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The frequencies ${\omega }_n(\mathbf{k})$ of electromagnetic Bloch modes form allowed bands in a periodic dielectric. Their gaps and curvature determine forbidden propagation, confinement and group velocity.

A monochromatic internal gravity wave in uniform stratification has a spatially linear phase. In the nonrotating inviscid Boussinesq approximation, its dispersion relation is ${\omega }^2=N^2|{\mathbf{k}}_h{|}^2/(|{\mathbf{k}}_h{|}^2+m^2)$. The wave vector is normal to its phase planes, while the group velocity lies along them.

A radio wave traversing a dilute cold plasma has a group velocity that depends on its frequency. In the ideal unmagnetized limit, the dispersion relation is ${\omega }^2=c^2{k}^2+{\omega }_p^2$, where ${\omega }_p$ is the angular plasma frequency. For $\omega \gg {\omega }_p$, differentiate to obtain $v_g=c\sqrt{1-{\omega }_p^2/{\omega }^2}$, then expand $1/v_g$ to first order. The excess travel time over a vacuum path isHere $f=\omega /(2\pi )$ is ordinary frequency, $e$ is the magnitude of the elementary charge, $m_e$ is the electron mass, ${ϵ}_0$ is the vacuum permittivity, $n_e$ is electron number density, and $c$ is the speed of light. This approximation neglects magnetic corrections, collisions and multipath scattering; its positive sign means lower frequencies arrive later.

For small quasi-geostrophic approximation waves on a locally uniform basic zonal flow with nonzero meridional potential-vorticity gradient $Q_y$, this signed quadratic wave activity obeys ${\mathcal{A}}_t+\nabla \cdot \mathbf{F}=\overline{q^{\prime }\mathcal{D}}/Q_y$, where $\mathbf{F}$ is the Eliassen–Palm flux and $\mathcal{D}$ is a PV source. Multiply the linear PV equation by $q^{\prime }/Q_y$ and use the Taylor identity for quasi-geostrophic flux to obtain it. In a locally uniform conservative wave packet, $\mathbf{F}=\mathcal{A}{\mathbf{c}}_g$, so the flux follows the group velocity, with its sign determined by $Q_y$.

For fixed vertical wavenumber $k_z$, the inertial-wave dispersion relation is ${\omega }^2=4{\Omega }^2{k}_z^2/(k_x^2+k_z^2)$. Its horizontal group velocity is the derivative displayed above, opposite in sign to $\omega k_x$. In a half-channel $x>0$, an outgoing propagating mode therefore requires $\omega k_x<0$. With time factor $e^{-i\omega t}$ and $q=|k_z|\sqrt{4{\Omega }^2/{\omega }^2-1}$, choose $e^{-i\mathrm{sgn}(\omega )qx}$. Above the inertial-frequency cutoff, the horizontal wavenumber is imaginary and the appropriate radiation condition instead selects the exponentially decaying solution. Energy transport, rather than the direction of phase travel alone, determines the outgoing sign.

Ray tracing follows the paths along which a slowly modulated wave packet propagates. For a local dispersion relation $\omega =\Omega (\mathbf{k};\mathbf{x},t)$, the ray velocity is the group velocity ${\nabla }_{\mathbf{k}}\Omega$.

The curvature tensor of a nondegenerate energy band, determining the linear relation between force and acceleration in semiclassical band dynamics. From group velocity $v_i={\hslash }^{-1}{\partial }_{k_i}E$ and $\hslash {\dot{k}}_j=F_j$, one obtains ${\dot{v}}_i=(M^{-1}{)}_{ij}{F}_j$. Near a parabolic minimum, $E-E_0={\hslash }^2\delta {\mathbf{k}}^T{M}^{-1}\delta \mathbf{k}/2$. The principal values are reciprocal curvature effective masses; a cyclotron experiment measures a direction-dependent combination rather than every tensor entry separately.

