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For a divalent square Bravais lattice, let $E_X={\hslash }^2{\pi }^2/(2m_e{a}^2)$ and real periodic potential coefficients $V_{10}<V_{11}<0$. The leading two-wave edge levels are $E_X\pm V_{10}$. The four-wave corner levels are $2E_X+V_{11}\pm 2V_{10}$ and a doublet at $2E_X-V_{11}$. Comparing the lowest corner level with the upper edge level gives the displayed indirect band gap. The first spin-degenerate energy band can hold the two electrons per primitive unit cell; it becomes isolated when $-E_X-V_{11}-3V_{10}>0$ in this truncation. This follows from resonant degenerate perturbation theory, and higher plane waves can shift the threshold.

Boosting a static spatially periodic magnetic field produces temporal oscillations together with an electric field. Its spatial-grating four-wavevector remains spacelike, rather than the null four-wavevector of a vacuum plane wave. For a boost along the grating axis, $|E^{\prime }|/|B^{\prime }|=v<c$. Ultrarelativistic motion can make the field approximately wave-like along a prescribed trajectory without making it an exact source-free radiation field.

For a time-independent Hamiltonian operator $H$, an energy eigenstate satisfiesFor a one-dimensional particle of mass $m$ in a real potential energy $U(x)$,Its full time-dependent wavefunction is $\Psi (x,t)=\psi (x)e^{-iEt/\hslash }$: substitution in the Schrödinger equation checks both the minus sign and the factor $\hslash$. Thus an energy eigenstate has stationary position probability density, even though its quantum phase evolves. The boundary conditions and operator domain are part of this eigenvalue problem, selecting physically allowed solutions; the infinite square well and Radial Schrodinger equation for the hydrogen atom give complementary confined examples.

For a complete discrete orthonormal basis of energy eigenstates, linearity gives $\Psi (x,t)={\sum }_n{c}_n{\psi }_n(x)e^{-i{E}_{n}t/\hslash }$, with ${\sum }_n|c_n{|}^2=1$. Cross terms in $|\Psi {|}^2$ oscillate at angular frequencies $(E_n-E_m)/\hslash$. A continuous spectrum requires the corresponding spectral integrals; for a free particle on the whole line, individual plane waves are generalized eigenfunctions, and a normalized Gaussian wave packet is their integral quantum superposition. Adding a spatially constant real $C$ to $U$ shifts every energy eigenvalue by $C$ and multiplies the full state by $e^{-iCt/\hslash }$, a global phase; energy differences, position probabilities and probability current are unchanged.

For $V(x)=g\delta (x-x_0)$, put $q=\sqrt{2mE}/\hslash >0$ and $\gamma =mg/({\hslash }^{2}q)$. Relate the coefficients of $A{e}^{iqx}+B{e}^{-iqx}$ immediately to the right and left by $v_R=D(x_0)v_L$. Continuity of the wavefunction and integration of the Schrödinger equation across the delta potential give ${\psi }^{\prime }(x_0^{+})-{\psi }^{\prime }(x_0^{-})=2mg\psi (x_0)/{\hslash }^2$. Solving these two matching equations givesFor scatterers at $x_1<\cdots <x_N$, the full transfer matrix is $M=D(x_N)\cdots D(x_1)$: the leftmost scatterer acts first on the coefficient column. With these globally phased plane waves, the propagation phases are already in each $D(x_j)$. For incidence only from the left, $t=1/M_{22}$ and $r=-M_{21}/M_{22}$ follow from $(t,0{)}^T=M(1,r{)}^T$ and $\det M=1$.

For a uniform on-axis plane wave reference, choose its constant wave phase to be zero, so $R=A_R>0$. The reconstructed cross terms are then proportional to $O$ and its complex conjugate $O^{\ast }$. This pair creates the twin image in holography. An outgoing curved wavefront and its conjugate have opposite curvature: under forward propagation, one can appear to diverge from a virtual image while the other converges to a real image. In in-line holography their depths differ, but a defocused member still overlaps the other. This is the same conjugation of spatial wave phase used in phase conjugation, although a transmission hologram here launches both components forward rather than automatically creating a returning phase-conjugate mirror.

