physics.bigb

= Physics
{wiki}

= Mathematical physics
{parent=Physics}
{wiki}

= Added mass
{parent=Mathematical physics}
{wiki}

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

= Added mass of a sphere
{parent=Added mass}

A sphere of volume $V$ translating through unbounded inviscid incompressible fluid has fluid kinetic energy
$$
K_f=\frac12\left(\frac{\rho V}{2}\right)U^2.
$$
Its added mass is therefore one half of the displaced fluid mass.

= Acceleration of a freely falling sphere with added mass
{parent=Added mass of a sphere}

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
$$
\dot U=\frac{\rho_s-\rho}{\rho_s+\rho/2}g.
$$

= Central force
{parent=Mathematical physics}
{wiki}

= Compton wavelength
{parent=Mathematical physics}
{c}
{wiki}

The Compton wavelength of a particle of mass $m$ is $\lambda_C=h/(mc)$.

= Compton scattering
{parent=Compton wavelength}
{c}
{wiki}

For a photon scattered through angle $\theta$ by an initially stationary electron,
$$
\lambda'-\lambda=\frac h{mc}(1-\cos\theta).
$$

= Conservation of energy
{parent=Mathematical physics}
{wiki}

= Dimensional analysis
{parent=Mathematical physics}
{wiki}

= Effective potential
{parent=Mathematical physics}
{wiki}

= Effective potential stability criterion
{parent=Effective potential}

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.

= Electromagnetic boundary condition
{parent=Mathematical physics}
{wiki}

= Electromagnetic damping
{parent=Mathematical physics}
{wiki}

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

= Flux conservation in a superconducting loop
{parent=Mathematical physics}
{wiki}

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

= Fermat principle
{parent=Mathematical physics}
{c}
{wiki}

= Spherical shell theorem
{parent=Mathematical physics}
{wiki}

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

= Hydrogen atom
{parent=Mathematical physics}
{wiki}

= Coulomb two-body reduced mass
{c}
{parent=Hydrogen atom}

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
$$
\mu=\frac{m_1m_2}{m_1+m_2}.
$$

= Bohr radius
{c}
{parent=Hydrogen atom}
{wiki}

For reduced mass $\mu$, the Coulomb length scale is
$$
a=\frac{4\pi\epsilon_0\hbar^2}{\mu e^2}.
$$

= Circular Coulomb bound state
{parent=Hydrogen atom}

A no-radial-node Coulomb eigenstate has principal quantum number $n=l+1$ and radial factor
$$
R(r)\propto r^l e^{-r/[a(l+1)]}.
$$
Its energy is $-e^2/[8\pi\epsilon_0a(l+1)^2]$.

= Mean radius of a circular Coulomb bound state
{parent=Circular Coulomb bound state}

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
$$
\langle r\rangle=\frac{(2l+3)(l+1)}2a.
$$

= Hydrogen energy eigenbasis
{parent=Hydrogen atom}

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

= Electric-dipole selection rules for hydrogen
{parent=Hydrogen energy eigenbasis}

For the $z$ component of position,
$$
\langle n',\ell',m'|z|n,\ell,m\rangle\ne0
$$
requires $\Delta m=0$ and $\Delta\ell=\pm1$. The first condition follows from $[L_z,z]=0$; the second follows from the vector-operator angular-momentum relation together with odd parity.

= Stark effect of the hydrogen ground state
{c}
{parent=Electric-dipole selection rules for hydrogen}
{wiki=Stark_effect}

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.

= First excited hydrogen eigenspace
{parent=Hydrogen energy eigenbasis}

The $n=2$ eigenspace has basis
$$
|2,0,0\rangle,\quad
|2,1,-1\rangle,\quad
|2,1,0\rangle,\quad
|2,1,1\rangle.
$$

= Orbital-angular-momentum-squared perturbation of hydrogen
{parent=First excited hydrogen eigenspace}

A perturbation $gL^2$ preserves the hydrogen eigenbasis and shifts $|n,\ell,m\rangle$ by
$$
g\hbar^2\ell(\ell+1).
$$
In the first excited eigenspace, the $2s$ state is unshifted and the three $2p$ states receive the common shift $2g\hbar^2$.

= Position-momentum selection rules in the first excited hydrogen eigenspace
{parent=First excited hydrogen eigenspace}

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 $|2,0,0\rangle$ and $|2,1,0\rangle$ can therefore have nonzero matrix elements.