A stationary straight meridional wall preserves frequency and tangential wavenumber $l$. The two normal wavenumbers obeyIncidence and reflection are distinguished by the normal group velocity. In the leading quasi-geostrophic approximation, a nonzero $l$ and the impermeability condition give reflected height amplitude $A_r=-A_i$. The $l=0$ no-normal-flow condition is degenerate in that reduced model; exact ageostrophic transport generally changes the amplitude ratio.

A lower ray reaches $H$ after ${\tau }_1=H/[U_1\sin {\theta }_1\cos {\theta }_1]$, at $x_H=H\cot {\theta }_1$. Conserved horizontal wavenumber gives $\cos {\theta }_2=(U_2{N}_1/U_1{N}_2)\cos {\theta }_1$. The transmitted front is $(x_H,H)+(t-{\tau }_1){\mathbf{c}}_{g2}$ on propagating branches; reflection reverses the vertical group velocity.

The refractive index compares the phase velocity of light in a medium with the vacuum speed of light $c$. For a transparent, uniform medium with isotropy and a positive-index propagation branch,Here $\omega >0$ is the angular frequency, $k>0$ is the wavenumber, and $v_{\mathrm{ph}}=\omega /k$ is the speed of a surface of constant wave phase. The refractive index is a dimensionless quantity; it is a property of a specified material response at a specified frequency, rather than a universal number attached to the word “glass”. More general absorbing or negative-index media require the conventions discussed in refractive index and electromagnetic response. ^[ref]

For an original model example, choose $n=3/2$ at vacuum wavelength ${\lambda }_0=600\mathrm{nm}$. The phase velocity is $v_{\mathrm{ph}}=2c/3\simeq 1.999\times 10^{8}m/s$, the frequency is $f=c/{\lambda }_0\simeq 4.997\times 10^{14}\mathrm{Hz}$, and the medium wavelength is $400\mathrm{nm}$. These are stipulated model values, not measured properties of glass or water. A value at this single frequency does not determine a pulse's group velocity.

The usefulness of refractive index is that one measured response connects several different observations: Snell's law predicts refraction, optical path length predicts accumulated wave phase, and changes with frequency describe optical dispersion. Even at normal incidence, where a light ray does not bend, a changed refractive index changes wavelength and wave phase. The following comparison isolates that effect; the oblique geometry is developed in wavefront.

For a linear stationary interface, phase matching at a planar wave interface preserves frequency: the incident and transmitted electric fields must oscillate together at each boundary point. Their phase velocities can differ, so their wavelengths differ. Writing $\omega =2\pi f$, the definitions giveThis explains the compressed wavefront spacing in the figure without changing the time between crests at the interface. Moving interfaces or time-dependent material responses need not preserve frequency. ^[ref]

The notation $n({\lambda }_0)$ labels refractive index by vacuum wavelength; it does not mean the wavelength inside the material. Dependence on ${\lambda }_0$, equivalently on frequency, is optical dispersion. The field driving the material induces oscillating electric charges; their emitted electromagnetic fields combine with the applied field to change the resulting wave phase. This collective response motivates refractive index, rather than a picture of independently delayed little pieces of light. ^[ref]

To distinguish three speeds, track a wavefront for phase velocity, a narrow wave envelope for group velocity, and the earliest newly introduced disturbance for front velocity. For a transparent differentiable propagation branch,The denominator is the group refractive index. It describes envelope transit when loss and reshaping are small. It need not equal $n$, and neither phase velocity nor a reshaped pulse peak generally measures signal arrival. The front velocity article explains the separate relativistic causality constraint and why a detector threshold is not an exact causal-front measurement.