A twisted boundary condition identifies the ends of an interval up to a fixed unit-modulus complex number. For the kinetic Hamiltonian operator $-{\hslash }^2{\partial }_x^2/(2m)$, one also imposes ${\psi }^{\prime }(x+L)=e^{i\theta }{\psi }^{\prime }(x)$. In integration by parts, the endpoint terms then cancel, giving the usual self-adjoint ring realization. The allowed plane waves have momenta $\hslash (2\pi n+\theta )/L$.

A constant magnetic vector potential can be removed from $(p-a{)}^2/(2m)$ locally by $\tilde{\psi }=e^{-iax/\hslash }\psi$. If $\psi$ is periodic, $\tilde{\psi }$ has twist $\theta =-aL/\hslash$. Thus the Aharonov-Bohm effect survives a locally vanishing vector potential as a global condition on the wavefunction.

For a real central potential and the outgoing convention $e^{i\pi /4}f(\theta )e^{ikr}/\sqrt{kr}$, the Partial-wave S-matrix is $S_m=e^{2i{\delta }_m}$. The cosine form of the Jacobi-Anger expansion fixes $w_0=1$ and $w_m=2$ for $m\ge 1$. Comparing the outgoing part of each phase-shifted radial cosine with the incident plane wave gives the displayed scattering amplitude. Integrating $|f{|}^2/k$ gives the displayed scattering cross-section, since the constant angular mode has squared norm $2\pi$ and each positive-order cosine has squared norm $\pi$. Reversing the multiplicities would already fail to reproduce the free plane wave at zero scattering phase shift.

Put $C={\hslash }^2/(2m)$, $v=|V_G|$, $\mathbf{G}=(2K,0)$, and $k_x=K+q$. The upper eigenvalue of the two coupled plane waves isAt fixed Fermi energy $C{k}_F^2$, its closed Fermi contour hasThus $A(v)=4{\int }_0^{q_m}y(q)dq$ can be inverted numerically to estimate the Bragg scattering coupling. The pocket disappears at $v=C(k_F^2-K^2)$. The estimate assumes a fixed chemical potential and neglects coupling to other plane waves; a measured pocket area alone does not determine all Fourier coefficients of an arbitrary lattice potential.

Wave absorption transfers energy from a propagating wave to other degrees of freedom of a medium. For a plane wave that is a monochromatic wave, with amplitude proportional to $e^{i(k^{\prime }+i{k}^{\prime \prime })z}$ and $k^{\prime \prime }>0$, its wave intensity decreases as $I(z)=I(0)e^{-2k^{\prime \prime }z}$. The intensity absorption coefficient is therefore $2k^{\prime \prime }$ when this loss is due to absorption. A decreasing transmitted wave intensity can also result from wave reflection or wave scattering, which redistribute energy rather than necessarily converting it into heat.

A wavefront connects points with the same wave phase at one specified time. It is a surface, rather than the trajectory of a particular piece of light. For a scalar monochromatic wave written as $u(\mathbf{r},t)=A(\mathbf{r})\cos \Phi (\mathbf{r},t)$, a wavefront is a connected level set of the phase function $\Phi$; different surfaces can share the same wave phase modulo $2\pi$. A plane wave has planar wavefronts. An outward wave from an ideal point source in a uniform medium has spheres as wavefronts, while an optical lens can turn nearly planar wavefronts into converging ones. The wavefront describes wave phase, not wave amplitude: drawing it does not say how bright the beam is. ^[ref]

The relation to light rays becomes precise in a stationary medium with isotropy, negligible loss and a real positive scalar refractive index $n(\mathbf{r},\omega )$. In the geometrical optics limit, spatial changes in the refractive index and wave amplitude are slow on the local wavelength scale. For a single smooth propagation branch, writeHere $\omega >0$ is the angular frequency, $c$ is the vacuum speed of light, $S$ has units of length, and the last relation is the optical eikonal equation. The local wavevector is $\mathbf{k}=\nabla \Phi =k_0\nabla S$. Its direction is normal to a smooth wavefront, since a gradient is normal to its level sets. A forward light ray has unit tangent vector $d\mathbf{r}/ds=\nabla S/n$, so it follows this normal under the stated assumptions. Along that light ray, $dS/ds=\nabla S\cdot (d\mathbf{r}/ds)=n$: the change in $S$ is the optical path length. This is the short geometric argument linking wavefronts, light rays and optical path length. ^[ref]

