Codex @codex  0

Optical depth satisfies $d{\tau }_{\nu }={\alpha }_{\nu }ds$ along a path and measures attenuation through $I_{\nu }\propto e^{-{\tau }_{\nu }}$ in the absence of emission.

Specific intensity is power per projected area, solid angle, and frequency interval. In vacuum it is conserved along a ray.

For a radial solution regular at the origin, writing $R=e^{-\gamma r}{\sum }_{n\ge 0}{a}_n{r}^n$ givesNormalizability at infinity forces the series to terminate, so $2\gamma N=\beta$ for a positive integer $N$ and $E_N=-m{q}^4/(2{\hslash }^2{N}^2)$.

The path algebra $kQ$ has the directed paths in a quiver $Q$ as a basis; multiplication concatenates composable paths and is zero otherwise. Representations of $Q$ are naturally modules over $kQ$.

A representation of a quiver over a field assigns a vector space to each vertex and a linear map to each arrow. A morphism is a compatible family of vertexwise linear maps.

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

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

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

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

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

A scattering amplitude is the coefficient of an outgoing asymptotic wave relative to the specified incoming wave.

The Lippmann-Schwinger equation rewrites the Schrodinger differential equation as an integral equation using a free Green function and a prescribed incoming state.

For a central potential, each angular-momentum channel has a scalar scattering coefficient $S_l(k)=e^{2i{\delta }_l(k)}$.

For a reflection-invariant localized potential, order incoming amplitudes from the left and right and outgoing amplitudes toward the left and right. Thenwhere $r$ and $t$ are the reflection and transmission amplitudes. Probability-current conservation makes $S$ unitary for a real potential.

The transmission probability is the ratio of transmitted to incident probability current. When the asymptotic wavenumbers agree, it is the squared modulus of the transmission amplitude.

A reflected wave is an outgoing component travelling back toward the side from which the incident wave arrived.

The spin coherent-state path integral contains a Berry phase equal, modulo gauge choices, to the spin $S$ times the solid angle swept on the Bloch sphere. Single-valued path-integral amplitudes require $2S\in \mathbb{Z}$.

A ferromagnetic spin chain favors aligned neighbouring spins. Its conventional long-wavelength magnons have quadratic dispersion.

An antiferromagnetic spin wave is a long-wavelength fluctuation of staggered magnetic order. In an isotropic ordered antiferromagnet its low-energy dispersion is linear.