For a one-dimensional free particle with $m>0$ and $p>0$, take the positive total energy from the relativistic energy-momentum relation:A fixed plane wave quantum phase obeys $kx-\omega t=\text{constant}$. Its phase velocity is thereforeFor a narrow wave packet centered at $p_0$, expand its dispersion relation as $E(p)\simeq E(p_0)+E^{\prime }(p_0)(p-p_0)$. In the Fourier inversion integral building the wave packet, the envelope then depends on $x-E^{\prime }(p_0)t$. Thus its group velocity isThe superluminal phase velocity follows a repeating phase marker; it does not describe a localized signal or superluminal transport of energy. Higher derivatives of the dispersion relation produce wave packet spreading, so the translated-envelope argument is a narrow-band approximation. For $v=0.6c$, the Lorentz factor is $1.25$, $p=0.75mc$ and $E=1.25m{c}^2$: the group velocity is $0.6c$ while the phase velocity is $5c/3$.

The total-energy convention matters. Define $\psi =e^{im{c}^{2}t/\hslash }\Psi$ to remove the common rest energy from a wavefunction $\Psi$ with total-energy time dependence. ThenThis is exactly the free-particle momentum convention used by the Schrödinger equation here. Removing a common time-dependent global phase changes the reported phase velocity, while leaving the de Broglie wavelength, relative spatial quantum phases and group velocity unchanged. Adding $m{c}^2$ again to a total energy would count the rest energy twice.

A physicist and author of teaching texts on classical mechanics, electromagnetism, quantum mechanics and statistical mechanics. His treatment of dispersive waves gives a useful example of the texts’ approach: a wave packet is built by adding sinusoidal waves, then expanding its dispersion relation to distinguish group velocity from phase velocity. This explains why the envelope of a water-wave packet can travel at a different speed from its crests.^[ref]

Choose $k,m>0$, $K=(k^2+m^2{)}^{1/2}$ and the positive angular frequency $\omega =Nk/K$. Then the phase velocity is $Nk(k,m)/K^3$ and the group velocity is $N(m^2,-km)/K^3$. Their dot product vanishes and their sum is horizontal with magnitude $N/K$. Draw the group velocity from the tip of the phase velocity: the two form the perpendicular sides of a right triangle. The horizontal hypotenuse is their sum, not the horizontal phase velocity $\omega /k$.

A free top and rigid base impose transverse wavenumbers $(n+1/2)\pi /h$ on guided SH modes between rigid and free planes. Their phase velocity and group velocity are $c={\beta }^{\prime }/\sqrt{1-{\omega }_n^2/{\omega }^2}$ and $U={\beta }^{\prime }\sqrt{1-{\omega }_n^2/{\omega }^2}$ for $\omega >{\omega }_n$. A real Love wave replaces the rigid base by an evanescent elastic load; the approximation becomes reliable sufficiently above the corresponding strong-dispersion transition, not at the actual mode cutoff.

For a sinusoidal small-amplitude bed, constant current $U>0$ and buoyancy frequency $N$, put $m=\sqrt{N^2/U^2-k^2}>0$. The linear vertical velocity amplitude is $W(z)=ikU{\eta }_0\sin [m(H-z)]/\sin (mH)$. The upward-wave coefficient is $A_{+}=-kU{\eta }_0{e}^{-imH}/[2\sin (mH)]$. At $mH=n\pi$, the prescribed nonzero bed motion and homogeneous rigid-lid condition are incompatible with a bounded stationary inviscid solution. With angle $\theta$ between intrinsic group velocity and vertical, $\tan \theta =m/k$, giving $kH\tan \theta =n\pi$.

The Yanai wave is the $n=0$ branch of the equatorial shallow-water dispersion relation with positive frequency:Its group velocity is $d\omega /dk=c{\omega }^2/({\omega }^2+\beta c)>0$. At low frequency it has westward phase velocity but transports energy eastward; at high frequency its phase and group velocities approach $c$.