The diagram shows incident and transmitted branches separately, with the reflected branch omitted. Its stipulated refractive indices are $n_1=1$ and $n_2=3/2$, and its incident angle is $30^{\circ }$ to the interface normal. At a stationary interface, the frequency and tangential wavevector agree on the two sides: Snell's law gives $\sin {\theta }_2=1/3$, hence ${\theta }_2\approx 19.4712^{\circ }$. The wavefronts turn while their normal spacing decreases from ${\lambda }_0$ to $2{\lambda }_0/3$, where ${\lambda }_0$ is the vacuum wavelength. More explicitly, with $x$ normal to the interface and $y$ tangential, take $S_1=\sqrt{3}x/2+y/2$ for $x<0$ and $S_2=\sqrt{2}x+y/2$ for $x>0$. These functions agree at $x=0$; the drawn wavefronts are $S_j=m{\lambda }_0$ at $t=0$ with ${\varphi }_0=0$. This is an exact piecewise plane wave phase construction for the stipulated transparent media, not measured material data. ^[ref]

The normal-to-wavefront rule requires its assumptions. In a medium with anisotropy, the wavevector remains normal to a smooth wavefront, but the group velocity and energy flux need not follow it. ^[ref] Near a wave caustic, where multiple light rays meet, one smooth phase function may no longer describe the field; diffraction and the full wave amplitude matter. A wavefront is also not generally a surface of equal pulse arrival time. In a medium with optical dispersion, a monochromatic wave phase moves with the phase velocity, while a sufficiently narrow wave packet follows the group velocity. Even identical optical path lengths at one frequency can give different group delays. The usual equal-travel-time picture needs a common launch reference and a nondispersive ray model; an arbitrary broadband disturbance has no single monochromatic wavefront.^[ref]

The wavelength $\lambda$ is the spatial period of a monochromatic wave, measured at a fixed time along the wavevector. For a plane wave with a sinusoid as its profile, it is the distance between consecutive wave crests, or between any consecutive points with the same wave phase modulo $2\pi$:Here $\mathbf{k}$ uses angular wavenumber, $f$ is ordinary frequency, and $v_p$ is phase speed. A distance between arbitrary zero crossings need not be a wavelength: consecutive zeros of a sinusoid are only $\lambda /2$ apart, with opposite slopes. In a standing wave, consecutive nodes of a standing wave are also $\lambda /2$ apart; $\lambda$ refers to the constituent travelling waves, even though the nodes of a standing wave do not travel.

Across a stationary interface between stationary linear media, the incident, reflected and transmitted monochromatic waves have the same frequency. The boundary conditions must hold at every time: terms oscillating with different angular frequencies cannot cancel throughout the entire time record. Each medium's dispersion relation then determines its own angular wavenumber and wavelength, soFor electromagnetic radiation in a transparent medium with refractive index $n$, $v_p=c_0/n$ and $\lambda ={\lambda }_0/n$, where $c_0$ is the vacuum speed of light. A synthetic example with ${\lambda }_0=600\mathrm{nm}$ and $n=1.5$ gives $\lambda =400\mathrm{nm}$ at the same frequency. This conservation argument assumes time-independent linear material response and a stationary interface in the chosen frame. A moving interface can produce a Doppler effect; time-dependent or nonlinear response can change or generate frequencies.^[ref]

A broad frequency spectrum generally supplies many component wavelengths, rather than one wavelength for an entire pulse. Even two tones have both their individual wavelengths and a different overall repetition distance: the $2$- and $8$-cycle-per-metre sum in spatial sampling of a wave repeats after $0.50\mathrm{m}$, but contains the finer $0.125\mathrm{m}$ structure. A narrow wave packet can have an approximate carrier wavelength; the wave envelope propagates approximately at group velocity, determined by the slope of the dispersion relation. Substituting that group velocity into $\lambda =v_p/f$ generally gives the wrong carrier wavelength. Where the medium varies slowly, local wavelength gives a more useful description.

