Editing Scientific law (section)

== Laws of physics ==
=== Conservation laws ===

==== Conservation and symmetry ====
{{Main|Symmetry (physics)}}

[[Conservation laws]] are fundamental laws that follow from the homogeneity of space, time and [[phase (waves)|phase]], in other words ''symmetry''.
* '''[[Noether's theorem]]:''' Any quantity with a continuously differentiable symmetry in the action has an associated conservation law.
* [[Conservation of mass]] was the first law to be understood since most macroscopic physical processes involving masses, for example, collisions of massive particles or fluid flow, provide the apparent belief that mass is conserved. Mass conservation was observed to be true for all chemical reactions. In general, this is only approximative because with the advent of relativity and experiments in nuclear and particle physics: mass can be transformed into energy and vice versa, so mass is not always conserved but part of the more general conservation of [[mass–energy equivalence|mass–energy]].
* '''[[Conservation of energy]]''', '''[[Conservation of momentum|momentum]]''' and '''[[Conservation of angular momentum|angular momentum]]''' for isolated systems can be found to be [[Time translation symmetry|symmetries in time]], translation, and rotation.
* '''[[Conservation of charge]]''' was also realized since charge has never been observed to be created or destroyed and only found to move from place to place.

==== Continuity and transfer ====

Conservation laws can be expressed using the general [[continuity equation]] (for a conserved quantity) can be written in differential form as:
: <math>\frac{\partial \rho}{\partial t}=-\nabla \cdot \mathbf{J} </math>
where ''ρ'' is some quantity per unit volume, '''J''' is the [[flux]] of that quantity (change in quantity per unit time per unit area). Intuitively, the [[divergence]] (denoted ∇⋅) of a [[vector field]] is a measure of flux diverging radially outwards from a point, so the negative is the amount piling up at a point; hence the rate of change of density in a region of space must be the amount of flux leaving or collecting in some region (see the main article for details). In the table below, the fluxes flows for various physical quantities in transport, and their associated continuity equations, are collected for comparison.

:{| class="wikitable" align="center"
|-
! scope="col" style="width:150px;"| Physics, conserved quantity
! scope="col" style="width:140px;"| Conserved quantity ''q'' 
! scope="col" style="width:140px;"| Volume density ''ρ'' (of ''q'') 
! scope="col" style="width:140px;"| Flux '''J''' (of ''q'')
! scope="col" style="width:10px;"| Equation
|-
| [[Hydrodynamics]], [[fluid]]s <br />
| ''m'' = [[mass]] (kg)
| ''ρ'' = volume [[mass density]] (kg m<sup>−3</sup>)
| ''ρ'' '''u''', where<br />
'''u''' = [[velocity field]] of fluid (m s<sup>−1</sup>)
| <math> \frac{\partial \rho}{\partial t} = - \nabla \cdot (\rho \mathbf{u}) </math>
|-
| [[Electromagnetism]], [[electric charge]]
| ''q'' = electric charge (C)
| ''ρ'' = volume electric [[charge density]] (C m<sup>−3</sup>)
| '''J''' = electric [[current density]] (A m<sup>−2</sup>)
| <math> \frac{\partial \rho}{\partial t} = - \nabla \cdot \mathbf{J} </math> 
|-
| [[Thermodynamics]], [[energy]]
| ''E'' = energy (J)
| ''u'' = volume [[energy density]] (J m<sup>−3</sup>)
| '''q''' = [[heat flux]] (W m<sup>−2</sup>)
| <math> \frac{\partial u}{\partial t}=- \nabla \cdot \mathbf{q} </math> 
|-
| [[Quantum mechanics]], [[probability]] 
| ''P'' = ('''r''', ''t'') = ∫|Ψ|<sup>2</sup>d<sup>3</sup>'''r''' = [[probability distribution]]
| ''ρ'' = ''ρ''('''r''', ''t'') = |Ψ|<sup>2</sup> = [[probability density function]] (m<sup>−3</sup>),<br />
Ψ = [[wavefunction]] of quantum system
| '''j''' = [[probability current]]/flux
| <math> \frac{\partial |\Psi|^2}{\partial t}=-\nabla \cdot \mathbf{j} </math> 
|}

More general equations are the [[convection–diffusion equation]] and [[Boltzmann transport equation]], which have their roots in the continuity equation.

