User:JMvanDijk/Sandbox 16/Box2

Generalized mechanics

Main articles: Analytical mechanics, Lagrangian mechanics, and Hamiltonian mechanics

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Generalized coordinates	q, Q		varies with choice	varies with choice
Generalized velocities	${\dot {q}},{\dot {Q}}$	${\dot {q}}\equiv \mathrm {d} q/\mathrm {d} t$	varies with choice	varies with choice
Generalized momenta	p, P	$p=\partial L/\partial {\dot {q}}$	varies with choice	varies with choice
Lagrangian	L	$L(\mathbf {q} ,\mathbf {\dot {q}} ,t)=T(\mathbf {\dot {q}} )-V(\mathbf {q} ,\mathbf {\dot {q}} ,t)$ where $\mathbf {q} =\mathbf {q} (t)$ and p = p(t) are vectors of the generalized coords and momenta, as functions of time	J	[M][L]2[T]−2
Hamiltonian	H	$H(\mathbf {p} ,\mathbf {q} ,t)=\mathbf {p} \cdot \mathbf {\dot {q}} -L(\mathbf {q} ,\mathbf {\dot {q}} ,t)$	J	[M][L]2[T]−2
Action, Hamilton's principal function	S, $\scriptstyle {\mathcal {S}}$	${\mathcal {S}}=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)\mathrm {d} t$	J s	[M][L]2[T]−1

Electromagnetism

Electric and Magnetic fields

General Classical Equations

Physical quantity	Electric	Magnetic
Potential gradient and field	$\mathbf {E} =-\nabla V$ $\Delta V=-\int _{r_{1}}^{r_{2}}\mathbf {E} \cdot d\mathbf {r} \,\!$	$\mathbf {B} =\nabla \times \mathbf {A}$
Electric:Point charge Magnetic: moment	$\mathbf {E} ={\frac {q}{4\pi \epsilon _{0}\left|\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} \,\!$	$\mathbf {A} ={\frac {\mu _{0}}{4\pi }}{\frac {\mathbf {m} \times \mathbf {r} }{\left|\mathbf {r} \right|^{3}}}$ $\mathbf {B} ({\mathbf {r} })=\nabla \times {\mathbf {A} }={\frac {\mu _{0}}{4\pi }}\left({\frac {3\mathbf {r} (\mathbf {m} \cdot \mathbf {r} )}{\left|\mathbf {r} \right|^{5}}}-{\frac {\mathbf {m} }{\left|\mathbf {r} \right|^{3}}}\right)$
Electric: At a point in a local array of point charges Magnetic: N/A	$\mathbf {E} =\sum \mathbf {E} _{i}={\frac {1}{4\pi \epsilon _{0}}}\sum _{i}{\frac {q_{i}}{\left|\mathbf {r} _{i}-\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} _{i}\,\!$	---
Electric: at a point due to a continuum of charge Magnetic: magnetic moment due to a current distribution	$\mathbf {E} ={\frac {1}{4\pi \epsilon _{0}}}\int _{V}{\frac {\mathbf {r} \rho \mathrm {d} V}{\left|\mathbf {r} \right|^{3}}}\,\!$	$\mathbf {m} ={\frac {1}{2}}\int _{V}\mathbf {r} \times \mathbf {J} \mathrm {d} V$
Torque and potential energy due to non-uniform fields and dipole moments	${\boldsymbol {\tau }}=\int _{V}\mathrm {d} \mathbf {p} \times \mathbf {E}$ $U=\int _{V}\mathrm {d} \mathbf {p} \cdot \mathbf {E}$	${\boldsymbol {\tau }}=\int _{V}\mathrm {d} \mathbf {m} \times \mathbf {B}$ $U=\int _{V}\mathrm {d} \mathbf {m} \cdot \mathbf {B}$