= Rotating-frame solution for a harmonically driven two-level system
{parent=Position-momentum selection rules in the first excited hydrogen eigenspace}

For
$$
H(t)=
\begin{pmatrix}
0&ce^{i\omega t/\hbar}\\
ce^{-i\omega t/\hbar}&0
\end{pmatrix},
$$
set $a=U(t)b$ with
$$
U(t)=\operatorname{diag}
\left(e^{i\omega t/(2\hbar)},e^{-i\omega t/(2\hbar)}\right).
$$
Then $b$ evolves under the constant Hamiltonian
$$
H_{\rm eff}=
\begin{pmatrix}
\omega/2&c\\
c&-\omega/2
\end{pmatrix},
\qquad
H_{\rm eff}^2=\left(c^2+\frac{\omega^2}{4}\right)I.
$$

= Radial Schrodinger equation for the hydrogen atom
{parent=Hydrogen atom}
{c}

For zero orbital angular momentum and Coulomb potential $-q^2/r$, the radial equation is
$$
R''+\frac2rR'+\left(\frac\beta r-\gamma^2\right)R=0,
\qquad
\beta=\frac{2mq^2}{\hbar^2},
\qquad
\gamma^2=-\frac{2mE}{\hbar^2}.
$$

= Radial Schrodinger equation
{synonym}

= Series-termination quantization of the Coulomb radial equation
{parent=Radial Schrodinger equation for the hydrogen atom}

Writing $R=e^{-\gamma r}\sum_{n\geq0}a_nr^n$ gives
$$
a_n=\frac{2\gamma n-\beta}{n(n+1)}a_{n-1}.
$$
Normalizability forces the series to terminate, so $2\gamma N=\beta$ for a positive integer $N$ and $E_N=-mq^4/(2\hbar^2N^2)$.

= Hydrogen ground-state energy
{parent=Series-termination quantization of the Coulomb radial equation}

The lowest Coulomb bound-state energy is
$$
E_1=-\frac{mq^4}{2\hbar^2}.
$$

= Radial normalization of the hydrogen ground state
{parent=Hydrogen atom}

With normalized $Y_{00}$, the radial ground state $R(r)=Ce^{-r/a_0}$ has
$$
C=2a_0^{-3/2},
\qquad
\langle r\rangle=\frac32a_0.
$$
If the angular factor is absorbed into the spherically symmetric wavefunction, its normalization constant is instead $(\pi a_0^3)^{-1/2}$.

= Kepler third law
{parent=Mathematical physics}
{c}
{wiki}

For two bodies with total mass $M$ and relative-orbit radius $r$, the Newtonian orbital period is
$$
T^2=\frac{4\pi^2r^3}{GM}.
$$

= Last-orbit frequency of a compact binary
{parent=Kepler third law}

Approximating the last orbit by $r=6GM/c^2$ gives
$$
T=12\pi\sqrt6\,\frac{GM}{c^3}.
$$
The dominant quadrupole gravitational-wave frequency is twice the orbital frequency, $f_{\rm GW}=2/T$.

= Magnetic monopole
{parent=Mathematical physics}
{wiki}

= Physical pendulum
{parent=Mathematical physics}
{wiki}

= Small oscillations of a hoop with a sliding bead
{parent=Physical pendulum}

For equal hoop and bead masses, with the hoop suspended from its circumference, the quadratic mass and stiffness matrices are proportional to
$$
M=\begin{pmatrix}3&1\\1&1\end{pmatrix},
\qquad
K=\begin{pmatrix}2&0\\0&1\end{pmatrix}.
$$
The dimensionless squared frequencies are the roots of $2c^2-5c+2$, namely $c=1/2$ and $c=2$.

= Planck mass
{parent=Mathematical physics}
{c}
{wiki}

= Proper time
{parent=Mathematical physics}
{wiki}

= Quantum transmission
{parent=Mathematical physics}
{wiki}

= Relativistic aberration
{parent=Mathematical physics}
{wiki}

= Rotating reference frame
{parent=Mathematical physics}
{wiki}

= Coriolis acceleration
{title2=$-2\boldsymbol\Omega\times\mathbf v$}
{c}
{parent=Rotating reference frame}
{wiki=Coriolis_force}

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

= Centrifugal acceleration
{title2=$-\boldsymbol\Omega\times(\boldsymbol\Omega\times\mathbf r)$}
{parent=Rotating reference frame}
{wiki=Centrifugal_force}

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

= Rotating conical pendulum with an offset pivot
{parent=Rotating reference frame}

A pendulum of length $a$ whose pivot rotates at radius $L$ can remain at angle $\phi$ from the downward vertical when
$$
\omega^2=\frac{g\tan\phi}{L+a\sin\phi}.
$$
If the connector is a spring of extension $x$, replace $a$ in the radius by $a+x$ and use vertical balance to determine $x$.

