View source for User:JMvanDijk/Sandbox 16/Box1

==Mechanics==


===Generalized mechanics===
{{main article|Analytical mechanics|Lagrangian mechanics|Hamiltonian mechanics|}}

{| class="wikitable"
|-
! scope="col" style="width:100px;"| Quantity (common name/s)
! scope="col" style="width:100px;"| (Common) symbol/s
! scope="col" style="width:300px;"| Defining equation
! scope="col" style="width:125px;"| SI units
! scope="col" style="width:100px;"| Dimension
|-
|[[Generalized coordinates]]
|| ''q, Q''
||
|| varies with choice
|| varies with choice
|-
|[[Generalized velocities]]
|| <math>\dot{q},\dot{Q} </math>
|| <math>\dot{q}\equiv \mathrm{d}q/\mathrm{d}t </math>
|| varies with choice
|| varies with choice
|-
|[[Canonical coordinates|Generalized momenta]]
|| ''p, P''
||<math> p = \partial L /\partial \dot{q} </math>
|| varies with choice
|| varies with choice
|-
| [[Lagrangian mechanics|Lagrangian]]
|| ''L''
|| <math> L(\mathbf{q},\mathbf{\dot{q}},t) = T(\mathbf{\dot{q}})-V(\mathbf{q},\mathbf{\dot{q}},t) </math>

where <math> \mathbf{q}=\mathbf{q}(t) </math> and '''p''' = '''p'''(''t'') are vectors of the generalized coords and momenta, as functions of time
|| J
|| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
| [[Hamiltonian mechanics|Hamiltonian]]
|| ''H''
|| <math> H(\mathbf{p},\mathbf{q},t) = \mathbf{p}\cdot\mathbf{\dot{q}} - L(\mathbf{q},\mathbf{\dot{q}},t) </math>
|| J
|| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
| [[Action (physics)|Action]], Hamilton's principal function
|| ''S'', <math> \scriptstyle{\mathcal{S}} </math>
|| <math> \mathcal{S} = \int_{t_1}^{t_2} L(\mathbf{q},\mathbf{\dot{q}},t) \mathrm{d}t </math>
|| J s
|| [M][L]<sup>2</sup>[T]<sup>−1</sup>
|}

==Electromagnetism==

===Electric fields===

General Classical Equations

:{| class="wikitable"
|-
! scope="col" width="200" | Physical situation 
! scope="col" width="10" | Equations
|-
!Electric potential gradient and field
| <math> \mathbf{E} = - \nabla V </math>

<math> \Delta V = -\int_{r_1}^{r_2} \mathbf{E} \cdot d\mathbf{r}\,\!</math>
|-
!Point charge
| <math> \mathbf{E} = \frac{q}{4 \pi \epsilon_0 \left | \mathbf{r} \right |^2 }\mathbf{\hat{r}} \,\!</math>
|-
!At a point in a local array of point charges
| <math>\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 \,\!</math>
|-
!At a point due to a continuum of charge
| <math> \mathbf{E} = \frac{1}{4\pi\epsilon_0} \int_V \frac{\mathbf{r} \rho \mathrm{d}V}{\left | \mathbf{r} \right |^3} \,\!</math>
|-
!Electrostatic torque and potential energy due to non-uniform fields and dipole moments
| <math> \boldsymbol{\tau} = \int_V \mathrm{d} \mathbf{p} \times \mathbf{E} </math>

<math> U = \int_V  \mathrm{d} \mathbf{p} \cdot \mathbf{E} </math>
|-
|}

===Magnetic fields and moments===

{{See also|Magnetic moment}}

General classical equations

:{| class="wikitable"
|-
! scope="col" width="200" | Physical situation 
! scope="col" width="10" | Equations
|-
!Magnetic potential,  EM vector potential
| <math> \mathbf{B} = \nabla \times \mathbf{A} </math> 
|-
!Due to a magnetic moment
| <math> \mathbf{A} = \frac{\mu_0}{4\pi}\frac{\mathbf{m}\times\mathbf{r}}{\left | \mathbf{r} \right |^3} </math>

<math>\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)</math>
|-
!Magnetic moment due to a current distribution
| <math> \mathbf{m} = \frac{1}{2}\int_V \mathbf{r}\times\mathbf{J} \mathrm{d} V </math>
|-
!Magnetostatic torque and potential energy due to non-uniform fields and dipole moments
| <math> \boldsymbol{\tau} = \int_V \mathrm{d} \mathbf{m} \times \mathbf{B} </math>

<math> U = \int_V \mathrm{d} \mathbf{m} \cdot \mathbf{B} </math>
|-
|}


===Formulation in SI units convention===

{| class="wikitable" style="text-align: center;"
|-
! scope="col" style="width: 15em;" | Name
! scope="col" | [[Integral]] equations
! scope="col" | [[Partial differential equation|Differential]] equations
|-
|  [[Gauss's law]]
| {{oiint
   | intsubscpt=<math>{\scriptstyle\partial \Omega }</math>
   | integrand=<math>\mathbf{E}\cdot\mathrm{d}\mathbf{S} = \frac{1}{\varepsilon_0} \iiint_\Omega \rho \,\mathrm{d}V</math>
   }}
| <math>\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}</math>
|-
| [[Gauss's law for magnetism]]
| {{oiint
   | intsubscpt = <math>{\scriptstyle \partial \Omega }</math>
   | integrand  = <math>\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 0</math>
  }}
| <math>\nabla \cdot \mathbf{B} = 0</math>
|-
| Maxwell–Faraday equation ([[Faraday's law of induction]])
|<math>\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{l}  = - \operatorname{\frac{d}{dt}} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} </math>
| <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
|-
| [[Ampère's circuital law]] (with Maxwell's addition)
| <math>
\begin{align}
\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{align}
</math>
| <math>\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right) </math>
|}