Maxwell's Equations

Main article: Planck units

Name	Integral equations -- SI Units	Differential equations -- SI Units	Integral equations -- Gaussian units	Differential equations -- Gaussian units
Gauss's law	${\scriptstyle \partial \Omega }$ $\mathbf {E} \cdot \mathrm {d} \mathbf {S} ={\frac {1}{\varepsilon _{0}}}\iiint _{\Omega }\rho \,\mathrm {d} V$	$\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$	${\scriptstyle \partial \Omega }$ $\mathbf {E} \cdot \mathrm {d} \mathbf {S} =4\pi \iiint _{\Omega }\rho \,\mathrm {d} V$	$\nabla \cdot \mathbf {E} =4\pi \rho$
Gauss's law for magnetism	${\scriptstyle \partial \Omega }$ $\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0$	$\nabla \cdot \mathbf {B} =0$	${\scriptstyle \partial \Omega }$ $\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0$	$\nabla \cdot \mathbf {B} =0$
Maxwell–Faraday equation (Faraday's law of induction)	$\oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {l}}=-\operatorname {\frac {d}{dt}} \iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S}$	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$	$\oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {1}{c}}{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S}$	$\nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}$
Ampère's circuital law (with Maxwell's addition)	${\begin{aligned}\oint _{\partial \Sigma }&\mathbf {B} \cdot \mathrm {d} {\boldsymbol {l}}=\mu _{0}\left(\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} +\varepsilon _{0}{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {S} \right)\\\end{aligned}}$	$\nabla \times \mathbf {B} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)$	${\begin{aligned}\oint _{\partial \Sigma }&\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}={\frac {1}{c}}\left(4\pi \iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} +{\frac {\mathrel {\mathrm {d} }}{\mathrm {d} t}}\iint _{\Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {S} \right)\end{aligned}}$	$\nabla \times \mathbf {B} ={\frac {1}{c}}\left(4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)$

Alternative formulations

For an overview, see Mathematical descriptions of the electromagnetic field.

For the equations in quantum field theory, see quantum electrodynamics.

For the use of a gauge see, see Gauge theory.

Vector calculus

Formulation	Homogeneous equations	Inhomogeneous equations
Fields 3D Euclidean space + time	${\begin{aligned}\nabla \cdot \mathbf {B} =0\end{aligned}}$ ${\begin{aligned}\nabla \times \mathbf {E} +{\frac {\partial \mathbf {B} }{\partial t}}=0\end{aligned}}$	${\begin{aligned}\nabla \cdot \mathbf {E} &={\frac {\rho }{\varepsilon _{0}}}\end{aligned}}$ ${\begin{aligned}\nabla \times \mathbf {B} -{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}&=\mu _{0}\mathbf {J} \end{aligned}}$
Potentials (any gauge) 3D Euclidean space + time	${\begin{aligned}\mathbf {B} &=\mathbf {\nabla } \times \mathbf {A} \end{aligned}}$ ${\begin{aligned}\mathbf {E} &=-\mathbf {\nabla } \varphi -{\frac {\partial \mathbf {A} }{\partial t}}\end{aligned}}$	${\begin{aligned}-\nabla ^{2}\varphi -{\frac {\partial }{\partial t}}\left(\mathbf {\nabla } \cdot \mathbf {A} \right)&={\frac {\rho }{\varepsilon _{0}}}\end{aligned}}$ ${\begin{aligned}\left(-\nabla ^{2}+{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right)\mathbf {A} +\mathbf {\nabla } \left(\mathbf {\nabla } \cdot \mathbf {A} +{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}\right)&=\mu _{0}\mathbf {J} \end{aligned}}$
Potentials (Lorenz gauge) 3D Euclidean space + time	${\begin{aligned}\mathbf {B} &=\mathbf {\nabla } \times \mathbf {A} \\\end{aligned}}$ ${\begin{aligned}\mathbf {E} &=-\mathbf {\nabla } \varphi -{\frac {\partial \mathbf {A} }{\partial t}}\\\end{aligned}}$ ${\begin{aligned}\mathbf {\nabla } \cdot \mathbf {A} &=-{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}\\\end{aligned}}$	${\begin{aligned}\left(-\nabla ^{2}+{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right)\varphi &={\frac {\rho }{\varepsilon _{0}}}\end{aligned}}$ ${\begin{aligned}\left(-\nabla ^{2}+{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right)\mathbf {A} &=\mu _{0}\mathbf {J} \end{aligned}}$

