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A morphism is a map between objects that preserves the structure under consideration. Its precise compatibility conditions depend on the category of objects.

A quiver is a directed multigraph, allowing loops and multiple arrows.

An algebra over a field $k$ is a vector space $A$ with a bilinear multiplication $A\times A\to A$. Bilinearity means $(ax+by)z=a(xz)+b(yz)$ and $z(ax+by)=a(zx)+b(zy)$ for $a,b\in k$. Associativity, commutativity, and a multiplicative identity are additional properties, not consequences of this definition.

The polynomial ring $k[X]$ and the matrix algebra $M_n(k)$ are associative unital examples; the latter is generally noncommutative. A Lie algebra is another kind of algebra over a field, with its bracket as multiplication, obeying antisymmetry and the Jacobi identity instead of associativity. Specifying which axioms are intended makes statements about all algebras precise.

A square matrix $A$ has row diagonal dominance when $|a_{ii}|\ge {\sum }_{j\ne i}|a_{ij}|$ in every row. Strict diagonal dominance requires a strict inequality in every row. Strictness guarantees nonsingularity, and a positive margin gives the inverse infinity-norm bound from diagonal dominance. Weak diagonal dominance alone does not: the matrix $\left(\begin{array}{cc}1& -1\\ -1& 1\end{array}\right)$ has diagonal dominance but has the nonzero kernel vector $(1,1{)}^T$.

A sphere of volume $V$ translating through unbounded inviscid incompressible fluid has fluid kinetic energyIts added mass is therefore one half of the displaced fluid mass.

In a commutative ring, an element $a$ is a non-zero-divisor when multiplication by $a$ is injective: $ab=0$ implies $b=0$. Equivalently, $ab=ac$ permits cancellation to $b=c$, since $a(b-c)=0$. In $\mathbb{Z}/6\mathbb{Z}$, the class of $2$ fails this condition because $2\cdot 3=0$, while the class of $5$ satisfies it because it is a unit in a ring.

For a noncommutative ring, multiplication on the left and on the right can have different null spaces. The implication $ab=0\Rightarrow b=0$ specifies the left-multiplication condition; the two-sided condition additionally requires $ba=0\Rightarrow b=0$. Stating the commutative setting or the side of multiplication prevents these distinct notions from being conflated.

In a chain $N_1\to N_2\to N_3$ with distinct decay constants, the Bateman equation expresses each population as a linear combination of the exponential functions $e^{-{\lambda }_{i}t}$. When two consecutive constants coalesce to $\lambda$, the repeated Laplace transform pole produces a term proportional to $t{e}^{-\lambda t}$.

A standing wave results from the superposition principle applied to equal-frequency, equal-amplitude waves travelling in opposite directions. Its spatial nodes of a standing wave remain fixed while its displacement oscillates in time.

For a pure standing wave, the two travelling waves also have equal amplitudes: their interference leaves a fixed mode shape, rather than a profile moving along the string. Choose the position origin and time origin so that their transverse displacements areHere $a>0$ is each constituent's amplitude, $k=2\pi /\lambda$ is the wavenumber, $\lambda$ is the wavelength, and $\omega =2\pi f$ is the angular frequency corresponding to frequency $f$. Their wave speed is $c=\omega /k$. The superposition principle and the sine addition formula giveThe spatial factor stays put; the temporal factor changes its size and sign. Other constant wave phases shift the position or time origin. Unequal constituent amplitudes generally leave moving displacement at the would-be nodes of a standing wave, rather than this exact factorization.^[ref]

For an original ideal-string example, set $L=1\mathrm{m}$, $c=20\mathrm{m}{\mathrm{s}}^{-1}$, and choose the third normal mode. Then $k=3\pi {\mathrm{m}}^{-1}$, $\omega =60\pi {\mathrm{s}}^{-1}$, and $f=30\mathrm{Hz}$ by fixed-end standing wave frequencies. With each constituent amplitude $a=5\mathrm{mm}$,Its nodes of a standing wave are at $x=0,1/3,2/3,1\mathrm{m}$ and its antinodes at $x=1/6,1/2,5/6\mathrm{m}$. These are constructed model parameters, not experimental readings.

