Quantum mechanics

Quantum mechanics models states by wavefunctions and observables by operators.

A quantum coordinate on a circle with kinetic Hamiltonian ${\ell }^2/(2I)$. Periodic wavefunctions have angular momentum $\hslash n$ and energy ${\hslash }^2{n}^2/(2I)$ for integer $n$. Its thermal trace has both a momentum-eigenstate sum and a topological winding-path representation. Twisted boundary conditions or flux would modify this spectrum and are separate models.

A chain with local rotor inertia and a phase-locking interaction $-J\cos ({\varphi }_{n+1}-{\varphi }_n)$. Its smooth-phase limit is an elastic phase field with inertia $I$ and stiffness $J$. Compact winding and phase-slip processes remain part of the full model even when absent from the Gaussian smooth-sector approximation.

Expanding the cosine gives a wave equation $I{\varphi }_{tt}=J{\varphi }_{xx}$ and a gapless phase mode with physical frequency $\sqrt{J/I}|q|$. The exact harmonic lattice frequency is $2\sqrt{J/I}|\sin (q/2)|$. It resembles superfluid sound. In one dimension, Gaussian phase fluctuations and possible compact phase slips require care before inferring true long-range order from this branch.

When imaginary time is divided by $\hslash$ and ranges from zero to $\beta$, integrating rotor momenta gives $I{\dot{\varphi }}^2/(2{\hslash }^2)$ but leaves the potential stiffness term $J({\partial }_x\varphi {)}^2/2$. Factoring $1/{\hslash }^2$ out of both terms requires spatial coefficient ${\hslash }^{2}J$ inside the bracket. Setting $\hslash =1$ hides this distinction, but restoring units must preserve it.

The thermal rotor path integral traces paths closed modulo $2\pi$. Lift them to the line and sum endpoints differing by $2\pi m$. In inverse-energy time $0<\tau <\beta$, the kinetic exponent is $I\int {\dot{\varphi }}^{2}d\tau /(2{\hslash }^2)$. Integer $m$ counts trajectory windings, not angular-momentum eigenstates.

Poisson resummation relates the momentum sum ${\sum }_n{e}^{-\beta {\hslash }^2{n}^2/(2I)}$ to the winding sum $\sqrt{2\pi I/(\beta {\hslash }^2)}{\sum }_m{e}^{-2{\pi }^{2}I{m}^2/(\beta {\hslash }^2)}$. Gaussian Fourier transformation proves equality, including the prefactor. The momentum form converges rapidly at low temperature and the winding form at high temperature.

Writing $\varphi ={\varphi }_0+2\pi m\tau /\beta +\eta$ with endpoint-fixed $\eta$ separates winding action and Gaussian fluctuations. The latter give the free return kernel on the lifted line, $Z_0=\sqrt{I/(2\pi {\hslash }^2\beta )}$. The starting-angle integral supplies $2\pi$. Zero winding is not the same as the zero-momentum ground-state sector.

Semiclassical quantization uses classical trajectories and actions to approximate quantum states in a regime where relevant actions are large compared with $\hslash$. Phase consistency on a periodic orbit leads to Bohr-Sommerfeld quantization. Soliton collective coordinates can be quantized this way, while fluctuations supply additional corrections to the leading classical mass.

Single-valued semiclassical phase on a closed classical orbit quantizes its action, with Maslov index $\mu$ accounting for turning-point phase corrections. A cyclic angle has no oscillator turning points: its wavefunctions are $e^{i\ell q}$, with signed $\ell \in \mathbb{Z}$ for a $2\pi$ period and canonical momentum $\hslash \ell$. There is no half-integer shift for that free angular coordinate.

The operator formalism describes a quantum system using states in a Hilbert space and observables represented by self-adjoint operators with suitable domains on that Hilbert space. Evolution and correlation functions are computed from Hamiltonian operators and their time-ordered products. For free quantized fields, the oscillator construction gives a Fock space. The operator-path-integral equivalence relates these matrix elements to integrals over field histories.

A quasiparticle is a collective excitation that can be described approximately by particle-like quantum numbers and dynamics, even though it is not an isolated elementary particle. Excitons and polaritons are examples.

A Bogoliubov quasiparticle is an excitation created by the diagonal mode operator obtained through a Bogoliubov transformation. Its vacuum can contain particles in the original modes because creation and annihilation operators mix. Examples include antiferromagnetic spin waves, Bose-gas phonons and paired-fermion excitations.