= Rotating-hoop bead
{parent=Rotating reference frame}

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
$$
\ddot\theta=(\omega^2\cos\theta-g/R)\sin\theta.
$$

= Pitchfork bifurcation of a rotating hoop bead
{parent=Rotating-hoop bead}

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

= Schrodinger equation
{parent=Mathematical physics}
{c}
{wiki}

= Time-independent Schrodinger equation
{title2=$H\psi=E\psi$}
{parent=Schrodinger equation}
{c}
{wiki=Schr%C3%B6dinger_equation#Time-independent_equation}

An energy eigenstate satisfies
$$
H\psi=E\psi.
$$
For a one-dimensional particle,
$$
-\frac{\hbar^2}{2m}\psi''+U(x)\psi=E\psi.
$$

= Bound-state quantization condition
{parent=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.

= Time-dependent Schrodinger equation
{title2=$i\hbar\partial_t\psi=H\psi$}
{c}
{parent=Schrodinger equation}
{wiki=Schr%C3%B6dinger_equation}

The time-dependent Schrodinger equation evolves a quantum state according to
$$
i\hbar\frac{\partial\psi}{\partial t}=H\psi.
$$

= Nondegeneracy of one-dimensional bound states
{parent=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 energy eigenvalue>[nondegenerate].

= Shallow water equations
{parent=Mathematical physics}
{wiki}

= Rossby deformation radius
{parent=Shallow water equations}
{c}
{wiki}

In a rotating shallow-water layer,
$$
L_R=\frac{\sqrt{gh_0}}{|f|}
$$
is the horizontal scale on which gravity and Coriolis effects balance.

= Geostrophic balance
{parent=Shallow water equations}
{wiki}

A steady horizontal pressure gradient can balance the Coriolis force:
$$
\mathbf f\times\mathbf u=-g\nabla\eta.
$$

= Vortex stretching
{parent=Mathematical physics}
{wiki}

= Vorticity equation
{parent=Mathematical physics}
{wiki}

= Beltrami flow
{parent=Vorticity equation}
{c}
{wiki}

A Beltrami flow has velocity parallel to vorticity.

= Conservation law
{parent=Mathematical physics}
{wiki}

A conservation law equates local time change of a density with the divergence of its flux.

= Conservation laws
{synonym}

= Mass flux
{parent=Conservation law}
{wiki=Mass_flow_rate}

Mass flux is the mass crossing a unit area per unit time. In a one-dimensional flow it is $\rho u$.

= Momentum flux
{parent=Conservation law}

Momentum flux includes advected momentum and stress. In a one-dimensional inviscid flow it is $p+\rho u^2$.

= Energy flux
{parent=Conservation law}

Energy flux is the energy crossing a unit area per unit time. For a one-dimensional inviscid gas with internal-energy density $W$, it is
$$
u\left(p+W+\frac12\rho u^2\right).
$$

= Continuity equation
{parent=Conservation law}
{wiki}

The continuity equation expresses local conservation as time change plus flux divergence equal to zero.

= Standing wave
{parent=Mathematical physics}
{wiki}

A standing wave is the superposition of equal-frequency waves travelling in opposite directions. Its spatial nodes remain fixed while its amplitude oscillates in time.

= Radioactive decay
{parent=Physics}
{wiki}

A population of unstable nuclei with decay constant $\lambda>0$ obeys $N'=-\lambda N$ and therefore decays as $N(t)=N(0)e^{-\lambda t}$.

= Sequential radioactive decay
{parent=Radioactive decay}
{wiki=Bateman_equation}

In a chain $N_1\to N_2\to N_3$ with distinct decay constants, each population is a linear combination of the exponentials $e^{-\lambda_i t}$. When two consecutive constants coalesce to $\lambda$, the repeated Laplace-transform pole produces a term proportional to $t e^{-\lambda t}$.

= Branch of physics
{parent=Physics}
{wiki=Branches_of_physics}

\Include[classical-mechanics]
\Include[continuum-mechanics]
\Include[cosmology]
\Include[dynamical-systems]
\Include[electromagnetism]
\Include[fluid-mechanics]
\Include[general-relativity]
\Include[integrable-systems]
\Include[mathematical-biology]
\Include[quantum-mechanics]
\Include[quantum-theory]
\Include[special-relativity]
\Include[statistical-physics]