At $t=1/120\mathrm{s}$, every point has zero displacement, but most have nonzero transverse velocity: the string is crossing its straight configuration. The black dots alone mark permanent nodes of a standing wave. At an antinode the greatest transverse speed is $2a\omega =0.6\pi \mathrm{m}{\mathrm{s}}^{-1}$, much smaller than the chosen $20\mathrm{m}{\mathrm{s}}^{-1}$ wave speed. One is motion of string material; the other is propagation of the constituent profiles.

The beauty of a standing wave is that motion reveals geometry: fixed nodes of a standing wave turn a moving disturbance into a spatial ruler. Their separation measures wavelength, and boundary conditions turn the continuous choice of wavenumber into a discrete set of normal modes. This is useful both for interpreting string vibrations and for understanding how a bounded wave equation selects its eigenfrequencies.

A local conservation law relates the change of a density to transport through the surrounding boundary. With density $q$ and flux $\mathbf{j}$, its source-free differential form is ${\partial }_{t}q+\nabla \cdot \mathbf{j}=0$.

Integrating over a fixed region $V$ and applying the divergence theorem gives $d({\int }_{V}qdV)/dt=-{\int }_{\partial V}\mathbf{j}\cdot \mathbf{n}dS$. The amount inside can therefore change without violating conservation: flux through the boundary accounts for the change. A closed boundary with zero net flux makes the integrated amount constant.

Conservation of mass and conservation of electric charge are examples with different densities and fluxes. A conservation law alone does not determine the flux; a constitutive relation such as Fick's first law supplies additional dynamics. Sources or sinks require an explicit term on the right-hand side rather than silently changing the source-free equation.

The vorticity equation describes how the curl $\mathbf{\omega }=\nabla \times \mathbf{u}$ of a fluid velocity evolves. For inviscid flow with mass density $\rho$ and conservative body forces, taking the curl of the momentum equation givesThe terms represent vortex stretching, change associated with local compression, and baroclinic vorticity generation. Misaligned pressure and mass density gradients can therefore create vorticity even when it initially vanishes.

For constant-density incompressible flow with constant kinematic viscosity $\nu$, the equation becomesIn a strictly two-dimensional velocity field independent of the third coordinate, the stretching term vanishes. Three-dimensional flow has an additional amplification mechanism; treating its vorticity as a passively transported scalar would miss it.

Vortex stretching changes the strength of vorticity when a fluid velocity varies along the direction of the local vorticity. For an inviscid, constant-density incompressible flow, the vorticity equation reduces to $D\mathbf{\omega }/Dt=(\mathbf{\omega }\cdot \nabla )\mathbf{u}$. Thus a flow that stretches a vortex in its axial direction can amplify its vorticity without an applied rotational body force.

An ideal thin vortex tube gives a geometric explanation. Incompressibility preserves the volume of a material segment, so stretching its length reduces its cross-sectional area. Conserved circulation then requires a larger mean axial vorticity. Compression reverses that effect. The Burgers vortex adds a competing mechanism: radial momentum diffusion balances the axial stretching and selects a finite core size.

The shallow water equations describe a free-surface fluid layer whose depth is small compared with its horizontal variation scale. The leading approximation is hydrostatic pressure, with a depth-averaged horizontal velocity $\mathbf{u}$. For a fixed bed elevation $b$, water depth $h$ and uniform gravitational acceleration $g$, an inviscid nonrotating model iswhere the gradient is horizontal. The first equation conserves water volume; the second relates acceleration to the free-surface slope. Additional physical effects require additional terms or a different approximation.

Linearizing about still water of uniform depth $h_0$ gives long-wave speed $\sqrt{g{h}_0}$. A tank experiment can therefore compare the travel times of small disturbances at different depths. The approximation predicts faster propagation in deeper water, but requires a wavelength large compared with the depth; it does not apply indiscriminately to every short ripple.

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.