A polariton is a hybrid light–matter excitation formed by strong coupling between a photon and a material excitation. For an exciton-polariton, the material component is an exciton.

An exciton is a bound state of an electron and an electron hole. Coupling this material excitation strongly to a photon can form a polariton.

An electron hole is the absence of an electron in an otherwise filled band, described as a mobile positive-charge excitation. It can form a bound state with an electron, giving an exciton.

When a Hamiltonian operator changes instantaneously by a finite potential step, the quantum state is continuous across the switching time. Its coefficients in the new energy eigenbasis are its overlaps with those eigenstates; the Born rule gives their squared moduli as probabilities. The state need not remain an energy eigenstate of the new Hamiltonian.

Canonical quantization promotes canonical variables to operators and their Poisson brackets to canonical commutation relations. Composite observables require an ordering prescription, and quantum anomalies can modify their algebra.

Weyl ordering symmetrizes position and momentum factors in a classical polynomial before promoting them to an operator. Its phase-space symbol corresponds to midpoint time slicing of a phase-space path integral. In particular, the Weyl symbol of $a^{†}a$ is $\bar{\psi }\psi -1/2$, whereas its normal ordering symbol is $\bar{\psi }\psi$.

The electromagnetic four-potential has a vanishing temporal momentum and a Gauss law constraint in gauge theory. Gauge reduction leaves two transverse canonical pairs. Their canonical commutation relations give the photon creation and annihilation operators. Covariant Gupta-Bleuler quantization instead retains auxiliary polarization modes and selects the same physical state space by a subsidiary condition and null-state quotient.

The free transverse photon field is a Hermitian sum of two transverse-polarization plane-wave modes with creation and annihilation operators. Their standard bosonic commutator gives the transverse equal-time commutator. The normal-ordered Hamiltonian is a sum of positive oscillator energies.

The reduced canonical commutation relation for a transverse field uses a transverse delta function instead of an unconstrained componentwise delta. Its Fourier transform is the spatial transverse projector. It can be derived from the two-polarization oscillator expansion or from the Dirac bracket after gauge fixing.

The Maxwell Lagrangian density contains no time derivative of $A_0$, so its canonical momentum vanishes. The condition is a first-class constraint. Its time preservation yields the Gauss law constraint in gauge theory; $A_0$ acts as the corresponding multiplier in the Hamiltonian.

Nonrelativistic quantum mechanics describes systems at speeds small compared with the speed of light, usually with time as an external parameter and evolution governed by the Schrodinger equation.

A Berry phase is the geometric phase acquired by a quantum state transported around a closed path in parameter space.

The classical limit is a regime in which quantum corrections or occupation-number discreteness become negligible and classical equations describe the relevant observables.

Parity is the eigenvalue $+1$ or $-1$ of a state under spatial reflection.

An interaction commuting with the parity operator preserves total parity. A two-body final state has the product of its particles' intrinsic parities times the orbital factor $(-1{)}^{\ell }$, giving the parity selection rule for a two-body decay.

The parity operator acts on a spatial wavefunction by $(P\psi )(x)=\psi (-x)$ and satisfies $P^2=I$. For a Hamiltonian operator with an even potential and parity-invariant domain, $HP=PH$. A nondegenerate energy eigenvalue has a one-dimensional eigenspace, so its eigenfunction is an eigenvector of $P$ with eigenvalue $+1$ or $-1$.

An orbital angular momentum eigenstate with quantum number $\ell$ changes by the factor $(-1{)}^{\ell }$ under spatial reflection. Equivalently,

The parity of a two-particle centre-of-mass state with intrinsic parities ${\eta }_1,{\eta }_2$ and relative orbital angular momentum $\ell$ is

If parity is conserved when a particle of intrinsic parity ${\eta }_i$ decays into two particles of intrinsic parities ${\eta }_1,{\eta }_2$, then their relative partial wave obeys

Spatial reflection reverses every spatial coordinate.

A pseudoscalar is invariant under proper rotations but changes sign under a spatial reflection. In three dimensions the scalar triple product is a pseudoscalar.

A photon is a quantum of the electromagnetic field with energy $E=h\nu =\hslash \omega$.

For a real on shell photon of null momentum $k$, a physical polarization satisfies $k\cdot ϵ=0$ and is defined modulo $ϵ\mapsto ϵ+\alpha k$. The two-dimensional quotient carries the two physical photon helicities. In a transverse gauge choose ${ϵ}^0=0$ and ${ϵ}^{\ast }\cdot ϵ=-1$. A Ward identity ensures that replacing an external polarization by its momentum gives zero in the summed physical amplitude.