Differential forms

Formulation	Homogeneous equations	Inhomogeneous equations
Fields Any space + time	${\begin{aligned}dB&=0\\dE+{\frac {\partial B}{\partial t}}&=0\\\end{aligned}}$	${\begin{aligned}d{*}E={\frac {\rho }{\epsilon _{0}}}\\d{*}B-{\frac {1}{c^{2}}}{\frac {\partial {*}E}{\partial t}}={\mu _{0}}J\\\end{aligned}}$
Potentials (any gauge) Any space (with topological restrictions) + time	${\begin{aligned}B&=dA\\E&=-d\phi -{\frac {\partial A}{\partial t}}\\\end{aligned}}$	${\begin{aligned}-d{*}\!\left(d\phi +{\frac {\partial A}{\partial t}}\right)&={\frac {\rho }{\epsilon _{0}}}\\d{*}dA+{\frac {1}{c^{2}}}{\frac {\partial }{\partial t}}{*}\!\left(d\phi +{\frac {\partial A}{\partial t}}\right)&=\mu _{0}J\\\end{aligned}}$
Potential (Lorenz Gauge) Any space (with topological restrictions) + time spatial metric independent of time	${\begin{aligned}B&=dA\\E&=-d\phi -{\frac {\partial A}{\partial t}}\\d{*}A&=-{*}{\frac {1}{c^{2}}}{\frac {\partial \phi }{\partial t}}\\\end{aligned}}$	${\begin{aligned}{*}\!\left(-\Delta \phi +{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\phi \right)&={\frac {\rho }{\epsilon _{0}}}\\{*}\!\left(-\Delta A+{\frac {1}{c^{2}}}{\frac {\partial ^{2}A}{\partial ^{2}t}}\right)&=\mu _{0}J\\\end{aligned}}$

Special Relativity: The metric and four-vectors

Main articles: metric tensor and four-vectors

Inner product (i.e. notion of length)

${\boldsymbol {\mathsf {a}}}\cdot {\boldsymbol {\mathsf {b}}}=\eta ({\boldsymbol {\mathsf {a}}},{\boldsymbol {\mathsf {b}}})$

where $\eta$ is known as the metric tensor. In special relativity, the metric tensor is the Minkowski metric:

$\eta ={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}$

Space-time interval

$ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}={\begin{pmatrix}cdt&dx&dy&dz\end{pmatrix}}{\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}{\begin{pmatrix}cdt\\dx\\dy\\dz\end{pmatrix}}$

ds² is invariant under the Lorentz transformation:

$\eta ({\boldsymbol {\mathsf {a}}}',{\boldsymbol {\mathsf {b}}}')=\eta \left(\Lambda {\boldsymbol {\mathsf {a}}},\Lambda {\boldsymbol {\mathsf {b}}}\right)=\eta ({\boldsymbol {\mathsf {a}}},{\boldsymbol {\mathsf {b}}})$

The sign of the metric and the placement of the ct, ct', cdt, and cdt′ time-based terms can vary depending on the author's choice. For instance, many times the time-based terms are placed first in the four-vectors, with the spatial terms following. Also, sometimes η is replaced with −η, making the spatial terms produce negative contributions to the dot product or spacetime interval, while the time term makes a positive contribution. These differences can be used in any combination, so long as the choice of standards is followed completely throughout the computations performed.

Lorentz transforms

It is possible to express the above coordinate transformation via a matrix. To simplify things, it can be best to replace t, t′, dt, and dt′ with ct, ct', cdt, and cdt′, which has the dimensions of distance. So:

$x'=\gamma x-\gamma \beta ct\,$

$y'=y\,$

$z'=z\,$

$ct'=\gamma ct-\gamma \beta x\,$

then in matrix form:

${\begin{pmatrix}ct'\\x'\\y'\\z'\end{pmatrix}}={\begin{pmatrix}\gamma &-\gamma \beta &0&0\\-\gamma \beta &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}{\begin{pmatrix}ct\\x\\y\\z\end{pmatrix}}$

The vectors in the above transformation equation are known as four-vectors, in this case they are specifically the position four-vectors. In general, in special relativity, four-vectors can be transformed from one reference frame to another as follows:

${\boldsymbol {\mathsf {a}}}'=\Lambda {\boldsymbol {\mathsf {a}}}$

So ${\boldsymbol {\mathsf {a}}}'$ can be a four-vector representing position, velocity, or momentum, and the same Λ can be used when transforming between the same two frames. The most general Lorentz transformation includes boosts and rotations; the components are complicated and the transformation requires spinors.

