increasing light intensity primarily increases the rate of events in the weak-illumination regime. Compton scattering additionally tests the associated momentum transfer. Interference still depends on probability amplitudes, so describing light in terms of photons does not discard its wave behavior.

For a quantum state that is an eigenstate of the parity operator, parity is its eigenvalue $+1$ or $-1$. In a scalar position representation in quantum mechanics, the parity operator acts by $(P\psi )(\mathbf{r})=\psi (-\mathbf{r})$. Since $P^2=I$, these are its only possible eigenvalues: even wavefunctions have parity $+1$ and odd wavefunctions have parity $-1$.

If the Hamiltonian commutes with the parity operator, its energy eigenstates can be chosen to have definite parity. In the quantum harmonic oscillator, parity alternates as $(-1{)}^n$. This helps establish selection rules: the position operator is odd under spatial reflection, so its matrix element between two states of the same parity vanishes.

A path integral represents quantum evolution or correlation functions as an integral over histories. Dividing time into short steps, inserting resolutions of the identity and multiplying short-time kernels gives the familiar oscillatory weight $e^{iS/\hslash }$, together with a measure and boundary data.

The weight is a complex amplitude, not an ordinary probability assigned independently to each classical trajectory. Histories interfere, and stationary-phase configurations reproduce classical equations when the approximation is controlled. A Euclidean path integral uses imaginary time and a decaying action weight under suitable convergence conditions, helping study tunnelling and thermal states. A formal continuum expression must retain a definition of its discretization, operator ordering and normalization where they affect the result.

The Large-N expansion organizes an $O(N)$-symmetric theory in powers of $1/N$. At leading order, an auxiliary constraint field is determined by a self-consistent saddle-point or gap equation.

Finite-temperature quantum field theory computes field observables in thermal equilibrium, typically $\rho =e^{-\beta H}/Z$ with $\beta =1/(k_{B}T)$. The thermal trace has an imaginary time representation of length $\beta \hslash$ in physical time units, or $\beta$ after setting $\hslash =1$.

Bosonic fields are periodic around this thermal interval, while fermionic fields are antiperiodic. Their Fourier expansions produce different Matsubara frequencies, encoding the quantum statistics. Imaginary time correlations are useful for equilibrium calculations, while a real-time response or relaxation problem needs an appropriate continuation or real-time formalism. Temperature therefore changes the state and boundary prescription, rather than simply replacing every zero-temperature energy by a classical thermal value.

A gauge-theory soliton is a finite-energy localized classical field configuration stabilized by topology or a balance among differently scaling energy terms.

Quantum field theory in curved spacetime treats matter fields quantum mechanically on a prescribed classical spacetime geometry. The equations of motion and covariant derivatives depend on that geometry, while the metric is not itself quantized in this approximation.

A stationary background can supply a preferred positive-frequency splitting through a timelike symmetry. On a time-dependent background, different mode choices can define different particle notions, as described by vacuum ambiguity in a nonstationary spacetime. Quantum states and local correlation observables still require a consistent construction; the absence of one universal particle basis is not an absence of field dynamics. Renormalized stress-energy expectation values can be used in semiclassical backreaction, with additional assumptions beyond a fixed-background calculation.

Perturbative quantum field theory approximates an interacting quantum field theory by expanding around a solvable reference theory, usually a free field theory. Powers of interaction couplings organize the contributions to correlation functions and scattering amplitudes. Feynman diagrams encode the terms, their internal propagators and their combinatorial factors.

At leading order, tree diagrams often reveal the main process; loop diagrams give quantum corrections. Divergent intermediate expressions require a specified regularization and renormalization prescription before comparison with physical quantities. A small coupling is helpful but is not the only validity condition: large logarithms, thresholds or long evolution can invalidate a fixed-order approximation. Some phenomena, including certain bound states and tunnelling effects, require resummation or methods beyond an ordinary power series.

A quantized field replaces a classical field's mode amplitudes by operators acting on a Hilbert space. For a free real scalar field, the mode expansion combines positive-frequency waves multiplied by annihilation operators and their conjugates multiplied by creation operators. Each independent momentum mode behaves like a quantum harmonic oscillator.