Two first-class constraints in Maxwell theory remove four phase-space variables from the eight variables of a four-component potential and its momenta. The remaining four phase-space variables form two canonical pairs. In radiation gauge these are transverse polarization modes; circular combinations have helicity $+1$ and $-1$. Auxiliary covariant-gauge components do not add physical photon states.

Photon absorption removes a photon and transfers its energy and momentum to matter. It contributes a negative photon-number collision source. It differs from Compton scattering, which redistributes a photon's energy and direction while preserving the count. The integrated source can vanish if photon emission and absorption balance.

Photon emission creates a photon, for example in bremsstrahlung or the decay of an excited material state. It produces a positive photon-number collision source when not balanced by inverse processes. The photon number current is conserved only if its integrated creation and destruction rates cancel.

A free particle has no potential energy, so its Hamiltonian is $H={\mathbf{p}}^2/(2m)$. On unbounded Euclidean space its energy spectrum has no normalizable lowest-energy eigenstate.

A ground state is an energy eigenstate with the smallest possible energy. For noninteracting particles, the many-body ground state is obtained by filling one-particle levels subject to the particles' exchange statistics.

The ground-state subspace is the eigenspace of a Hamiltonian operator at its lowest eigenvalue $E_0$. A computational history state construction can have many valid quantum witnesses, hence a degenerate ground space. Perturbation bounds must optimize on this entire subspace, rather than on a chosen basis of ground states.

The ground-state degeneracy counts independent states minimizing the energy. For a classical continuous-symmetry system it can instead be described by a manifold of minimizing configurations, such as the sphere of orientations of a Néel state in an isotropic Heisenberg antiferromagnet.

The ground-state energy is the smallest energy eigenvalue of a Hamiltonian operator.

For every normalized trial state $\psi$ in the domain of a Hamiltonian operator $H$ that is bounded below,where $E_0$ is the ground-state energy. Equality holds precisely when the trial state lies in the ground-state eigenspace.

For $V=-A{e}^{-\mu r}/r$, the normalized exponential with scale $\alpha$ gives $E(\alpha )={\hslash }^2{\alpha }^2/(2m)-4A{\alpha }^3/(\mu +2\alpha {)}^2$. Choosing $\alpha =\mu /2$ proves binding if $\mu <Am/{\hslash }^2$. Failure to obtain a negative trial expectation outside this range does not prove absence of a bound state. At equality a small positive admixture of the scale-$\mu$ exponential has a negative cross matrix element and proves binding too.

For a bound stationary state in a homogeneous potential $V(\lambda x)={\lambda }^{n}V(x)$, scaling a normalized wavefunction gives stationarity of ${\lambda }^2⟨T⟩+{\lambda }^{-n}⟨V⟩$ at $\lambda =1$. Hence $2⟨T⟩=n⟨V⟩$.

An energy eigenvalue $E$ is an eigenvalue of the Hamiltonian: $H|\psi ⟩=E|\psi ⟩$ for some nonzero state $|\psi ⟩$.

An energy eigenstate is a nonzero quantum state represented by an eigenvector of the Hamiltonian operator. Its time evolution changes only its overall phase.

The energy eigenspace for $E$ is $\ker (H-EI)$.

An excited state is an energy eigenstate whose energy exceeds the ground state energy.

The first excited state has the smallest energy strictly above the ground-state energy.

An eigenstate of an observable is a quantum state represented by an eigenvector of that observable.

A momentum eigenstate is an eigenstate of the momentum operator. In one dimension it is a plane wave $e^{ikx}$ with momentum eigenvalue $\hslash k$.

A quantum system is modeled by a complex Hilbert space; its quantum states describe preparations, while observables describe measurable quantities.

A pure quantum state is a ray in a complex Hilbert space and may be represented by a normalized state vector.

A flavor eigenstate is a state with a definite flavor label in the specified interaction or flavor-charge basis. For neutral kaons, $K^0=d\bar{s}$ and ${\bar{K}}^0=s\bar{d}$ have definite opposite strangeness. Weak interactions mix them, so these flavor states are not the mass eigenstates. The precise flavor basis must be specified when discussing mixing.

A normalized quantum state is a linear combination of orthonormal basis vectors in a Hilbert space. Its density operator is $|\psi ⟩⟨\psi |$. When at least two basis amplitudes are nonzero, it has off-diagonal terms and differs from the mixed state ${\sum }_j|a_j{|}^2|j⟩⟨j|$. Quantum gates act linearly on this superposition and preserve its phase information.