===Formulation in Gaussian units convention===
{{main|Gaussian units}}

The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbing [[dimensional analysis|dimensioned]] factors of {{math|''ε''<sub>0</sub>}} and {{math|''μ''<sub>0</sub>}} into the units of calculation, by convention. With a corresponding change in convention for the [[Lorentz force]] law this yields the same physics, i.e. trajectories of charged particles, or [[work (physics)|work]] done by an electric motor. These definitions are often preferred in theoretical and high energy physics where it is natural to take the electric and magnetic field with the same units, to simplify the appearance of the [[electromagnetic tensor]]: the Lorentz covariant object unifying electric and magnetic field would then contain components with uniform unit and dimension.<ref name=Jackson>{{cite book|author=J. D. Jackson|title=Classical Electrodynamics|edition=3rd|isbn=978-0-471-43132-9|date=1975-10-17|url=https://archive.org/details/classicalelectro00jack_0}}</ref>{{rp|vii}}  Such modified definitions are conventionally used with the Gaussian ([[Centimetre gram second system of units#Alternate derivations of CGS units in electromagnetism|CGS]]) units. Using these definitions and conventions, colloquially "in Gaussian units",<ref name=Littlejohn>
{{cite web
 | url=http://bohr.physics.berkeley.edu/classes/221/0708/notes/emunits.pdf
 | title=Gaussian, SI and Other Systems of Units in Electromagnetic Theory
 | work=Physics 221A, University of California, Berkeley lecture notes
 | author=Littlejohn, Robert|authorlink1=Robert Grayson Littlejohn
 | date=Fall 2007
 | accessdate=2008-05-06
}}</ref>
the Maxwell equations become:<ref name=Griffiths>{{cite book
 | author=David J Griffiths
 | title=Introduction to electrodynamics
 | year=1999
 | edition=Third
 | pages=[https://archive.org/details/introductiontoel00grif_0/page/559 559–562]
 | publisher=Prentice Hall
 | isbn=978-0-13-805326-0
 | url=https://archive.org/details/introductiontoel00grif_0/page/559
 }}</ref>

{| class="wikitable" style="text-align: center;"
|-
! scope="col" style="width: 15em;" | Name
! scope="col" | Integral equations
! scope="col" | Differential equations
|-
| Gauss's law
|{{oiint
   | intsubscpt=<math>{\scriptstyle\partial \Omega }</math>
   | integrand=<math>\mathbf{E}\cdot\mathrm{d}\mathbf{S} = 4\pi \iiint_\Omega \rho \,\mathrm{d}V</math>
   }}
| <math>\nabla \cdot \mathbf{E} = 4\pi\rho </math>
|-
| Gauss's law for magnetism
|{{oiint
   | intsubscpt = <math>{\scriptstyle \partial \Omega }</math>
   | integrand  = <math>\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 0</math>
  }}
| <math>\nabla \cdot \mathbf{B} = 0</math>
|-
| Maxwell–Faraday equation (Faraday's law of induction)
| <math>\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}</math>
| <math>\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}</math>
|-
| Ampère's circuital law (with Maxwell's addition)
| <math>
\begin{align}
\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{align}
</math>
| <math>\nabla \times \mathbf{B} = \frac{1}{c}\left( 4\pi\mathbf{J} + \frac{\partial \mathbf{E}}{\partial t}\right)</math>
|}

The equations are particularly readable when length and time are measured in compatible units like seconds and lightseconds i.e. in units such that c = 1 unit of length/unit of time. Ever since 1983 (see [[International System of Units]]), metres and seconds are compatible except for historical legacy since ''by definition'' c = 299 792 458 m/s (≈ 1.0 feet/nanosecond).

Further cosmetic changes, called rationalisations, are possible by absorbing factors of {{math|4''π''}} depending on whether we want [[Coulomb's law]] or [[Gauss's law]] to come out nicely, see [[Lorentz-Heaviside units]] (used mainly in [[particle physics]]). In [[theoretical physics]] it is often useful to choose units such that [[Planck's constant]], the [[elementary charge]], and even [[Newton's constant]] are 1. See [[Planck units]].


===Alternative formulations===
<!--In the table below: Lorenz is the correct name (not Lorentz).-->

{{For|an overview|Mathematical descriptions of the electromagnetic field}}
{{For|the equations in [[quantum field theory]]|quantum electrodynamics}}

Following is a summary of some of the numerous other mathematical formalisms to write the microscopic Maxwell's equations, with the columns separating the two homogeneous Maxwell equations from the two inhomogeneous ones involving charge and current. Each formulation has versions directly in terms of the electric and magnetic fields, and indirectly in terms of the [[electrical potential]] {{math|''φ''}} and the [[vector potential]] {{math|'''A'''}}. Potentials were introduced as a convenient way to solve the homogeneous equations, but it was thought that all observable physics was contained in the electric and magnetic fields (or relativistically, the Faraday tensor). The potentials play a central role in quantum mechanics, however, and act quantum mechanically with observable consequences even when the electric and magnetic fields vanish ([[Aharonov–Bohm effect]]).

Each table describes one formalism. See the [[Mathematical descriptions of the electromagnetic field|main article]] for details of each formulation.  SI units are used throughout.

{|class="wikitable" style="text-align: center;"
|+ [[Vector calculus]]
! scope="column" | Formulation
! scope="column" | Homogeneous equations
! scope="column" | Inhomogeneous equations
|-
! Fields
3D Euclidean space + time
| <math>\begin{align}
                                            \nabla\cdot\mathbf{B} = 0 
\end{align}</math>
<math>\begin{align}
  \nabla\times\mathbf{E} + \frac{\partial \mathbf{B}}{\partial t} = 0
\end{align}</math>
| <math>\begin{align}
                                                         \nabla\cdot\mathbf{E} &= \frac{\rho}{\varepsilon_0} 
\end{align}</math>
<math>\begin{align}
  \nabla\times\mathbf{B} - \frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t} &= \mu_0\mathbf{J}
\end{align}</math>
|-
! Potentials (any [[Gauge theory|gauge]])
3D Euclidean space + time
| <math>\begin{align}
  \mathbf B &= \mathbf \nabla \times \mathbf A 
\end{align}</math>
<math>\begin{align}
  \mathbf E &= - \mathbf \nabla \varphi - \frac{\partial \mathbf A}{\partial t}
\end{align}</math>
| <math>\begin{align}
  -\nabla^2 \varphi - \frac{\partial}{\partial t} \left( \mathbf \nabla \cdot \mathbf A \right) &= \frac{\rho}{\varepsilon_0} 
\end{align}</math>
<math>\begin{align}
  \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{align}</math>
|-
! Potentials ([[Lorenz gauge]])
3D Euclidean space + time
| <math>\begin{align}
                       \mathbf B &= \mathbf \nabla \times \mathbf A \\
\end{align}</math>
<math>\begin{align}
                       \mathbf E &= - \mathbf \nabla \varphi - \frac{\partial \mathbf A}{\partial t} \\
\end{align}</math>

