Mathematical physics

Compactification makes some spatial dimensions compact, so modes along them acquire discrete momenta. At energies below the first nonzero compact-mode mass, a dimensional reduction to the zero modes gives a lower-dimensional effective field theory. The compact volume fixes the gravitational kinetic coefficient in an unwarped product background.

For an unwarped product metric with fixed compact volume $V_n$, integrating the higher-dimensional Einstein-Hilbert action gives $M_{\mathrm{Pl}}^2=M_{\ast }^{n+2}{V}_n$ when the action coefficient is $M_D^{D-2}/2$. The reduced Planck mass convention must be distinguished from the unreduced one, and stabilizing the internal metric is a separate dynamical issue.

A Kaluza-Klein mode is a coefficient in a field expansion in eigenfunctions on compact extra dimensions. On a circle of circumference $L$, momentum is $2\pi \ell /L$, and a higher-dimensional massless field has lower-dimensional squared mass $(2\pi \ell /L{)}^2$. Nonzero modes give exponentially suppressed static-potential corrections at distances much larger than $L$.

On three noncompact spatial dimensions times a circle of circumference $L$, the method of images gives ${\sum }_{j\in \mathbb{Z}}[r^2+j^2{L}^2{]}^{-1}=\pi \coth (\pi r/L)/(Lr)$ at equal compact coordinates. This interpolates from $1/r^2$ at short distances to $\pi /(Lr)$ at long distances. A gravitational circle-radius scalar can change the long-range coefficient if it remains unstabilized.

Dimensional reduction retains fields independent of chosen internal coordinates. In a flat-torus zero-mode reduction, a metric supplies a metric, internal-component vectors and internal metric scalars; a massless p-form gauge field supplies lower-rank fields with internal indices. Fluxes, twists or projections can alter the spectrum and preserved supercharges.

A spacetime diagram plots one spatial coordinate against time. Curves on it represent trajectories, wavefronts, or characteristics, and their slopes encode propagation speeds.

A continuum limit replaces lattice variables that vary slowly over many lattice spacings by smooth fields. Lattice sums become spatial integrals and finite differences become derivatives, while short-distance information survives in effective coefficients and higher-derivative corrections.

Added mass is the effective inertia contributed by fluid accelerated together with an immersed body.

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.

A sphere of density ${\rho }_s$ falling in inviscid fluid of density $\rho$ has effective inertia $({\rho }_s+\rho /2)V$ and net weight $({\rho }_s-\rho )Vg$. Hence

The Compton wavelength of a particle of mass $m$ is ${\lambda }_C=h/(mc)$.

For a photon scattered through angle $\theta$ by an initially stationary electron,

For a charged Dirac particle, incoming momenta $p,k$ and outgoing momenta $p^{\prime },k^{\prime }$ obey $p+k=p^{\prime }+k^{\prime }$. The two QED diagrams have internal fermion momenta $p+k$ and $p-k^{\prime }$, giving $s$ and $u$ channels. The amplitude is proportional to $\bar{u}(p^{\prime })[/{ϵ}^{\prime \ast }R(p+k)/ϵ+/ϵR(p-k^{\prime })/{ϵ}^{\prime \ast }]u(p)$, where $R(q)=(/q+m)/(q^2-m^2+i0)$. It is related to particle-antiparticle two-photon annihilation by crossing symmetry, with appropriate changes to external wavefunctions and conjugations.

Double Compton scattering is $e^{-}+\gamma \to e^{-}+\gamma +\gamma$, together with its inverse. The forward reaction has one incoming and two outgoing photons, so it changes their number and can source the divergence of the photon number current. Ordinary Compton scattering has one incoming and one outgoing photon and conserves photon number. Forward and inverse rates can balance without either reaction individually conserving it.

The physical dimension of a quantity records its powers of chosen base dimensions, independently of units. For example, velocity has dimension $L{T}^{-1}$ and force has dimension $ML{T}^{-2}$. Dimensional analysis compares these expressions in physical equations.

Natural units choose units so that selected fundamental constants have numerical value one. In relativistic quantum field theory the standard choice is $\hslash =c=1$: energy, mass and inverse length then have the same units. This is a convention, not a physical change of the constants; other natural-unit systems may also normalize other constants.