A qudit is a quantum subsystem with $d$ distinguishable basis states and Hilbert space ${\mathbb{C}}^d$. A qubit is the special case $d=2$. Tensor products of qudits give a natural setting for local Hamiltonians and finite-dimensional quantum circuits.

For an $n$-level qudit and primitive root of unity $\omega$, these unitary operators obey $X^n=Z^n=I$ and $ZX=\omega XZ$. The inverse of $X^s{Z}^{-r}$ is $Z^r{X}^{-s}$, with the order reversed. These operators implement the receiver corrections in qudit teleportation.

A qubit is a two-dimensional quantum system. Its pure state is a ray in a two-dimensional complex Hilbert space and may be written $\alpha |0⟩+\beta |1⟩$ with $|\alpha {|}^2+|\beta {|}^2=1$.

A qutrit is a three-dimensional quantum system, with pure states $\alpha |0⟩+\beta |1⟩+\gamma |2⟩$ satisfying $|\alpha {|}^2+|\beta {|}^2+|\gamma {|}^2=1$. A qubit-qutrit state is a bipartite state on ${\mathbb{C}}^2\otimes {\mathbb{C}}^3$.

A quantum number is a discrete label, usually an observable's eigenvalue, used to distinguish quantum states.

Multiplying a quantum state by one global complex phase does not change any measurement probability; relative phases between components can affect interference.

A global phase multiplies an entire quantum state by one complex number of modulus one. It does not change any measurement probability, unlike a relative phase between components.

A quantum observable is represented by a self-adjoint operator; its possible measured values are spectral values of that operator.

For a normalized state with finite second moment of a quantum observable $Q$, its uncertainty is the standard deviation of the measurement outcome: $(\Delta Q{)}^2=⟨Q^2⟩-⟨Q{⟩}^2$. This equals $\Vert (Q-⟨Q⟩)\psi {\Vert }^2$. The Robertson uncertainty principle bounds products of uncertainties using the expectation of the commutator.

In a normalized state $|\psi ⟩$, the expectation of an observable $A$ is $⟨\psi |A|\psi ⟩$.

The Hamiltonian operator represents the total energy and generates time evolution through the Schrödinger equation.

For $Q=\hat{p}-isA(x)$ with real $A$ and positive $s$, the factorization gives potential $(s^2{A}^2-s\hslash A^{\prime })/(2m)$ and a nonnegative energy form. Its zero mode is proportional to $\exp [-s\int Adx/\hslash ]$. Taking $A=x$ and optimizing positivity over $s$ gives the Heisenberg uncertainty principle for states with finite second moments.

A unitary operator $U$ is a symmetry of a time-independent Hamiltonian $H$ whenequivalently $[U,H]=0$. A differentiable one-parameter symmetry $U(s)=e^{-isQ/\hslash }$ has a self-adjoint generator $Q$ satisfying $[Q,H]=0$, so $Q$ is conserved.

If $U(s)=e^{-isQ/\hslash }$ and $U(s{)}^{†}HU(s)=H$, differentiation at $s=0$ givesIn the Heisenberg picture,

A time-independent Hamiltonian evolves a state by the unitary operator $U(t)=e^{-iHt/\hslash }$.

In the Schrödinger picture, states evolve with the Hamiltonian operator while observables without explicit time dependence are fixed. For time-independent $H$, the corresponding Heisenberg picture operator is $O_H(t)=e^{iHt/\hslash }{O}_S{e}^{-iHt/\hslash }$.

The time-evolution operator maps an initial quantum state to its state at a later time. For a time-independent Hamiltonian operator $H$, it is the unitary operator $U(t)=e^{-iHt/\hslash }$.

A quantum-mechanical propagator is the position-space kernel $K(q_f,t_f;q_i,t_i)=⟨q_f|U(t_f,t_i)|q_i⟩$ of the time-evolution operator. It is the amplitude to evolve between the specified endpoint configurations.

For unit mass and $\hslash =1$, this prepoint kernel tends to the identity and generates the Time-dependent Schrodinger equation. The square-root branch follows the damped Fresnel integral. Its normalized moments are $1$, $0$, and $i\Delta t$ at orders zero, one and two; these fix both the normalization and the kinetic sign.