4-vectors and frame-invariant results

Invariance and unification of physical quantities both arise from four-vectors. The inner product of a 4-vector with itself is equal to a scalar (by definition of the inner product), and since the 4-vectors are physical quantities their magnitudes correspond to physical quantities also.

Property/effect	3-vector	4-vector	Invariant result
Space-time events	3-position: r = (x1, x2, x3) $\mathbf {r} \cdot \mathbf {r} \equiv r^{2}\equiv x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\,\!$	4-position: X = (ct, x1, x2, x3)	${\boldsymbol {\mathsf {X}}}\cdot {\boldsymbol {\mathsf {X}}}=\left(c\tau \right)^{2}\,\!$ ${\begin{aligned}&\left(ct\right)^{2}-\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)\\&=\left(ct\right)^{2}-r^{2}\\&=-\chi ^{2}=\left(c\tau \right)^{2}\end{aligned}}\,\!$ τ = proper time χ = proper distance
Momentum-energy invariance	$\mathbf {p} =\gamma m\mathbf {u} \,\!$ 3-momentum: p = (p1, p2, p3) $\mathbf {p} \cdot \mathbf {p} \equiv p^{2}\equiv p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\,\!$	4-momentum: P = (E/c, p1, p2, p3) ${\boldsymbol {\mathsf {P}}}=m{\boldsymbol {\mathsf {U}}}\,\!$	${\boldsymbol {\mathsf {P}}}\cdot {\boldsymbol {\mathsf {P}}}=\left(mc\right)^{2}\,\!$ ${\begin{aligned}&\left({\frac {E}{c}}\right)^{2}-\left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\right)\\&=\left({\frac {E}{c}}\right)^{2}-p^{2}\\&=\left(mc\right)^{2}\end{aligned}}\,\!$ which leads to: $E^{2}=\left(pc\right)^{2}+\left(mc^{2}\right)^{2}\,\!$ E = total energy m = invariant mass
Velocity	3-velocity: u = (u1, u2, u3) $\mathbf {u} ={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\,\!$	4-velocity: U = (U0, U1, U2, U3) ${\boldsymbol {\mathsf {U}}}={\frac {\mathrm {d} {\boldsymbol {\mathsf {X}}}}{\mathrm {d} \tau }}=\gamma \left(c,\mathbf {u} \right)$	${\boldsymbol {\mathsf {U}}}\cdot {\boldsymbol {\mathsf {U}}}=c^{2}\,\!$
Acceleration	3-acceleration: a = (a1, a2, a3) $\mathbf {a} ={\frac {\mathrm {d} \mathbf {u} }{\mathrm {d} t}}\,\!$	4-acceleration: A = (A0, A1, A2, A3) ${\boldsymbol {\mathsf {A}}}={\frac {\mathrm {d} {\boldsymbol {\mathsf {U}}}}{\mathrm {d} \tau }}=\gamma \left(c{\frac {\mathrm {d} \gamma }{\mathrm {d} t}},{\frac {\mathrm {d} \gamma }{\mathrm {d} t}}\mathbf {u} +\gamma \mathbf {a} \right)$	${\boldsymbol {\mathsf {A}}}\cdot {\boldsymbol {\mathsf {U}}}=0\,\!$
Force	3-force: f = (f1, f2, f3) $\mathbf {f} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}\,\!$	4-force: F = (F0, F1, F2, F3) ${\boldsymbol {\mathsf {F}}}={\frac {\mathrm {d} {\boldsymbol {\mathsf {P}}}}{\mathrm {d} \tau }}=\gamma m\left(c{\frac {\mathrm {d} \gamma }{\mathrm {d} t}},{\frac {\mathrm {d} \gamma }{\mathrm {d} t}}\mathbf {u} +\gamma \mathbf {a} \right)$	${\boldsymbol {\mathsf {F}}}\cdot {\boldsymbol {\mathsf {U}}}=0\,\!$