For uniform rotation, the Newtonian gravitational potential plus the centrifugal potential generates the non-Coriolis part of the equation of motion in a rotating frame. Stationary particles obey $\nabla {\Phi }_{\mathrm{eff}}=0$. The Coriolis force changes perturbation dynamics, so curvature of this potential alone is not a general rotating-frame stability criterion.

Effective gravity in a uniformly rotating frame is minus the gradient of the sum of the gravitational and centrifugal potentials. A fluid surface in rotational equilibrium is an equipotential surface and is therefore normal to this effective gravity.

After reducing conserved cyclic momenta, an equilibrium at a strict local minimum of the resulting effective potential is stable in the reduced one-degree-of-freedom dynamics.

Motion-dependent induced current can produce a magnetic force opposing motion and dissipating energy resistively.

An ideal superconducting loop supports persistent current that preserves linked magnetic flux when resistance vanishes.

A uniformly charged spherical shell produces zero field inside and the field of a point charge outside.

The Lyman-alpha line is the hydrogen $2p\leftrightarrow 1s$ electric-dipole transition, with photon energy about $10.2\mathrm{eV}$ and vacuum wavelength about $1216$ angstroms. Its emission cools collisionally excited gas, while absorption traces neutral hydrogen along a background-source sightline.

A neutral hydrogen atom absorbs a photon near the Lyman-alpha resonance by excitation from $1s$ to $2p$. Absorption lines in a quasar spectrum can therefore locate hydrogen along the sightline by their redshift, subject to peculiar-velocity and line-profile effects.

For a population with comoving number density $n_c$ and proper interception area ${\sigma }_p$, the absorption incidence is the proper density multiplied by cross-section and proper light-path length $cdz/[H(z)(1+z)]$. The displayed relation assumes one counted absorption system per interception. Completeness, covering fraction, evolution and the distinction between observed and rest-frame wavelength separation must be stated in an inference from a spectrum.

A circular, opaque, geometrically thin disk has projected area $\pi R^2|\cos i|$. For isotropically oriented normals, $|\cos i|$ is uniform on $[0,1]$, so the mean interception area is half the face-on area. A nonunit covering fraction multiplies this cross-section. Finite thickness, orientation-dependent absorption thresholds and incomplete coverage can change it; this is an average over randomly placed disks, not the inclination distribution conditional on an observed interception.

Radiative decay from the hydrogen $2p$ level to $1s$ produces a Lyman-alpha photon. In an optically thin collisionally excited gas, escape of this photon removes the excitation energy deposited by an Electron collision. The excitation threshold is responsible for the rapid temperature sensitivity of this cooling channel.

The Bohr model assumes circular electron orbits with quantized orbital angular momentum $N\hslash$. It reproduces the hydrogen energy levels, but wavefunctions in quantum mechanics describe spatial probability distributions rather than definite trajectories. In particular, spherically symmetric Coulomb bound states have zero orbital angular momentum.

For a hydrogenic bound state, the principal quantum number is the positive integer $n$ that determines the unperturbed energy. Its allowed orbital quantum numbers are $\ell =0,\dots ,n-1$.

The ionization energy is the minimum energy needed to remove an electron from an isolated atom or molecule in a specified state. Ground-state hydrogen has ionization energy $13.6\mathrm{eV}$.

For two particles of masses $m_1,m_2$ interacting through a central Coulomb potential, separation of the centre of mass leaves the relative-coordinate Schrödinger equation with

For reduced mass $\mu$, the Coulomb length scale is

A no-radial-node Coulomb eigenstate has principal quantum number $n=l+1$ and radial factorIts energy is $-e^2/[8\pi {ϵ}_{0}a(l+1{)}^2]$.

For $R(r)\propto r^l{e}^{-r/[a(l+1)]}$, the radial probability density is proportional to $r^{2l+2}{e}^{-2r/[a(l+1)]}$ and

Neglecting spin, bound hydrogen eigenstates are labelled $|n,\ell ,m⟩$, where $\ell =0,\dots ,n-1$ and $m=-\ell ,\dots ,\ell$. Their unperturbed energy depends only on $n$.