<math>\begin{align}
  \mathbf \nabla \cdot \mathbf A &=  - \frac{1}{c^2}\frac{\partial \varphi}{\partial t} \\
\end{align}</math>
| <math>\begin{align}
   \left( -\nabla^2 +\frac{1}{c^2}\frac{\partial^2}{\partial t^2} \right) \varphi &= \frac{\rho}{\varepsilon_0} 
\end{align}</math>
<math>\begin{align}
   \left( -\nabla^2 +\frac{1}{c^2}\frac{\partial^2}{\partial t^2} \right) \mathbf A &= \mu_0 \mathbf J
\end{align}</math>
|}

{|class="wikitable" style="text-align: center;"
|+ [[Tensor calculus]]
! scope="column" | Formulation
! scope="column" | Homogeneous equations
! scope="column" | Inhomogeneous equations
|-
!Fields
space + time

spatial metric independent of time
|<math>
\begin{align}
                 \partial_{[i} B_{jk]} &=  \\
                  \nabla_{[i} B_{jk]} &= 0 \\
          \partial_{[i} E_{j]} + \frac{\partial B_{ij}}{\partial t} &=  \\
          \nabla_{[i} E_{j]} + \frac{\partial B_{ij}}{\partial t} &= 0
\end{align}
</math>
|<math>
\begin{align}
             \frac{1}{\sqrt{h}}  \partial_i  \sqrt{h}   E^i &=  \\
             \nabla_i E^i &= \frac{\rho}{\epsilon_0} \\
            -\frac{1}{\sqrt{h}}\partial_i \sqrt{h} B^{ij} - \frac{1}{c^2} \frac{\partial}{\partial t}  E^j &= & \\
             -\nabla_iB^{ij}  - \frac{1}{c^2} \frac{\partial E^j}{\partial t}  &= \mu_0 J^j  \\
\end{align}
</math>
|-
!Potentials
space (with topological restrictions) + time

spatial metric independent of time
|<math>
\begin{align}
               B_{ij} &= \partial_{[i} A_{j]}   \\
                        &= \nabla_{[i} A_{j]}   
\end{align}</math>
<math>\begin{align}
               E_i    &= -\frac{\partial A_i}{\partial t}  - \partial_i \phi \\
                        &= -\frac{\partial A_i}{\partial t}  - \nabla_i \phi \\
\end{align}
</math>
|<math>
\begin{align}
  -\frac{1}{\sqrt{h}} \partial_i \sqrt{h} \left( \partial^i \phi  + \frac{\partial A^i}{\partial t} \right)  &= \\
  -\nabla_i \nabla^i \phi   -   \frac{\partial}{\partial t} \nabla_i  A^i &= \frac{\rho}{\epsilon_0} \\
  - \frac{1}{\sqrt{h}} \partial_i \left( \sqrt{h}h^{im}h^{jn} \partial_{[m} A_{n]} \right) + \frac{1}{c^2} \frac{\partial}{\partial t} \left( \frac{\partial A^j}{\partial t}  + \partial^j \phi \right) &=  \\
  - \nabla_i \nabla^i  A^j  + \frac{1}{c^2} \frac{\partial^2 A^j}{\partial t^2}  + R_i^j A^i + \nabla^j \left( \nabla_i A^i  + \frac{1}{c^2} \frac{\partial \phi}{\partial t} \right) &= \mu_0 J^j\\
\end{align}
</math> 
|-      
!Potentials (Lorenz gauge)
space (with topological restrictions) + time

spatial metric independent of time
|<math>
\begin{align}
               B_{ij} &= \partial_{[i} A_{j]}   \\
                        &= \nabla_{[i} A_{j]}   \\
               E_i    &= -\frac{\partial A_i}{\partial t}  - \partial_i \phi \\
                        &= -\frac{\partial A_i}{\partial t}  - \nabla_i \phi \\
              \nabla_i A^i &= - \frac{1}{c^2} \frac{\partial\phi}{\partial t}\\
\end{align}
</math>
|<math>
\begin{align}
                -\nabla_i \nabla^i \phi   + \frac{1}{c^2}  \frac{\partial^2 \phi }{\partial t^2} &= \frac{\rho}{\epsilon_0} \\
               - \nabla_i \nabla^i  A^j  + \frac{1}{c^2} \frac{\partial^2 A^j}{\partial t^2}  + R_i^j A^i  &= \mu_0 J^j\\
\end{align}
</math> 
|}

{|class="wikitable" style="text-align: center;"
|+ [[Exterior calculus|Differential forms]]
! scope="column" | Formulation
! scope="column" | Homogeneous equations
! scope="column" | Inhomogeneous equations
|-
!Fields
Any space + time
|<math>\begin{align}
d B &= 0 \\
d E + \frac{\partial B}{\partial t}  &= 0\\
\end{align}
</math>
|<math>\begin{align}
d{*}E = \frac{\rho}{\epsilon_0}  \\
d{*}B  - \frac{1}{c^2} \frac{\partial{*}E}{\partial t}  =  {\mu_0} J \\
\end{align}
</math>
|-
!Potentials (any gauge)
Any space (with topological restrictions) + time
|<math>\begin{align}
B &= d A \\
E &= -d\phi - \frac{\partial A}{\partial t} \\
\end{align}
</math>
|<math>\begin{align}
-d{*}\!\left( d\phi + \frac{\partial A}{\partial t} \right) &= \frac{\rho}{\epsilon_0} \\
d{*}d A + \frac{1}{c^2} \frac{\partial}{\partial t}{*}\!\left( d\phi + \frac{\partial A}{\partial t} \right) &= \mu_0 J \\
\end{align}
</math>
|-
!Potential (Lorenz Gauge)
Any space (with topological restrictions) + time

spatial metric independent of time
|<math>\begin{align}
B   &= d A \\
E   &= -d\phi - \frac{\partial A}{\partial t} \\
d{*}A &= -{*}\frac{1}{c^2} \frac{\partial \phi}{\partial t} \\ 
\end{align}
</math>
|<math>\begin{align}
{*}\!\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{align}
</math>
|-
<!-- Please don't re-add a geometric calculus version, the table is long enough as it is. For an overview article, the geometric calculus version is not mainstream enough and does not give enough additional physical insight to warrant inclusion in this table. Also it is just one click away as an additional alternative formulation -->
|}


=== Special Relativity: The metric and four-vectors ===

{{main|metric tensor|four-vectors}}

In what follows, bold sans serif is used for [[4-vector]]s while normal bold roman is used for ordinary 3-vectors.