Quantum Mechanics

Wave–particle duality and time evolution

Property or effect	Nomenclature	Equation
Planck–Einstein equation and de Broglie wavelength relations	P = (E/c, p) is the four-momentum, K = (ω/c, k) is the four-wavevector, E = energy of particle ω = 2πf is the angular frequency and frequency of the particle ħ = h/2π are the Planck constants c = speed of light	$\mathbf {P} =(E/c,\mathbf {p} )=\hbar (\omega /c,\mathbf {k} )=\hbar \mathbf {K}$
Schrödinger equation	Ψ = wavefunction of the system Ĥ = Hamiltonian operator, E = energy eigenvalue of system i is the imaginary unit t = time	General time-dependent case: $i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi$ Time-independent case: ${\hat {H}}\Psi =E\Psi$
Heisenberg equation	 = operator of an observable property [ ] is the commutator $\langle \,\rangle$ denotes the average	${\frac {d}{dt}}{\hat {A}}(t)={\frac {i}{\hbar }}[{\hat {H}},{\hat {A}}(t)]+{\frac {\partial {\hat {A}}(t)}{\partial t}},$
Time evolution in Heisenberg picture (Ehrenfest theorem)	m = mass, V = potential energy, r = position, p = momentum, of a particle.	${\frac {d}{dt}}\langle {\hat {A}}\rangle ={\frac {1}{i\hbar }}\langle [{\hat {A}},{\hat {H}}]\rangle +\left\langle {\frac {\partial {\hat {A}}}{\partial t}}\right\rangle$ For momentum and position; $m{\frac {d}{dt}}\langle \mathbf {r} \rangle =\langle \mathbf {p} \rangle$ ${\frac {d}{dt}}\langle \mathbf {p} \rangle =-\langle \nabla V\rangle$

Non-relativistic time-independent Schrödinger equation

Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions. Notice in the case of one spatial dimension, for one particle, the partial derivative reduces to an ordinary derivative.

	One particle	N particles
One dimension	${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(x)=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)$	${\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\frac {{\hat {p}}_{n}^{2}}{2m_{n}}}+V(x_{1},x_{2},\cdots x_{N})\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}+V(x_{1},x_{2},\cdots x_{N})\end{aligned}}$ where the position of particle n is xn.
	$E\Psi =-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi +V\Psi$	$E\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}\Psi +V\Psi \,.$
	$\Psi (x,t)=\psi (x)e^{-iEt/\hbar }\,.$ There is a further restriction — the solution must not grow at infinity, so that it has either a finite L2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): $\|\psi \|^{2}=\int |\psi (x)|^{2}\,dx.\,$	$\Psi =e^{-iEt/\hbar }\psi (x_{1},x_{2}\cdots x_{N})$ for non-interacting particles $\Psi =e^{-i{Et/\hbar }}\prod _{n=1}^{N}\psi (x_{n})\,,\quad V(x_{1},x_{2},\cdots x_{N})=\sum _{n=1}^{N}V(x_{n})\,.$
Three dimensions	${\begin{aligned}{\hat {H}}&={\frac {{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}}{2m}}+V(\mathbf {r} )\\&=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )\end{aligned}}$ where the position of the particle is r = (x, y, z).	${\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\frac {{\hat {\mathbf {p} }}_{n}\cdot {\hat {\mathbf {p} }}_{n}}{2m_{n}}}+V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}+V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})\end{aligned}}$ where the position of particle n is r n = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is $\nabla _{n}^{2}={\frac {\partial ^{2}}{{\partial x_{n}}^{2}}}+{\frac {\partial ^{2}}{{\partial y_{n}}^{2}}}+{\frac {\partial ^{2}}{{\partial z_{n}}^{2}}}$
	$E\Psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi$	$E\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}\Psi +V\Psi$
	$\Psi =\psi (\mathbf {r} )e^{-iEt/\hbar }$	$\Psi =e^{-iEt/\hbar }\psi (\mathbf {r} _{1},\mathbf {r} _{2}\cdots \mathbf {r} _{N})$ for non-interacting particles $\Psi =e^{-i{Et/\hbar }}\prod _{n=1}^{N}\psi (\mathbf {r} _{n})\,,\quad V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})=\sum _{n=1}^{N}V(\mathbf {r} _{n})$

Non-relativistic time-dependent Schrödinger equation

Again, summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of solutions.