A wavenumber measures how rapidly a wave phase changes with distance. Two conventions coexist: angular wavenumber counts radians per unit length, while ordinary spatial frequency counts cycles per unit length. For a monochromatic wave with wavelength $\lambda >0$,Here $K\ge 0$ is the magnitude of the angular wavenumber. The symbol $k$ can instead denote a signed one-dimensional angular wavenumber: a factor $e^{ikx}$ has wavelength $2\pi /|k|$ for real $k\ne 0$. Both conventions have dimensions of inverse length; writing $\mathrm{rad}{\mathrm{m}}^{-1}$ or “cycles per metre” exposes the convention. A quoted value in ${\mathrm{m}}^{-1}$ alone does not resolve the factor of $2\pi$. Spectroscopic wavenumber usually uses the cycles-per-length convention.^[ref]

For a real monochromatic plane wave, fix the conventionThe wavevector $\mathbf{k}$ supplies both the spatial wave phase slope and the normal to the wavefronts; its dot product with position makes the wave phase constant on parallel planes. The angular frequency is $\omega =2\pi f$, with $f>0$ the ordinary frequency. Moving a distance $\lambda$ along $\mathbf{k}$ changes the wave phase by $2\pi$. Holding a wave crest's wave phase fixed gives its phase velocity $\omega \mathbf{k}/K^2$ and phase speedIn one dimension the signed phase velocity is $\omega /k$. This identifies a moving pattern; it does not identify the motion of the medium's particles or a wave packet's group velocity.

For the synthetic plane wave below, choose $\lambda =0.40\mathrm{m}$ and $f=5\mathrm{Hz}$. Then $K=5\pi \mathrm{rad}{\mathrm{m}}^{-1}$, ${\nu }_x=2.5$ cycles per metre, $\omega =10\pi \mathrm{rad}{\mathrm{s}}^{-1}$, and $v_p=2\mathrm{m}{\mathrm{s}}^{-1}$. The wave crests are $0.40\mathrm{m}$ apart at one instant; after a quarter oscillation period, each has advanced $0.10\mathrm{m}$. The lower panel retains whole turns of wave phase, making its slope visible instead of reducing it modulo $2\pi$.

Wave scattering at a planar interface separates an incident plane wave into reflected and transmitted plane waves. Translation invariance parallel to the interface conserves the tangential component of the wavevector.

The isotropic Weyl Hamiltonian acts on a two-component spinor as $H=v\mathbf{\sigma }\cdot \mathbf{p}$. For a plane wave of momentum $\hslash \mathbf{k}$, its two energy eigenvalues are $E_{\pm }=\pm \hslash |v||\mathbf{k}|$. For $v>0$ the corresponding Bloch vectors point along and opposite $\mathbf{k}$, respectively. If the Pauli matrices describe physical spin one-half, the mean spin is $\pm (\hslash /2)\hat{\mathbf{k}}$; in an effective two-band model the same algebra can describe an internal pseudospin instead. The bands meet linearly at the origin. ^[ref]

Here $m=\sqrt{1-p^2-q^2}$ has nonnegative imaginary part. This representation of the outgoing Helmholtz equation kernel separates propagating plane waves from evanescent waves. The absolute vertical separation is required for fields on either side of a source; choosing its sign fixes the upward or downward angular spectrum.

In the inertia-free, constant-density electron magnetohydrodynamics model with uniform guide field $B_0\hat{\mathbf{z}}$, a transverse magnetic plane wave obeys $\omega \mathbf{b}=i{v}_A{d}_i{k}_{\parallel }\mathbf{k}\times \mathbf{b}$. The helicity decomposition of a transverse Fourier mode gives the displayed dispersive branches with opposite circular polarizations. This electron-current mode is called a whistler wave; see Lyutikov's analysis of electron magnetohydrodynamics. Its pressure-free induction equation should not be identified with a general kinetic Alfvén model.

A concentric diffractive focusing element with successive zone boundaries chosen to differ by half a wavelength of optical path to a focus. Alternating transmitting and absorbing zones suppress cancellation between neighboring half-period zones. For an incident plane wave, the paraxial boundaries satisfy $r_j^2=j\lambda f$. The outermost zone width controls the finest transverse scale, and the focal length varies with wavelength.