;[[Inner product]] (i.e. notion of [[length]]):

:<math> \boldsymbol{\mathsf{a}} \cdot \boldsymbol{\mathsf{b}} =\eta (\boldsymbol{\mathsf{a}} , \boldsymbol{\mathsf{b}})</math>

where <math>\eta</math> is known as the [[metric tensor]].  In special relativity, the metric tensor is the [[Minkowski metric]]:

:<math>\eta = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0  & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}</math>

;[[Space-time interval]]

:<math>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}</math>

In the above, ''ds''<sup>2</sup> is known as the spacetime interval.  This inner product is invariant under the Lorentz transformation, that is,

:<math> \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}} )</math>

The sign of the metric and the placement of the ''ct'', ''ct''', ''cdt'', and ''cdt&prime;'' 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&prime;'', ''dt'', and ''dt&prime;'' with ''ct'', ''ct''', ''cdt'', and ''cdt&prime;'', which has the dimensions of distance.  So:

:<math>x' = \gamma x - \gamma \beta c t \,</math>
:<math>y' = y \,</math>
:<math>z' = z \,</math>
:<math>c t' = \gamma c t - \gamma \beta x \,</math>

then in matrix form:

:<math>\begin{pmatrix} c t' \\ 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} c t \\ x \\ y \\ z \end{pmatrix}</math>

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:

:<math>\boldsymbol{\mathsf{a}}' = \Lambda \boldsymbol{\mathsf{a}}</math>

In the above, <math>\boldsymbol{\mathsf{a}}'</math> and <math>\boldsymbol{\mathsf{a}}</math> are the four-vector and the transformed four-vector, respectively, and Λ is the transformation matrix, which, for a given transformation is the same for all four-vectors one might want to transform.  So <math>\boldsymbol{\mathsf{a}}'</math> 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 [[spinor]]s.

===4-vectors and frame-invariant results===

Invariance and unification of physical quantities both arise from [[four-vector]]s.<ref>Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Manchester Physics Series, John Wiley & Sons, 2009, {{ISBN|978-0-470-01460-8}}</ref> 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.

{| class="wikitable"
! scope="col" width="100" | Property/effect
! scope="col" width="200" | 3-vector
! scope="col" width="200" | 4-vector
! scope="col" width="200" | Invariant result
|-
![[Space-time]] [[Event (relativity)|events]]
|3-position: '''r''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>) <br />
<math> \mathbf{r} \cdot \mathbf{r} \equiv r^2 \equiv x_1^2 + x_2^2 + x_3^2 \,\!</math> 
|4-position: '''X''' = (''ct'', ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)
|<math> \boldsymbol{\mathsf{X}} \cdot \boldsymbol{\mathsf{X}} = \left ( c \tau \right )^2 \,\!</math> <br />
<math> \begin{align} & \left ( c t \right )^2 - \left ( x_1^2 + x_2^2 + x_3^2 \right ) \\
& = \left ( c t \right )^2 - r^2 \\
& = -\chi^2 = \left ( c\tau \right )^2 \end{align} \,\!</math><br />
''τ'' = proper time <br />
''χ'' = proper distance <br />
|-
![[invariant mass|Momentum-energy invariance]]
| 
<math> \mathbf{p} = \gamma m\mathbf{u} \,\!</math>

3-momentum: '''p''' = (''p''<sub>1</sub>, ''p''<sub>2</sub>, ''p''<sub>3</sub>) <br />
<math> \mathbf{p} \cdot \mathbf{p} \equiv p^2 \equiv p_1^2 + p_2^2 + p_3^2 \,\!</math> 
| 4-momentum: '''P''' = (''E/c'', ''p''<sub>1</sub>, ''p''<sub>2</sub>, ''p''<sub>3</sub>) <br />
<math> \boldsymbol{\mathsf{P}} = m \boldsymbol{\mathsf{U}} \,\!</math>
|<math> \boldsymbol{\mathsf{P}} \cdot \boldsymbol{\mathsf{P}} = \left ( m c \right )^2 \,\!</math> 
<math> \begin{align} & \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{align} \,\!</math>

which leads to:<br />
<math> E^2 = \left ( pc \right )^2 + \left ( mc^2 \right )^2 \,\!</math>

''E'' = total energy <br />
''m'' = invariant mass <br />
|- style="border-top: 3px solid;"
!Velocity
| 3-velocity: '''u''' = (''u''<sub>1</sub>, ''u''<sub>2</sub>, ''u''<sub>3</sub>) <br />
<math> \mathbf{u} = \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} \,\!</math> 
| 4-velocity: '''U''' = (''U''<sub>0</sub>, ''U''<sub>1</sub>, ''U''<sub>2</sub>, ''U''<sub>3</sub>) <br />
<math>\boldsymbol{\mathsf{U}} =\frac{\mathrm{d}\boldsymbol{\mathsf{X}} }{\mathrm{d} \tau} = \gamma \left( c, \mathbf{u} \right) </math>
|<math> \boldsymbol{\mathsf{U}} \cdot \boldsymbol{\mathsf{U}} = c^2 \,\!</math><br />
|-
!Acceleration
| 3-acceleration: '''a''' = (''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>) <br />
<math> \mathbf{a} = \frac{\mathrm{d}\mathbf{u}}{\mathrm{d}t} \,\!</math>
| 4-acceleration: '''A''' = (''A''<sub>0</sub>, ''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>) <br />
<math>\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)</math>
|<math> \boldsymbol{\mathsf{A}} \cdot \boldsymbol{\mathsf{U}} = 0 \,\!</math><br />
|-
!Force
| 3-force: '''f''' = (''f''<sub>1</sub>, ''f''<sub>2</sub>, ''f''<sub>3</sub>) <br />
<math> \mathbf{f} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} \,\!</math>
| 4-force: '''F''' = (''F''<sub>0</sub>, ''F''<sub>1</sub>, ''F''<sub>2</sub>, ''F''<sub>3</sub>) <br />
<math>\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)</math>
|<math> \boldsymbol{\mathsf{F}} \cdot \boldsymbol{\mathsf{U}} = 0 \,\!</math><br />
|-
|}