For constant buoyancy frequency $N$, the log-pressure quasi-geostrophic potential vorticity equation has modes $\psi =e^{z/(2H)}{e}^{i(kx+mz-\omega t)}$ withThe prefactor compensates the density decrease in the energy weight. For $\beta ,k>0$, vertical propagation occurs for $-\beta k/[k^2+f_0^2/(4N^2{H}^2)]<\omega <0$. An outgoing radiation condition selects the root with positive vertical group velocity; bounded weighted amplitude alone does not select between two propagating roots.

A spatially localized forcing, such as persistent tropical heating, can excite Rossby waves that propagate wave activity into distant regions. Mean winds Doppler-shift the waves, and background potential vorticity gradients and jets guide or refract the packets. A stationary pattern can thus connect remote ridges and troughs even though the underlying wave energy propagates. Group velocity, rather than phase velocity alone, diagnoses packet propagation; damping, seasonality and eddy feedbacks affect the response.

The Schrödinger equation governs the evolution of a quantum state between instances of measurement in quantum mechanics. For a particle of mass $m>0$ moving in a prescribed real potential energy $V(\mathbf{r},t)$, its position wavefunction satisfiesHere $h$ is the Planck constant, ${\nabla }^2$ is the Laplace operator, and the Hamiltonian operator combines kinetic energy with potential energy. Specify an initial wavefunction, an operator domain and any boundary conditions: the differential expression alone does not completely specify a physical model. The Born rule then turns the evolving probability amplitude into position probabilities through $\rho =|\psi {|}^2$ and integration over a region.

This scalar Schrödinger operator describes nonrelativistic quantum mechanics, with relevant momenta much smaller than $mc$, where $c$ is the speed of light. It omits the constant rest energy $m{c}^2$, assumes fixed particle number, and suppresses active spin dynamics and coupling to a magnetic vector potential. A free electron with a fixed, uncoupled spin state can still use this scalar model; the Pauli equation supplies the additional spin components and magnetic coupling when needed. A prescribed real $V$ describes a closed evolution model or an approximation in which the source of that potential energy is treated as external. Dissipation, particle creation and a general measurement in quantum mechanics need further modelling.

For a time-independent self-adjoint Hamiltonian operator, the spectral theorem defines unitary time evolution byInitial states in the operator domain satisfy the differential equation in the Hilbert space sense; the unitary operator also evolves all other states in that Hilbert space. Time-dependent Hamiltonian operators need suitable regularity and domain hypotheses; their evolution is not generally the exponential of $-itH(t)/\hslash$.

The local reason for conservation is especially tangible. Multiply the Schrödinger equation by the complex conjugate ${\psi }^{\ast }$, subtract its conjugate multiplied by $\psi$, and use the fact that the real potential energy terms cancel. For smooth wavefunctions this gives the probability continuity equationThe divergence theorem identifies the change in probability inside a region with minus the outward probability current. Decay at infinity, or appropriate probability-conserving boundary conditions, preserves wavefunction normalization. Conservation of total probability does not require a stationary probability density.

Its usefulness comes from being a linear partial differential equation: quantum superpositions of solutions are solutions, while different quantum phases can produce interference. For a fixed Hamiltonian operator, the time-independent Schrödinger equation finds the energy eigenstates; each then acquires just its own time-dependent quantum phase. One eigenvalue problem therefore organizes both allowed energies and the evolution of arbitrary quantum superpositions, with a continuous spectrum included when required. This is the attractive bridge between a spatial wave problem and a clock: energy differences determine the rate at which relative quantum phases change. It makes atomic spectral lines, quantum tunnelling and quantum scattering calculable from a specified model.