	One particle	N particles
One dimension	${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(x,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V(x,t)$	${\hat {H}}=\sum _{n=1}^{N}{\frac {{\hat {p}}_{n}^{2}}{2m_{n}}}+V(x_{1},x_{2},\cdots x_{N},t)=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}+V(x_{1},x_{2},\cdots x_{N},t)$ where the position of particle n is xn.
	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\Psi +V\Psi$	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}\Psi +V\Psi \,.$
	$\Psi =\Psi (x,t)$	$\Psi =\Psi (x_{1},x_{2}\cdots x_{N},t)$
Three dimensions	${\begin{aligned}{\hat {H}}&={\frac {{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}}{2m}}+V(\mathbf {r} ,t)\\&=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)\\\end{aligned}}$	${\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\frac {{\hat {\mathbf {p} }}_{n}\cdot {\hat {\mathbf {p} }}_{n}}{2m_{n}}}+V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N},t)\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}+V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N},t)\end{aligned}}$
	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi$	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}\Psi +V\Psi$ This last equation is in a very high dimension, so the solutions are not easy to visualize.
	$\Psi =\Psi (\mathbf {r} ,t)$	$\Psi =\Psi (\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N},t)$

Photoemission

Property/Effect	Nomenclature	Equation
Photoelectric equation	Kmax = Maximum kinetic energy of ejected electron (J) h = Planck's constant f = frequency of incident photons (Hz = s−1) φ, Φ = Work function of the material the photons are incident on (J)	$K_{\mathrm {max} }=hf-\Phi \,\!$
Threshold frequency and	φ, Φ = Work function of the material the photons are incident on (J) f0, ν0 = Threshold frequency (Hz = s−1)	Can only be found by experiment. The De Broglie relations give the relation between them: $\phi =hf_{0}\,\!$
Photon momentum	p = momentum of photon (kg m s−1) f = frequency of photon (Hz = s−1) λ = wavelength of photon (m)	The De Broglie relations give: $p=hf/c=h/\lambda \,\!$

Quantum uncertainty

Property or effect	Nomenclature	Equation
Heisenberg's uncertainty principles	n = number of photons φ = wave phase [, ] = commutator	Position-momentum $\sigma (x)\sigma (p)\geq {\frac {\hbar }{2}}\,\!$ Energy-time $\sigma (E)\sigma (t)\geq {\frac {\hbar }{2}}\,\!$ Number-phase $\sigma (n)\sigma (\phi )\geq {\frac {\hbar }{2}}\,\!$
Dispersion of observable	A = observables (eigenvalues of operator)	${\begin{aligned}\sigma (A)^{2}&=\langle (A-\langle A\rangle )^{2}\rangle \\&=\langle A^{2}\rangle -\langle A\rangle ^{2}\end{aligned}}$
General uncertainty relation	A, B = observables (eigenvalues of operator)	$\sigma (A)\sigma (B)\geq {\frac {1}{2}}\langle i[{\hat {A}},{\hat {B}}]\rangle$

Probability Distributions

Property or effect	Nomenclature	Equation
Density of states		$N(E)=8{\sqrt {2}}\pi m^{3/2}E^{1/2}/h^{3}\,\!$
Fermi–Dirac distribution (fermions)	P(Ei) = probability of energy Ei g(Ei) = degeneracy of energy Ei (no of states with same energy) μ = chemical potential	$P(E_{i})=g(E_{i})/(e^{(E-\mu )/kT}+1)\,\!$
Bose–Einstein distribution (bosons)		$P(E_{i})=g(E_{i})/(e^{(E_{i}-\mu )/kT}-1)\,\!$