==Quantum Mechanics==

===Wave–particle duality and time evolution===

{| class="wikitable"
|-
! scope="col" width="150" | Property or effect 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equation 
|-
![[Planck–Einstein equation]] and [[de Broglie wavelength]] relations
|<div class="plainlist">
*'''P''' = (''E/c'', '''p''') is the [[four-momentum]],
*'''K''' = (ω/''c'', '''k''') is the [[four-vector#Four-wavevector|four-wavevector]],
*''E'' = energy of particle
*ω = 2π''f'' is the angular frequency and frequency of the particle
*''ħ'' = ''h''/2π are the [[Planck constant]]s
*''c'' = [[speed of light]]
</div>
|<math>\mathbf{P} = (E/c, \mathbf{p}) = \hbar(\omega /c ,\mathbf{k}) = \hbar \mathbf{K}</math>
|-
![[Schrödinger equation]]
|<div class="plainlist">
*''Ψ'' = [[wavefunction]] of the system
*''Ĥ'' = [[Hamiltonian (quantum mechanics)|Hamiltonian operator]],
*''E'' = energy eigenvalue of system
*''i'' is the [[imaginary unit]]
*''t'' = time
</div>
|General time-dependent case:
<math>i\hbar\frac{\partial}{\partial t} \Psi = \hat{H}\Psi</math>

Time-independent case:
<math>\hat{H}\Psi = E\Psi</math>
|-
![[Heisenberg picture|Heisenberg equation]]
|<div class="plainlist">
*''Â'' = operator of an observable property
*[ ] is the [[commutator]]
*<math>\langle \, \rangle</math> denotes the average
</div>
|<math>\frac{d}{dt}\hat{A}(t)=\frac{i}{\hbar}[\hat{H},\hat{A}(t)]+\frac{\partial \hat{A}(t)}{\partial t},</math>
|-
!Time evolution in Heisenberg picture ([[Ehrenfest theorem]])
|<div class="plainlist">
*''m'' = [[mass]],
*''V'' = [[potential energy]], 
*'''r''' = [[position space|position]],
*'''p''' = [[momentum]],
</div>
of a particle.
|<math>\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</math>

For momentum and position;

<math>m\frac{d}{dt}\langle \mathbf{r}\rangle = \langle \mathbf{p} \rangle</math>

<math>\frac{d}{dt}\langle \mathbf{p}\rangle = -\langle \nabla V \rangle</math>
|-

|-
|}

====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]].

{| class="wikitable"
|-
| scope="col" style="width:10px;"|
| scope="col" style="width:450px;"| '''One particle''' 
| scope="col" style="width:450px;"| ''' ''N'' particles'''
|-
|rowspan="3"| '''One dimension''' 
| <math> \hat{H} = \frac{\hat{p}^2}{2m} + V(x) = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2} + V(x)</math>
| <math> \begin{align}\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{align}</math>

where the position of particle ''n'' is ''x<sub>n</sub>''.
|-
| <math> E\Psi = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\Psi + V\Psi </math>
| <math> E\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\frac{\partial^2}{\partial x_n^2}\Psi + V\Psi \, .</math>
|-
|<math> \Psi(x,t)=\psi(x) e^{-iEt/\hbar} \, . </math>

There is a further restriction — the solution must not grow at infinity, so that it has either a finite '''''L''<sup>2</sup>-norm''' (if it is a [[bound state]]) or a slowly diverging norm (if it is part of a [[continuum (theory)|continuum]]):<ref>
{{cite book
 |last=Feynman |first=R.P.
 |last2=Leighton |first2=R.B.
 |last3=Sand |first3=M.
 |year=1964
 |chapter=Operators
 |title=[[The Feynman Lectures on Physics]]
 |volume=3 |pages=20–7
 |publisher=[[Addison-Wesley]]
 |isbn=0-201-02115-3
}}</ref>
<math>\| \psi \|^2 = \int |\psi(x)|^2\, dx.\,</math>
| <math> \Psi = e^{-iEt/\hbar}\psi(x_1,x_2\cdots x_N) </math>

for non-interacting particles

<math> \Psi = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(x_n) \, , \quad V(x_1,x_2,\cdots x_N) = \sum_{n=1}^N V(x_n) \, .</math>
|-
|rowspan="3"| '''Three dimensions''' 
| <math> \begin{align}\hat{H} & = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r}) \\
& = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) 
\end{align}</math>

where the position of the particle is '''r''' = (''x, y, z'').
| <math> \begin{align} \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{align}</math>

where the position of particle ''n'' is '''r''' ''<sub>n</sub>'' = (''x<sub>n</sub>'', ''y<sub>n</sub>'', ''z<sub>n</sub>''), and the Laplacian for particle ''n'' using the corresponding position coordinates is

<math>\nabla_n^2=\frac{\partial^2}{{\partial x_n}^2} + \frac{\partial^2}{{\partial y_n}^2} + \frac{\partial^2}{{\partial z_n}^2}</math>
|-
| <math> E\Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi </math>
| <math> E\Psi = -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi + V\Psi </math>
|-
|<math> \Psi = \psi(\mathbf{r}) e^{-iEt/\hbar} </math>
| <math> \Psi = e^{-iEt/\hbar}\psi(\mathbf{r}_1,\mathbf{r}_2\cdots \mathbf{r}_N) </math>

for non-interacting particles

<math> \Psi = e^{-i{E t/\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) </math>
|}

====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.