For an explicit example without confinement, take a free particle on the whole line, $V=0$, initially in a Gaussian wave packet with position standard deviation ${\sigma }_0$ and mean momentum $\hslash k_0$:Unlike a single plane wave, this wavefunction satisfies wavefunction normalization. A Fourier transform writes it as a quantum superposition of plane waves. Their dispersion relation is $\omega (k)=\hslash k^2/(2m)$, so each Fourier component gains the factor $e^{-i\hslash k^{2}t/(2m)}$. Completing the square in the resulting Gaussian integral giveswith the square root chosen continuously from $t=0$. Its position probability density and standard deviation areThe centre moves with the group velocity ${\omega }^{\prime }(k_0)=v_0$ and the wave packet spreads because different wavenumbers have different group velocities. This is unitary wave dispersion, without a random force or loss of probability. Independently, the initial momentum standard deviation is $\Delta p=\hslash /(2{\sigma }_0)$ and the initial position-momentum covariance vanishes. Free evolution gives $x(t)=x(0)+p(0)t/m$ in the Heisenberg picture, henceThis independently checks the spreading law. The Gaussian integral of $\rho$ is one at every time; the decreasing peak is exactly offset by increasing width.

As hypothetical preparation parameters, choose a free electron with ${\sigma }_0=1\mathrm{nm}$ and $k_0=2{\mathrm{nm}}^{-1}$. Using $\hslash \simeq 1.054571817\times 10^{-34}\mathrm{J}\mathrm{s}$ and $m\simeq 9.1093837\times 10^{-31}\mathrm{kg}$ givesThe rounded electron mass uses the 2022 adjustment.^[ref] At $t=t_s$, the centre is $4\mathrm{nm}$ and ${\sigma }_t=\sqrt{2}\mathrm{nm}$; at $t=2t_s$, they are $8\mathrm{nm}$ and $\sqrt{5}\mathrm{nm}$. The mean kinetic energy staysIt includes both mean motion and the momentum spread. The mean speed is about $7.72\times 10^{-4}c$ and the momentum spread corresponds to $1.93\times 10^{-4}c$, making the nonrelativistic model accurate for the appreciably occupied part of this ideal Gaussian distribution. Its mathematically unbounded tails are an idealization.

The complex width also explains where the probability goes. Write $\psi =\sqrt{\rho }{e}^{i\theta }$, where $\theta$ is the dimensionless quantum phase. Differentiating the exact solution givesRelative to the moving centre, the probability current $j_{\mathrm{rel}}=j-v_0\rho$ points outward on both sides for $t>0$. This is a flux statement, not an additional assumption about individual particle trajectories. At $t=t_s$ and $x-v_{0}t=\pm {\sigma }_0$, the relative flow speeds $j_{\mathrm{rel}}/\rho$ are $\pm {\sigma }_0/(2t_s)$, about $\pm 2.8942\times 10^4\mathrm{m}{\mathrm{s}}^{-1}$. The probability within a moving interval of half-width ${\sigma }_0$ falls from about $0.6827$ initially to $0.5205$ at $t_s$, exactly because of this outward probability current.

Historically, the 1926 Quantisierung als Eigenwertproblem (Erste Mitteilung) by Erwin Schrödinger began with the stationary hydrogen atom problem. It distinguishes discrete negative energies from a positive continuous spectrum; it is not yet the general evolution equation displayed above.^[ref] The opening of quantum mechanics discusses its original vibrating-string analogy.

In the 12 December 1933 lecture The fundamental idea of wave mechanics by Erwin Schrödinger, Erwin Schrödinger explains the move from independent rays to waves through diffraction, then applies that analogy to atomic physics. His closing pages explicitly acknowledge the difficulty of reconciling wave language with observed particle tracks. This retrospective physical argument motivates the model; it does not derive the modern Born rule from classical mechanics.^[ref]

The dates separate proposal from measurement: the de Broglie wavelength belongs to the 1924 thesis, the Schrödinger equation to 1926, and the Davisson–Germer experiment's electron diffraction beams to 1927.^[ref] That linked experiment describes the prepared nickel crystal, evacuated electron gun, angular collector scans, retarding-potential selection and heating controls. Its measured collector electric current tests wave-dependent scattering probabilities; it neither measures a wavefunction directly nor establishes this equation uniquely.