Neglecting spin, the hydrogen level with principal quantum number $n$ has degeneracy

For the $z$ component of position,requires $\Delta m=0$ and $\Delta \ell =\pm 1$. The first condition follows from $[L_z,z]=0$; the second follows from the vector-operator angular-momentum relation together with odd parity.

The even, spherically symmetric hydrogen ground state has no permanent electric dipole, so a uniform electric field gives no first-order energy shift. Odd-parity excited states mix in at second order, yielding a negative shift quadratic in the field.

The $n=2$ eigenspace has basis

A perturbation $g{L}^2$ preserves the hydrogen eigenbasis and shifts $|n,\ell ,m⟩$ byIn the first excited eigenspace, the $2s$ state is unshifted and the three $2p$ states receive the common shift $2g{\hslash }^2$.

The operators $X_3$ and $P_3$ commute with $L_3$, so they preserve $m$. Both are odd under parity, so they connect only states of opposite orbital parity. Within the $n=2$ eigenspace, only the pair $|2,0,0⟩$ and $|2,1,0⟩$ can therefore have nonzero matrix elements.

Forset $a=U(t)b$ withThen $b$ evolves under the constant Hamiltonian

For zero orbital angular momentum and Coulomb potential $-q^2/r$, the radial equation is

For a normalized maximal-angular-momentum hydrogen state, the probability inside a small radius $r_p$ is the integral of $|R{|}^2{r}^2$. Since $R$ behaves as $r^l$ near zero, it equals $2^{2l+3}(r_p/a{)}^{2l+3}/[(2l+3)!(l+1{)}^{2l+3}]$ times $1+O(r_p/a)$. Increasing angular momentum strongly suppresses penetration toward the Coulomb centre.

A hydrogen atom state with principal quantum number $n=l+1$ has no radial nodes and radial factor $R(r)=A{r}^l{e}^{-r/[a(l+1)]}$, where $a$ is the Bohr radius. Its energy is $-{\hslash }^2/[2m{a}^2(l+1{)}^2]$. With unit-normalized spherical harmonics, the radial normalization gives $|A{|}^2=2^{2l+3}/[a^{2l+3}(l+1{)}^{2l+3}(2l+2)!]$.

After the logarithmic radial coordinate $r=e^x$, the Langer replacement $l(l+1)\mapsto (l+1/2{)}^2$ turns the radial Coulomb problem into a two-turning-point WKB approximation problem. Its action integral giveswhich in this Coulomb case agrees with the exact dependence on the principal quantum number.

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 lowest Coulomb bound-state energy is

With normalized $Y_{00}$, the radial ground state $R(r)=C{e}^{-r/a_0}$ hasIf the angular factor is absorbed into the spherically symmetric wavefunction, its normalization constant is instead $(\pi a_0^3{)}^{-1/2}$.

For two bodies with total mass $M$ and relative-orbit radius $r$, the Newtonian orbital period is

Approximating the last orbit by $r=6GM/c^2$ givesThe dominant quadrupole gravitational-wave frequency is twice the orbital frequency, $f_{\mathrm{GW}}=2/T$.

The radius vector from the force centre to an orbiting body sweeps out equal areas in equal times. Its areal velocity is constant because a central force conserves angular momentum.

A uniform circular disc of radius $a$ and mass $m$, pivoted on a perpendicular axis through its rim, has moment of inertia $I_A=3m{a}^2/2$ by the parallel axis theorem. Its hanging small oscillations have period $2\pi \sqrt{3a/(2g)}$. Measured from the upright position, the falling equation is $\ddot{\theta }=2g\sin \theta /(3a)$. Exact upright rest remains an unstable equilibrium; the energy level of that state describes a limiting separatrix after perturbation, with ${\dot{\theta }}^2=4g(1-\cos \theta )/(3a)$. The center acceleration and pivot reaction follow by resolving radial and tangential components.

In nondimensional time the quadratic damping removes energy from the pendulum at rate $-c|\omega {|}^3$. Downward resting positions are nonhyperbolic asymptotically stable spirals, even though the linearized equation has a centre. Upward resting positions remain saddle equilibria. Each fixed-sign branch of $\omega$ has a linear equation for ${\omega }^2$ as a function of angle.