Angular momentum

Main articles: angular momentum operator and quantum number

Property or effect	Nomenclature	Equation
Angular momentum quantum numbers	s = spin quantum number ms = spin magnetic quantum number ℓ = Azimuthal quantum number mℓ = azimuthal magnetic quantum number mj = total angular momentum magnetic quantum number j = total angular momentum quantum number	Spin projection: $m_{s}\in \{-s,-s+1\cdots s-1,s\}\,\!$ Orbital: $m_{\ell }\in \{-\ell ,-\ell +1\cdots \ell -1,\ell \}\,\!$ $m_{\ell }\in \{0\cdots n-1\}\,\!$ Total: ${\begin{aligned}&j=\ell +s\\&j\in \{|\ell -s|,|\ell -s|+1\cdots |\ell +s|-1,|\ell +s|\}\\\end{aligned}}\,\!$
Angular momentum magnitudes	angular momementa: S = Spin, L = orbital, J = total	Spin magnitude: $|\mathbf {S} |=\hbar {\sqrt {s(s+1)}}\,\!$ Orbital magnitude: $|\mathbf {L} |=\hbar {\sqrt {\ell (\ell +1)}}\,\!$ Total magnitude: $\mathbf {J} =\mathbf {L} +\mathbf {S} \,\!$ $|\mathbf {J} |=\hbar {\sqrt {j(j+1)}}\,\!$
Angular momentum components		Spin: $S_{z}=m_{s}\hbar \,\!$ Orbital: $L_{z}=m_{\ell }\hbar \,\!$

Magnetic moments

In what follows, B is an applied external magnetic field and the quantum numbers above are used.

Property or effect	Nomenclature	Equation
orbital magnetic dipole moment	e = electron charge me = electron rest mass L = electron orbital angular momentum gℓ = orbital Landé g-factor μB = Bohr magneton	${\boldsymbol {\mu }}_{\ell }=-e\mathbf {L} /2m_{e}=g_{\ell }{\frac {\mu _{B}}{\hbar }}\mathbf {L} \,\!$ z-component: $\mu _{\ell ,z}=-m_{\ell }\mu _{B}\,\!$
spin magnetic dipole moment	S = electron spin angular momentum gs = spin Landé g-factor	${\boldsymbol {\mu }}_{s}=-e\mathbf {S} /m_{e}=g_{s}{\frac {\mu _{B}}{\hbar }}\mathbf {S} \,\!$ z-component: $\mu _{s,z}=-eS_{z}/m_{e}=g_{s}eS_{z}/2m_{e}\,\!$
dipole moment potential	U = potential energy of dipole in field	$U=-{\boldsymbol {\mu }}\cdot \mathbf {B} =-\mu _{z}B\,\!$

The Hydrogen atom

Main article: Hydrogen atom

Property or effect	Nomenclature	Equation
Energy level :p≈	En = energy eigenvalue n = principal quantum number e = electron charge me = electron rest mass ε0 = permittivity of free space h = Planck's constant	$E_{n}=-me^{4}/8\epsilon _{0}^{2}h^{2}n^{2}=-13.61eV/n^{2}\,\!$
Spectrum	λ = wavelength of emitted photon, during electronic transition from Ei to Ej	${\frac {1}{\lambda }}=R\left({\frac {1}{n_{j}^{2}}}-{\frac {1}{n_{i}^{2}}}\right),\,n_{j}<n_{i}\,\!$

Fundamental forces

These equations need to be refined such that the notation is defined as has been done for the previous sets of equations.