{| class="wikitable"
|-
| scope="col" style="width:10px;"|
| scope="col" style="width:450px;"| '''One particle''' 
| scope="col" style="width:450px;"| ''' ''N'' particles'''
|-
|rowspan="3"| '''One dimension''' 
| <math> \hat{H} = \frac{\hat{p}^2}{2m} + V(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)</math>
| <math> \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) 
</math>

where the position of particle ''n'' is ''x<sub>n</sub>''.
|-
| <math> i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\Psi + V\Psi </math>
| <math> 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 \, .</math>
|-
| <math> \Psi = \Psi(x,t) </math>
| <math> \Psi = \Psi(x_1,x_2\cdots x_N,t) </math>
|-
|rowspan="3"| '''Three dimensions''' 
| <math> \begin{align}\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{align}</math>
| <math> \begin{align} \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{align}</math> 
|-
| <math> i\hbar\frac{\partial}{\partial t}\Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi </math>
| <math> 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 </math>

This last equation is in a very high dimension,<ref>
{{cite book
 |last=Shankar |first=R.
 |year=1994
 |title=Principles of Quantum Mechanics
 |page=141
 |publisher=[[Kluwer Academic]]/[[Plenum Publishers]]
 |isbn=978-0-306-44790-7
}}</ref> so the solutions are not easy to visualize. 
|-
| <math> \Psi = \Psi(\mathbf{r},t) </math>
| <math> \Psi = \Psi(\mathbf{r}_1,\mathbf{r}_2,\cdots\mathbf{r}_N,t) </math>
|}

===Photoemission===

{| class="wikitable"
!Property/Effect
!Nomenclature
!Equation
|-
![[Photoelectric]] equation
|<div class="plainlist">
*''K''<sub>max</sub> = Maximum kinetic energy of ejected electron (J)
*''h'' = [[Planck's constant]]
*''f'' = frequency of incident photons (Hz = s<sup>−1</sup>)
*''φ'', ''Φ'' = [[Work function]] of the material the photons are incident on (J)
</div>
|<math>K_\mathrm{max} = hf - \Phi\,\!</math>
|-
|[[Photoelectric effect|Threshold frequency]] and 
| <div class="plainlist">
*''φ'', ''Φ'' = Work function of the material the photons are incident on (J)
*''f''<sub>0</sub>, ''ν''<sub>0</sub> = Threshold frequency (Hz = s<sup>−1</sup>)
</div>
| Can only be found by experiment.

The De Broglie relations give the relation between them:

<math>\phi = hf_0\,\!</math>
|-
![[Photon]] momentum
| <div class="plainlist">
*''p'' = momentum of photon (kg m s<sup>−1</sup>)
*''f'' = frequency of photon (Hz = s<sup>−1</sup>)
*''λ'' = wavelength of photon (m)
</div>
|
The De Broglie relations give:

<math>p = hf/c = h/\lambda\,\!</math>
|-
|}

===Quantum uncertainty===

{| class="wikitable"
|-
! scope="col" width="100" | Property or effect 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equation 
|-
![[Heisenberg's uncertainty principle]]s
|<div class="plainlist">
*''n'' = number of photons
*''φ'' = wave phase
*[, ] = [[commutator]]
</div>

|Position-momentum

<math>\sigma(x) \sigma(p) \ge \frac{\hbar}{2} \,\!</math>

Energy-time
<math>\sigma(E) \sigma(t) \ge \frac{\hbar}{2} \,\!</math>

Number-phase
<math>\sigma(n) \sigma(\phi) \ge \frac{\hbar}{2} \,\!</math>
|-
!Dispersion of observable
|<div class="plainlist">
*''A'' = observables (eigenvalues of operator)
</div>
|
<math>\begin{align}
\sigma(A)^2 & = \langle(A-\langle A \rangle)^2\rangle \\
& = \langle A^2 \rangle - \langle A \rangle^2
\end{align}</math>
|-
!General uncertainty relation
|<div class="plainlist">
*''A'', ''B'' = observables (eigenvalues of operator)
</div>
|<math>\sigma(A)\sigma(B) \geq \frac{1}{2}\langle i[\hat{A}, \hat{B}] \rangle</math>
|-
|}

{| class="wikitable"
|+Probability Distributions
! scope="col" width="100" | Property or effect 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equation 
|-
![[Density of states]]
|
|<math>N(E) = 8\sqrt{2}\pi m^{3/2}E^{1/2}/h^3\,\!</math>
|-
! [[Fermi–Dirac distribution]] (fermions)
|<div class="plainlist">
*''P''(''E<sub>i</sub>'') = probability of energy ''E<sub>i</sub>''
*''g''(''E<sub>i</sub>'') = degeneracy of energy ''E<sub>i</sub>'' (no of states with same energy)
*μ = chemical potential
</div>
|<math>P(E_i) = g(E_i)/(e^{(E-\mu)/kT}+1)\,\!</math>
|-
! [[Bose–Einstein distribution]] (bosons) 
|
|<math>P(E_i) = g(E_i)/(e^{(E_i-\mu)/kT}-1)\,\!</math>
|-
|}

===Angular momentum===

{{main|angular momentum operator|quantum number}}

{| class="wikitable"
|-
! scope="col" width="100" | Property or effect 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equation 
|-
![[Angular momentum]] [[quantum number]]s
|<div class="plainlist">
*''s'' = [[spin quantum number]]
*''m<sub>s</sub>'' = spin magnetic quantum number
*''ℓ'' = [[Azimuthal quantum number]]
*''m<sub>ℓ</sub>'' = azimuthal magnetic quantum number
*''m<sub>j</sub>'' = total angular momentum magnetic quantum number
*''j'' = total angular momentum quantum number
</div>

|Spin projection:
<math>m_s \in \{-s,-s+1\cdots s-1,s\}\,\!</math>

Orbital:
<math>m_\ell \in \{-\ell,-\ell+1\cdots \ell-1,\ell\}\,\!</math><br />
<math>m_\ell \in \{0 \cdots n-1\}\,\!</math>

Total:
<math>\begin{align}& j = \ell +s \\
& j \in \{|\ell-s|,|\ell-s|+1 \cdots |\ell+s|-1,|\ell+s| \} \\
\end{align}\,\!</math>
|-
![[Angular momentum]] magnitudes
|angular momementa:<div class="plainlist">
*''S'' = Spin,
*''L'' = orbital,
*''J'' = total
</div>

|Spin magnitude:

<math>|\mathbf{S}| = \hbar\sqrt{s(s+1)}\,\!</math>

Orbital magnitude:
<math>|\mathbf{L}| = \hbar\sqrt{\ell(\ell+1)}\,\!</math>

Total magnitude:
<math>\mathbf{J} = \mathbf{L} + \mathbf{S}\,\!</math>

<math>|\mathbf{J}| = \hbar\sqrt{j(j+1)}\,\!</math>
|-
![[Angular momentum]] components
|
|Spin:
<math>S_z = m_s \hbar\,\!</math>

Orbital:
<math>L_z = m_\ell \hbar\,\!</math>
|-
|}

;Magnetic moments

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

{| class="wikitable"
|-
! scope="col" width="100" | Property or effect 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equation 
|-
![[Electron magnetic dipole moment#Orbital magnetic dipole moment|orbital magnetic dipole moment]]
|<div class="plainlist">
*''e'' = [[electron charge]]
*''m<sub>e</sub>'' = electron rest mass
*'''L''' = electron orbital angular momentum
*''g<sub>{{ell}}</sub>'' = orbital [[Landé g-factor]]
*μ<sub>''B''</sub> = [[Bohr magneton]]
</div>
|<math>\boldsymbol{\mu}_\ell = -e\mathbf{L}/2m_e = g_\ell \frac{\mu_B}{\hbar} \mathbf{L}\,\!</math>
z-component:
<math>\mu_{\ell,z} = -m_\ell\mu_B\,\!</math>
|-
![[Electron magnetic dipole moment#Orbital magnetic dipole moment|spin magnetic dipole moment]]
|<div class="plainlist">
*'''S''' = electron spin angular momentum
*''g<sub>s</sub>'' = spin [[Landé g-factor]]
</div>
|<math>\boldsymbol{\mu}_s = -e\mathbf{S}/m_e = g_s \frac{\mu_B}{\hbar} \mathbf{S}\,\!</math>

z-component:
<math>\mu_{s,z} = -e S_z/m_e = g_seS_z/2m_e\,\!</math>
|-
![[magnetic moment|dipole moment]] potential
|<div class="plainlist">
*''U'' = potential energy of dipole in field
</div> 
|<math>U = -\boldsymbol{\mu}\cdot\mathbf{B} = -\mu_z B\,\!</math>
|-
|}

===The Hydrogen atom===
{{main|Hydrogen atom}}

{| class="wikitable"
|-
! scope="col" width="100" | Property or effect 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equation 
|-
! Energy level :p≈
|<div class="plainlist">
*''E<sub>n</sub>'' = energy [[eigenvalue]]
*''n'' = [[principal quantum number]]
*''e'' = [[electron charge]]
*''m<sub>e</sub>'' = electron rest mass
*ε<sub>0</sub> = permittivity of free space
*''h'' = [[Planck's constant]]
</div>
|<math>E_n = -me^4/8\epsilon_0^2h^2n^2 = -13.61eV/n^2\,\!</math>
|-
! Spectrum
|λ = wavelength of emitted photon, during [[Excited state|electronic transition]] from ''E<sub>i</sub>'' to ''E<sub>j</sub>'' 
|<math>\frac{1}{\lambda} = R\left(\frac{1}{n_j^2} - \frac{1}{n_i^2}\right), \, n_j<n_i\,\!</math>
|-
|}


==Fundamental forces==
'''These equations need to be refined such that the notation is defined as has been done for the previous sets of equations.'''
:{| class="wikitable"
|-
! Name
! Equations
|-
![[Strong force]]
||<math>
\begin{align}
\mathcal{L}_\mathrm{QCD}
& = \bar{\psi}_i\left(i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \\
& = \bar{\psi}_i (i \gamma^\mu \partial_\mu  -  m )\psi_i - g G^a_\mu \bar{\psi}_i \gamma^\mu T^a_{ij} \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \,,\\
\end{align}
\,\!</math>
|-
![[Electroweak interaction]]
||:<math>\mathcal{L}_{EW} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y.\,\!</math>
:<math>\mathcal{L}_g = -\frac{1}{4}W_a^{\mu\nu}W_{\mu\nu}^a - \frac{1}{4}B^{\mu\nu}B_{\mu\nu}\,\!</math>
:<math>\mathcal{L}_f =   \overline{Q}_i iD\!\!\!\!/\; Q_i+ \overline{u}_i^c iD\!\!\!\!/\; u^c_i+ \overline{d}_i^c iD\!\!\!\!/\; d^c_i+ \overline{L}_i iD\!\!\!\!/\; L_i+ \overline{e}^c_i iD\!\!\!\!/\; e^c_i \,\!</math>
:<math>\mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2\,\!</math>
:<math>\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^c_j - y_{e\,ij} \,h\, \overline{L}_i e^c_j + h.c.\,\!</math>
|-
![[Quantum electrodynamics]]
||<math>\mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\;,\,\!</math>
|}

==General Relativity==

{{Spacetime|cTopic=Mathematics}}

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


The Einstein field equations (EFE) may be written in the form:<ref>{{cite book |title=Einstein's General Theory of Relativity: With Modern Applications in Cosmology |edition=illustrated |first1=Øyvind |last1=Grøn |first2=Sigbjorn |last2=Hervik |publisher=Springer Science & Business Media |year=2007 |isbn=978-0-387-69200-5 |page=180 |url=https://books.google.com/books?id=IyJhCHAryuUC&pg=PA180}}</ref><ref name="ein"/>

{{Equation box 1
|indent =:
|equation = <math>R_{\mu \nu} - \tfrac{1}{2}R g_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G }{c^4} T_{\mu \nu}</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

where {{mvar|R{{sub|μν}}}} is the [[Ricci curvature|Ricci curvature tensor]], {{mvar|R}} is the [[scalar curvature]], {{mvar|g{{sub|μν}}}} is the [[metric tensor (general relativity)|metric tensor]], {{mvar|Λ}} is the [[cosmological constant]], {{mvar|G}} is [[gravitational constant|Newton's gravitational constant]], {{mvar|c}} is the [[speed of light]] in vacuum, and {{mvar|T{{sub|μν}}}} is the [[stress–energy tensor]].