The Fock vacuum contains no mode excitations, and creation operators construct multiparticle states in Fock space. Appropriate equal-time canonical commutation relations implement bosonic statistics; fermionic fields require canonical anticommutation relations. At a precise mathematical level a local quantum field is an operator-valued distribution, so smearing and definitions of composite operators matter. Quantization does not make a field a random ordinary numerical function; its noncommuting observables are essential.

A relativistic quantum field is a quantum field whose transformation and dynamics are compatible with special relativity. A scalar field, a Dirac field and a gauge field transform differently under Lorentz transformations, reflecting their different spin content.

Relativistic consistency also constrains locality. Microcausality requires local physical observables at spacelike separation to commute, so an operation in one region cannot immediately change measurement statistics in the other. Vacuum correlations across spacelike separation do not contradict that requirement. Field excitations can describe particles and antiparticles, allowing creation and annihilation processes when conserved quantities and available energy permit them. An interacting theory's observables therefore cannot generally be organized using one permanently fixed particle number.

Classical field theory describes physical quantities assigned throughout space and time, with field values that are ordinary numerical or geometric objects. A scalar field assigns a scalar to each point; the electromagnetic field assigns electric and magnetic components. Initial and boundary conditions together with field equations determine their evolution.

A local action $S=\int \mathcal{L}(\varphi ,\partial \varphi ,x)d^{d}x$ provides a systematic formulation. Varying the fields gives the Euler-Lagrange equations, while continuous symmetries lead through Noether theorem to conserved currents. Maxwell equations and the classical Klein-Gordon equation are examples. The field description makes finite-speed propagation and local energy transfer explicit. In quantum field theory, the corresponding fields become operator-valued objects and their correlations determine quantum predictions.

The Standard Model is the quantum field theory of quarks, leptons and their strong, weak and electromagnetic interactions, organized by the gauge group $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$. Quantum chromodynamics describes the color sector; the electroweak interaction combines the remaining gauge fields with the Higgs field.

Electroweak symmetry breaking leaves the electromagnetic $U(1{)}_Q$ unbroken. The W bosons and Z boson become massive, while the photon stays massless; Yukawa interactions generate charged-fermion masses. This structure relates scattering, decay and particle-production processes using a common set of couplings. Predictions require specifying parameters and the approximation used, not only listing the group.

The minimal renormalizable field content has massless neutrinos. Observed neutrino oscillations require masses and mixing, motivating extensions such as the Weinberg operator or additional neutrino fields. The Standard Model also does not supply a quantum description of gravity.

A discrete symmetry maps fields and states to new configurations through transformations not continuously connected to the identity within the stated symmetry group. Important examples are charge conjugation, parity symmetry in quantum field theory and time-reversal symmetry.

Testing such a symmetry requires transforming the fields, their arguments and all relevant interaction terms. A free kinetic term may be invariant even when an interaction is not. A weak charged current, for example, distinguishes left- and right-chiral matter and violates parity. Products such as CP symmetry can behave differently from their factors. The CPT theorem supplies a combined symmetry under its relativistic locality and quantum-theory hypotheses; it does not assert separate conservation of each factor.

Spontaneous symmetry breaking occurs when the action and equations have a symmetry but a chosen ground state is invariant only under a proper subgroup. Acting with the broken symmetry generates a degenerate vacuum manifold. An order parameter distinguishes the selected states: the equations do not prefer its direction, yet a particular state does. This lets one use the symmetry of the laws to predict excitations around a state that has less symmetry, notably the Goldstone bosons of a broken continuous symmetry.

A finite system with symmetry-preserving boundary conditions can have a symmetric ground state; a symmetric statistical mixture of broken states also has vanishing mean order parameter. A nonzero mean therefore requires a stated selection procedure. The symmetry-breaking order of limits takes the thermodynamic limit in a small applied source before removing that source. The finite-volume angular source selection example makes the distinction explicit for a continuous symmetry.