Name	Equations
Strong force	${\begin{aligned}{\mathcal {L}}_{\mathrm {QCD} }&={\bar {\psi }}_{i}\left(i\gamma ^{\mu }(D_{\mu })_{ij}-m\,\delta _{ij}\right)\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\\&={\bar {\psi }}_{i}(i\gamma ^{\mu }\partial _{\mu }-m)\psi _{i}-gG_{\mu }^{a}{\bar {\psi }}_{i}\gamma ^{\mu }T_{ij}^{a}\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\,,\\\end{aligned}}\,\!$
Electroweak interaction	:${\mathcal {L}}_{EW}={\mathcal {L}}_{g}+{\mathcal {L}}_{f}+{\mathcal {L}}_{h}+{\mathcal {L}}_{y}.\,\!$ ${\mathcal {L}}_{g}=-{\frac {1}{4}}W_{a}^{\mu \nu }W_{\mu \nu }^{a}-{\frac {1}{4}}B^{\mu \nu }B_{\mu \nu }\,\!$ ${\mathcal {L}}_{f}={\overline {Q}}_{i}iD\!\!\!\!/\;Q_{i}+{\overline {u}}_{i}^{c}iD\!\!\!\!/\;u_{i}^{c}+{\overline {d}}_{i}^{c}iD\!\!\!\!/\;d_{i}^{c}+{\overline {L}}_{i}iD\!\!\!\!/\;L_{i}+{\overline {e}}_{i}^{c}iD\!\!\!\!/\;e_{i}^{c}\,\!$ ${\mathcal {L}}_{h}=|D_{\mu }h|^{2}-\lambda \left(|h|^{2}-{\frac {v^{2}}{2}}\right)^{2}\,\!$ ${\mathcal {L}}_{y}=-y_{u\,ij}\epsilon ^{ab}\,h_{b}^{\dagger }\,{\overline {Q}}_{ia}u_{j}^{c}-y_{d\,ij}\,h\,{\overline {Q}}_{i}d_{j}^{c}-y_{e\,ij}\,h\,{\overline {L}}_{i}e_{j}^{c}+h.c.\,\!$
Quantum electrodynamics	${\mathcal {L}}={\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-m)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\;,\,\!$

General Relativity

see: General relativity, Einstein field equations, List of equations in gravitation

$R_{\mu \nu }-{\tfrac {1}{2}}Rg_{\mu \nu }+\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }$

where R_μν is the Ricci curvature tensor, R is the scalar curvature, g_μν is the metric tensor, Λ is the cosmological constant, G is Newton's gravitational constant, c is the speed of light in vacuum, and T_μν is the stress–energy tensor.

One can write the EFE in a more compact form by defining the Einstein tensor

$G_{\mu \nu }=R_{\mu \nu }-{\tfrac {1}{2}}Rg_{\mu \nu },$

which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as

$G_{\mu \nu }+\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }.$

In standard units, each term on the left has units of 1/length². With this choice of Einstein constant as 8πG/c⁴, then the stress-energy tensor on the right side of the equation must be written with each component in units of energy-density (i.e., energy per volume = pressure).

Using geometrized units where G = c = 1, this can be rewritten as

$G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi T_{\mu \nu }\,.$

The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how matter/energy determines the curvature of spacetime.

These equations, together with the geodesic equation, which dictates how freely-falling matter moves through space-time, form the core of the mathematical formulation of general relativity.

Sign convention

The above form of the EFE is the standard established by Misner, Thorne, and Wheeler. The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):

${\begin{aligned}g_{\mu \nu }&=[S1]\times \operatorname {diag} (-1,+1,+1,+1)\\[6pt]{R^{\mu }}_{\alpha \beta \gamma }&=[S2]\times \left(\Gamma _{\alpha \gamma ,\beta }^{\mu }-\Gamma _{\alpha \beta ,\gamma }^{\mu }+\Gamma _{\sigma \beta }^{\mu }\Gamma _{\gamma \alpha }^{\sigma }-\Gamma _{\sigma \gamma }^{\mu }\Gamma _{\beta \alpha }^{\sigma }\right)\\[6pt]G_{\mu \nu }&=[S3]\times {\frac {8\pi G}{c^{4}}}T_{\mu \nu }\end{aligned}}$

The third sign above is related to the choice of convention for the Ricci tensor:

$R_{\mu \nu }=[S2]\times [S3]\times {R^{\alpha }}_{\mu \alpha \nu }$

With these definitions Misner, Thorne, and Wheeler classify themselves as (+ + +), whereas Weinberg (1972) and Peacock (1994) are (+ − −), Peebles (1980) and Efstathiou et al. (1990) are (− + +), Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) are (− + −).

Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative

$R_{\mu \nu }-{\tfrac {1}{2}}Rg_{\mu \nu }-\Lambda g_{\mu \nu }=-{\frac {8\pi G}{c^{4}}}T_{\mu \nu }.$

The sign of the (very small) cosmological term would change in both these versions, if the (+ − − −) metric sign convention is used rather than the MTW (− + + +) metric sign convention adopted here.