The EFE is a tensor equation relating a set of [[symmetric tensor|symmetric 4&nbsp;×&nbsp;4 tensors]]. Each tensor has 10 independent components. The four [[Bianchi identities]] reduce the number of independent equations from 10 to 6, leaving the metric with four [[gauge fixing]] [[Degrees of freedom (physics and chemistry)|degrees of freedom]], which correspond to the freedom to choose a coordinate system.

The EFE is a tensor equation relating a set of [[symmetric tensor|symmetric 4&nbsp;×&nbsp;4 tensors]]. Each tensor has 10 independent components. The four [[Bianchi identities]] reduce the number of independent equations from 10 to 6, leaving the metric with four [[gauge fixing]] [[Degrees of freedom (physics and chemistry)|degrees of freedom]], which correspond to the freedom to choose a coordinate system.

Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in {{mvar|n}} dimensions.<ref name="Stephani et al">{{cite book | last = Stephani | first = Hans |first2=D. |last2=Kramer |first3=M. |last3=MacCallum |first4=C. |last4=Hoenselaers |first5=E. |last5=Herlt | title = Exact Solutions of Einstein's Field Equations | publisher = [[Cambridge University Press]] | year = 2003 | isbn = 0-521-46136-7 }}</ref> The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when {{mvar|T}} is identically zero) define [[Einstein manifold]]s.

Despite the simple appearance of the equations they are actually quite complicated. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor {{mvar|g{{sub|μν}}}}, as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. In fact, when fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic [[partial differential equation]]s.<ref>Rendall, Alan D. "Theorems on existence and global dynamics for the Einstein equations." ''Living Reviews in Relativity'' 8.1 (2005): 6.</ref>

One can write the EFE in a more compact form by defining the [[Einstein tensor]]
:<math>G_{\mu \nu} = R_{\mu \nu} - \tfrac{1}{2} R g_{\mu \nu},</math>

which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as
:<math>G_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}.</math>

In standard units, each term on the left has units of 1/length<sup>2</sup>. With this choice of [[Einstein constant]] as 8πG/c<sup>4</sup>, 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 {{math|''G'' {{=}} ''c'' {{=}} 1}}, this can be rewritten as
:<math>G_{\mu \nu} + \Lambda g_{\mu \nu} = 8 \pi T_{\mu \nu}\,.</math>

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 (general relativity)|geodesic equation]],<ref name="SW1993">{{cite book|last=Weinberg |first=Steven|title=Dreams of a Final Theory: the search for the fundamental laws of nature|year=1993|publisher=Vintage Press|pages=107, 233|isbn=0-09-922391-0}}</ref> which dictates how freely-falling matter moves through space-time, form the core of the [[mathematics of general relativity|mathematical formulation]] of [[general relativity]].

===Sign convention===
The above form of the EFE is the standard established by [[Gravitation (book)|Misner, Thorne, and Wheeler]].{{sfnp|Misner|Thorne|Wheeler|1973|p=501ff}} The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):

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

The third sign above is related to the choice of convention for the Ricci tensor:
:<math>R_{\mu \nu}=[S2]\times [S3] \times {R^\alpha}_{\mu\alpha\nu} </math>

With these definitions [[Gravitation (book)|Misner, Thorne, and Wheeler]] classify themselves as {{math|(+ + +)}}, whereas Weinberg (1972){{sfnp|Weinberg|1972}} and Peacock (1994){{sfnp|Peacock|1994}} are {{math|(+ − −)}}, Peebles (1980)<ref>{{cite book |last=Peebles |first=Phillip James Edwin |title=The Large-scale Structure of the Universe |location= |publisher=Princeton University Press |year=1980 |isbn=0-691-08239-1 }}</ref> and Efstathiou et al. (1990)<ref>{{cite journal |last=Efstathiou |first=G. |first2=W. J. |last2=Sutherland |first3=S. J. |last3=Maddox |title=The cosmological constant and cold dark matter |journal=[[Nature (journal)|Nature]] |volume=348 |issue=6303 |year=1990 |pages=705 |doi=10.1038/348705a0 }}</ref> are {{math|(− + +)}},  Rindler (1977){{citation needed|date=October 2014}}, Atwater (1974){{citation needed|date=October 2014}}, Collins Martin & Squires (1989)<ref>{{cite book |last=Collins |first=P. D. B. |last2=Martin |first2=A. D. |last3=Squires |first3=E. J. |year=1989 |title=Particle Physics and Cosmology |location=New York |publisher=Wiley |isbn=0-471-60088-1 }}</ref> are {{math|(− + −)}}.

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
:<math>R_{\mu \nu} - \tfrac{1}{2} R g_{\mu \nu} - \Lambda g_{\mu \nu} = -\frac{8 \pi G}{c^4} T_{\mu \nu}.</math>

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

===Equivalent formulations===
Taking the [[Scalar curvature#Definition|trace with respect to the metric]] of both sides of the EFE one gets
:<math>R - \frac{D}{2} R + D \Lambda = \frac{8 \pi G}{c^4} T \,</math>

where {{mvar|D}} is the spacetime dimension. This expression can be rewritten as
:<math>-R + \frac{D \Lambda}{\frac{D}{2} -1} = \frac{8 \pi G}{c^4} \frac{T}{\frac{D}{2}-1} \,.</math>

If one adds {{math|−{{sfrac|1|2}}''g{{sub|μν}}''}} times this to the EFE, one gets the following equivalent "trace-reversed" form
:<math>R_{\mu \nu} - \frac{ \Lambda g_{\mu \nu}}{\frac{D}{2}-1} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \frac{1}{D-2}Tg_{\mu \nu}\right) .\,</math>

For example, in {{math|''D'' {{=}} 4}} dimensions this reduces to
:<math>R_{\mu \nu} - \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} \left(T_{\mu \nu} - \tfrac{1}{2}T\,g_{\mu \nu}\right) .\,</math>

Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace {{mvar|g{{sub|μν}}}} in the expression on the right with the [[Minkowski metric]] without significant loss of accuracy).