For equal hoop and bead masses, with the hoop suspended from its circumference, the quadratic mass and stiffness matrices are proportional toThe dimensionless squared frequencies are the roots of $2c^2-5c+2$, namely $c=1/2$ and $c=2$.

The reduced Planck mass isIn natural units, Einstein's gravitational coupling is $M_{\mathrm{Pl}}^{-2}=8\pi G$.

The Planck length is ${\ell }_P=\sqrt{\hslash G/c^3}$. Together with the Planck mass $m_P=\sqrt{\hslash c/G}$ it satisfies $G{m}_P/c^2={\ell }_P$.

For isotropic planar photon emission in a rest frame, aberration of light maps its forward half of the angular directions into a lab cone of semi-angle $\alpha$ satisfying the displayed equation. This follows from $\cos \theta =(\cos {\theta }^{\prime }+\beta )/(1+\beta \cos {\theta }^{\prime })$ at ${\theta }^{\prime }=\pm \pi /2$. For a fast positive Lorentz boost, $\alpha \sim 1/\gamma$. The half fraction concerns photon count or directions: the relativistic Doppler effect changes the angular energy weights, so the cone need not contain half the radiant energy.

A reference frame assigns spatial coordinates and time coordinates to physical events.

An inertial frame is a reference frame in which a force-free particle moves with constant velocity.

A local inertial frame is a coordinate frame in which, at one chosen spacetime event, the metric tensor is Minkowskian and the Levi-Civita connection vanishes. Freely falling experiments confined to a sufficiently small neighbourhood therefore obey the laws of special relativity up to tidal corrections.

For uniform angular velocity $\omega$, potential $\varphi$, and additional forces doing no work in the rotating frame,The Coriolis force does no work; the centrifugal potential accounts for the remaining apparent force. This energy is conserved precisely when the potential has no explicit time dependence in the rotating coordinates. An inertial time-independent conservative force need not meet that condition unless its potential is also invariant under the frame's rotation. The balance prevents applying a rotating-frame conservation law to a nonaxisymmetric stationary inertial potential.

For a smooth plane at inclination $\theta$ rotating about a vertical axis at constant angular velocity $\omega$, choose rotating coordinates $\xi$ horizontally and $\eta$ uphill, with their cross product the upward normal vector. A sliding particle under gravity satisfiesThe normal reaction is $N=m(g\cos \theta -2\omega \dot{\xi }\sin \theta +{\omega }^2\eta \sin \theta \cos \theta )$ while contact is maintained. The Coriolis force does no work in the rotating frame, giving the conserved specific energyThe last term is the centrifugal potential; this quantity need not equal the inertial mechanical energy.

For a smooth horizontal circular hoop of radius $a$, rotated at constant angular velocity $\omega$ about a vertical axis through a rim point, measure $\theta$ at that pivot from the diameter toward the bead. In rotating Cartesian coordinate system coordinates, $\mathbf{r}=a(1+\cos 2\theta ,\sin 2\theta ,0)$. The tangential component of the centrifugal acceleration gives the displayed equation; the Coriolis acceleration is normal to the hoop. The farthest point $\theta =0$ is stable for $\omega \ne 0$, with small oscillation angular frequency $|\omega |$. The pivot itself is unstable. This geometry differs from a vertical rotating hoop and has no tangential gravitational term.

If a frame rotates with angular velocity $\mathbf{\Omega }$ relative to an inertial frame, the derivatives of any vector $\mathbf{a}(t)$ satisfyApplying this formula twice to a position gives the Coriolis acceleration, Euler acceleration, and centrifugal acceleration terms.

In a frame rotating with angular velocity $\mathbf{\Omega }$, a moving particle of mass $m$ has apparent Coriolis force $-2m\mathbf{\Omega }\times \mathbf{v}$.

In a frame rotating with angular velocity $\mathbf{\Omega }$, a moving particle has apparent Coriolis acceleration $-2\mathbf{\Omega }\times \mathbf{v}$.