A rotating reference frame uses spatial axes whose orientation changes relative to an inertial frame. Coordinates in those axes can simplify an apparatus rotating at a known angular velocity $\mathbf{\Omega }$, but the equations of motion acquire inertial terms because the basis vectors themselves change.

For a shared fixed origin and constant $\mathbf{\Omega }$, a particle at rotating-frame position $\mathbf{r}$ and velocity $\mathbf{u}$ satisfiesThe additional terms are the Coriolis force and centrifugal force. A time-dependent $\mathbf{\Omega }$ adds an Euler term $-m\dot{\mathbf{\Omega }}\times \mathbf{r}$; a translating origin can add another term. The rotating-axes theorem derives these contributions by differentiating the position twice, rather than treating them as new interactions with another body.^[ref]

Relativistic aberration is the change of an incoming ray's apparent direction between relatively moving inertial frames. It follows by applying a Lorentz transformation to a photon's four-momentum, so the effect changes angles even when the source and receiver are observing the same ray.

If a photon propagates at angle $\theta$ to the positive $x$ axis in frame $S$, and $S^{\prime }$ moves at velocity $v=\beta c$ along that axis, thenThese are propagation-direction angles; the sky position from which the light appears to arrive points in the opposite direction. That distinction avoids sign errors when applying the formula to stellar observations. The annual stellar aberration amplitude uses Earth's changing orbital velocity, whereas stellar parallax uses a changing viewing position and depends on the source's distance.

Quantum transmission is the passage of an incident quantum particle into an outgoing region through a specified interaction or potential. A wave function can have both reflected and transmitted components; the measurable transmission probability compares outgoing and incoming probability currents, rather than comparing wave amplitudes alone.

For a one-dimensional, constant-mass Schrödinger equation, write an incident amplitude of one and transmitted amplitude $t$, with positive asymptotic wavenumbers $k_L$ and $k_R$. The probability current givesOnly when the asymptotic wavenumbers agree does this become $|t{|}^2$. A real, static potential with no additional loss channels has reflection plus transmission probability equal to one. Quantum tunnelling is an important case in which the transmitted component crosses a region forbidden by the corresponding classical energy condition.

Proper time is the time measured by a clock following a timelike worldline. With metric signature $(-,+,+,+)$ and units $c=1$,along that worldline.

Restoring the speed of light, the same metric signature gives $d\tau =\sqrt{-d{s}^2}/c=dt\sqrt{1-|\mathbf{v}{|}^2/c^2}$ for a future-directed timelike curve. Integrating along a clock's world line gives its elapsed proper time, even when its velocity varies. For uniform motion this becomes the time dilation formula; it is not generally the square root of the interval between endpoints of a curved world line.

The Planck mass combines the reduced Planck constant, speed of light and gravitational constant into a mass scale, $m_P=\sqrt{\hslash c/G}$. Its associated Compton wavelength, using the reduced convention $\hslash /(m_{P}c)$, equals the Planck length ${\ell }_P=\sqrt{\hslash G/c^3}$. It therefore identifies a scale at which these particular quantum and gravitational length estimates coincide.

This dimensional comparison motivates the use of Planck units in quantum gravity; it is not an experimental observation of a particle with that mass. The reduced Planck mass instead uses an additional factor $1/\sqrt{8\pi }$, so a gravitational equation written with either convention must state which one is meant.

A physical pendulum is an extended rigid body that swings under gravity about a fixed horizontal axis. Let $M$ be its mass, $d$ the perpendicular distance from the axis to its centre of mass, and $I$ its moment of inertia about that axis. Neglecting friction and other applied torques,for small amplitude about the stable hanging orientation. The distribution of mass matters through $I$, unlike the ideal simple pendulum where all the mass is concentrated at one point.

A laboratory comparison can suspend a rod at different offsets from its centre of mass, release it at the same small amplitude, and measure several oscillation periods at each offset. The parallel axis theorem predicts $I=I_{\mathrm{CM}}+M{d}^2$: moving the pivot changes both the restoring torque and the rotational inertia, so the period need not decrease monotonically with $d$.^[ref]