Expand a smooth wavefunction and potential in the normalized short-time Schrödinger kernel. Its odd moment vanishes and its second moment is $i\Delta t$, so one step changes the wavefunction by $i\Delta t{\psi }^{\prime \prime }/2-i\Delta tV\psi +O(\Delta t^2)$. Division by the time step gives the Time-dependent Schrodinger equation.

For a free particle of mass $m$ on the line,

A semiclassical propagator sums $e^{i{S}_{\mathrm{cl}}/\hslash }$ over classical paths joining its endpoints, weighted by their quadratic fluctuation determinants and phase indices. It is exact for quadratic actions.

The Van Vleck determinant is the determinant of mixed endpoint derivatives of a classical action. It measures how nearby classical trajectories focus and supplies the fluctuation prefactor of the semiclassical propagator.

In the Heisenberg picture, states are fixed and an observable evolves asFor a time-independent Schrödinger-picture observable,

For an operator $O_H=U^{†}{O}_{S}U$ in the Heisenberg picture, differentiating the evolution operators gives $d{O}_H/dt=i[H_H,O_H]+U^{†}(\partial O_S/\partial t)U$ in units $\hslash =1$. With no explicit operator time dependence, the second term vanishes. Applying this commutator identity to the canonical momentum and real scalar field gives their operator field equations.

For $H=H_0+H_{\mathrm{int}}$, the interaction picture moves the free evolution into operators and leaves states to evolve with $H_I(t)=e^{i{H}_{0}t}{H}_{\mathrm{int}}{e}^{-i{H}_{0}t}$. Its evolution operator is the time-ordered Dyson series.

A time-reversal operator $T$ intertwines forward and backward evolution:It must be antiunitary in ordinary quantum mechanics. Its conjugate linearity sends $i$ to $-i$, allowing it to commute with a time-reversal-invariant Hamiltonian rather than reversing the sign of its energy.

A quantum time-reversal operator is an antiunitary operator reversing spatial momentum and angular momentum. For a time-reversal-invariant Hamiltonian operator, its conjugation of $i$ makes it intertwine forward and backward unitary time evolution without reversing the energy spectrum. A scalar momentum eigenstate may transform as $\hat{T}|\mathbf{p}⟩=e^{i\chi (\mathbf{p})}|-\mathbf{p}⟩$. With unit-modulus phases and a complete normalized basis, expansion of arbitrary states proves antiunitarity.

If a unitary $T$ satisfied $U(t)T=TU(-t)$, differentiation at zero would giveIt would map every energy $E$ to $-E$. A Hamiltonian unbounded above would consequently be unbounded below and have no stable ground state.

For antiunitary $T$, conjugate linearity gives $T(i\psi )=-iT\psi$. Differentiatingtherefore yields $HT=TH$. Time reversal maps an energy eigenspace to itself and creates no negative-energy instability.

A bound state is a normalizable energy eigenstate spatially confined by a potential.

A Coulomb bound state is a physical normalizable eigenfunction for an attractive potential $V(r)=-\gamma /r$, $\gamma >0$. For the usual regular boundary condition at the origin its energies are the displayed values, $N=1,2,\dots$. The spherically symmetric states have orbital angular momentum zero; series-termination quantization of the Coulomb radial equation constructs them as a polynomial times a decaying exponential. Square integrability on the punctured radial interval alone does not replace the physical condition at the origin.

A stationary state is an energy eigenstate whose time dependence is only an overall phase.

A quantum particle constrained to a circle has integer angular-momentum eigenvalues and periodic position eigenfunctions. Its propagator can be written either as a sum over angular momenta or as a sum over winding number sectors.

The reduced Planck constant is $\hslash =h/(2\pi )$ and sets the scale of quantum commutators and phases.

A wavefunction is a complex amplitude whose squared modulus gives a probability density.

Wavefunction normalization makes the total Born rule probability equal to one. In an orthonormal basis of energy eigenstates, $\psi ={\sum }_n{c}_n{\varphi }_n$ is normalized precisely when ${\sum }_n|c_n{|}^2=1$.

A matter wave is the quantum wave associated with a material particle. Its phase can interfere after coherent paths recombine.

Matter-wave interference occurs when coherent spatial branches of a particle's wavefunction recombine and their relative quantum phase changes the detection probabilities.

The de Broglie wavelength of a particle with momentum magnitude $p$ is $\lambda =h/p$.

A neutron interferometer coherently splits and recombines neutron matter waves, allowing path-dependent phases to be measured through the output intensity.

The Colella–Overhauser–Werner experiment observed the relative phase acquired by neutron paths at different heights in the Earth's gravitational field.