When a reference frame has time-dependent angular velocity $\mathbf{\Omega }(t)$, a particle at position $\mathbf{r}$ has apparent Euler acceleration $-\dot{\mathbf{\Omega }}\times \mathbf{r}$.

In a uniformly rotating frame, centrifugal acceleration points away from the rotation axis and equals $-\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{r})$.

For uniform rotation about an axis, centrifugal acceleration is the negative gradient of the potential $-{\Omega }^2{s}^2/2$, where $s$ is distance from the rotation axis.

A pendulum of length $a$ whose pivot rotates at radius $L$ can remain at angle $\varphi$ from the downward vertical whenIf the connector is a spring of extension $x$, replace $a$ in the radius by $a+x$ and use vertical balance to determine $x$.

For a smooth circular hoop of radius $R$ rotating about its vertical diameter at angular speed $\omega$, a bead at angle $\theta$ from the downward vertical obeys

At ${\omega }^2=g/R$, the stable downward equilibrium of a rotating-hoop bead loses stability and two stable equilibria withemerge symmetrically.

The Schrodinger equation governs unitary quantum evolution under a Hamiltonian.

The free Schrodinger equation describes linear dispersive evolution without a potential term. In this normalization a spectral mode is $e^{ikx-i{k}^{2}t}$. On a half-line, boundary data enter through the global relation for the half-line free Schrodinger equation.

An energy eigenstate satisfiesFor a one-dimensional particle,

A Schrodinger operator is a differential operator of the formwhere $\Delta$ is the Laplace operator and $V$ is a potential function. Its eigenvalue problem is the Time-independent Schrodinger equation.

Matching a wavefunction and its derivative across finite potential discontinuities, together with decay or wall conditions, permits only discrete bound-state energies.

The time-dependent Schrodinger equation evolves a quantum state according to

Writing the wavefunction in modulus and dimensionless quantum phase gives a probability continuity equation and a Hamilton-Jacobi equation with a quantum potential:The phase equations hold locally away from wavefunction nodes. They follow by equating real and imaginary parts of the Time-dependent Schrodinger equation.

Two bound-state solutions of the same one-dimensional time-independent Schrodinger equation have constant Wronskian. Their decay at infinity makes this constant zero, so the two solutions are linearly dependent. Every bound-state energy is therefore nondegenerate.

For linear surface gravity waves with wavelength much larger than the depth, ${\omega }^2=gh{\kappa }^2$. It follows from ${\omega }^2=g\kappa \tanh (\kappa h)$ when $\kappa h\ll 1$, and the wave speed is $\sqrt{gh}$.

The rigid-lid approximation suppresses free surface displacement in a fluid layer while retaining a variable pressure that enforces the mass constraint. At constant depth it gives horizontally nondivergent motion. In a one-layer quasi-geostrophic approximation, removing free-surface stretching eliminates the $-\psi /R_D^2$ term from shallow-water quasi-geostrophic potential vorticity. This is also the small-horizontal-scale limit compared with the Rossby deformation radius, when that term is negligible rather than exactly absent.

The rigid-lid approximation removes external surface displacement from volume conservation while retaining its pressure as a constraint multiplier. It does not mean zero horizontal pressure gradient. In two-layer geostrophic flow, $f_0{\psi }_1=\Pi$ and $f_0{\psi }_2=\Pi -g\prime \chi$, so the upper-layer depth anomaly is $\chi =f_0({\psi }_1-{\psi }_2)/g\prime$. This pressure difference supplies internal potential-vorticity coupling.

A vertically mixed gravity current with horizontally varying reduced gravity $b$ obeys $h_t+(hu{)}_x=0$ and $(hu{)}_t+(h{u}^2+b{h}^2/2{)}_x=0$. Consequently the material acceleration is $Du/Dt=-b{h}_x-h{b}_x/2$. Advected density variables add repeated contact characteristic curves of speed $u$ to the gravity-wave characteristic curves of speeds $u\pm \sqrt{bh}$.

The shallow-water approximation applies when a fluid layer's vertical scale is much smaller than its horizontal scale. Vertical acceleration is then negligible to leading order, pressure is hydrostatic pressure, and the horizontal velocity can be represented by a depth average.