=== Laws of classical mechanics ===

==== Principle of least action ====

{{Main|Principle of least action}}

Classical mechanics, including [[Newton's laws]], [[Lagrangian mechanics|Lagrange's equations]], [[Hamiltonian mechanics|Hamilton's equations]], etc., can be derived from the following principle:
: <math> \delta \mathcal{S} = \delta\int_{t_1}^{t_2} L(\mathbf{q}, \mathbf{\dot{q}}, t) \, dt = 0 </math>
where <math> \mathcal{S} </math> is the [[action (physics)|action]]; the integral of the [[Lagrangian mechanics|Lagrangian]]
: <math> L(\mathbf{q}, \mathbf{\dot{q}}, t) = T(\mathbf{\dot{q}}, t)-V(\mathbf{q}, \mathbf{\dot{q}}, t)</math>
of the physical system between two times ''t''<sub>1</sub> and ''t''<sub>2</sub>. The kinetic energy of the system is ''T'' (a function of the rate of change of the [[Configuration space (physics)|configuration]] of the system), and [[potential energy]] is ''V'' (a function of the configuration and its rate of change). The configuration of a system which has ''N'' [[Degrees of freedom (mechanics)|degrees of freedom]] is defined by [[generalized coordinates]] '''q''' = (''q''<sub>1</sub>, ''q''<sub>2</sub>, ... ''q<sub>N</sub>'').

There are [[Canonical coordinates|generalized momenta]] conjugate to these coordinates, '''p''' = (''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p<sub>N</sub>''), where:
: <math>p_i = \frac{\partial L}{\partial \dot{q}_i}</math>

The action and Lagrangian both contain the dynamics of the system for all times. The term "path" simply refers to a curve traced out by the system in terms of the [[generalized coordinates]] in the [[Configuration space (physics)|configuration space]], i.e. the curve '''q'''(''t''), parameterized by time (see also [[parametric equation]] for this concept).

The action is a ''[[functional (mathematics)|functional]]'' rather than a ''[[function (mathematics)|function]]'', since it depends on the Lagrangian, and the Lagrangian depends on the path '''q'''(''t''), so the action depends on the ''entire'' "shape" of the path for all times (in the time interval from ''t''<sub>1</sub> to ''t''<sub>2</sub>). Between two instants of time, there are infinitely many paths, but one for which the action is stationary (to the first order) is the true path. The stationary value for the ''entire continuum'' of Lagrangian values corresponding to some path, ''not just one value'' of the Lagrangian, is required (in other words it is ''not'' as simple as "differentiating a function and setting it to zero, then solving the equations to find the points of [[maxima and minima]] etc", rather this idea is applied to the entire "shape" of the function, see [[calculus of variations]] for more details on this procedure).<ref>{{cite book | last1=Feynman | first1=Richard Phillips | last2=Leighton | first2=Robert B. | last3=Sands | first3=Matthew Linzee | title=The Feynman Lectures on Physics | publisher=Addison Wesley Longman | publication-place=Reading/Mass. | date=1963 | isbn=0-201-02117-X}}</ref>

Notice ''L'' is ''not'' the total energy ''E'' of the system due to the difference, rather than the sum:
: <math>E=T+V</math>

The following<ref>{{cite book | last1=Lerner | first1=Rita G. |authorlink1=Rita G. Lerner| last2=Trigg | first2=George L. | title=Encyclopedia of Physics | publisher=VCH Publishers | publication-place=New York Weinheim Cambridge Basel | date=1991 | isbn=0-89573-752-3 }}</ref><ref>{{cite book | last=Kibble | first=T. W. B. | title=Classical Mechanics | publisher=McGraw Hill | publication-place=London; New York | date=1973 | isbn=0-07-084018-0}}</ref> general approaches to classical mechanics are summarized below in the order of establishment. They are equivalent formulations. Newton's is commonly used due to simplicity, but Hamilton's and Lagrange's equations are more general, and their range can extend into other branches of physics with suitable modifications.

: {| class="wikitable" align="center"
|-
! scope="col" style="width:600px;" colspan="2"| '''Laws of motion'''
|-
|colspan="2" |'''[[Principle of least action]]:'''

<math> \mathcal{S} = \int_{t_1}^{t_2} L \,\mathrm{d}t \,\!</math>
|- valign="top"
| rowspan="2" scope="col" style="width:300px;"|'''The [[Euler–Lagrange equation]]s are:'''
: <math> \frac{\mathrm{d}}{\mathrm{d} t} \left ( \frac{\partial L}{\partial \dot{q}_i } \right ) = \frac{\partial L}{\partial q_i} </math>
Using the definition of generalized momentum, there is the symmetry:
: <math> p_i = \frac{\partial L}{\partial \dot{q}_i}\quad \dot{p}_i = \frac{\partial L}{\partial {q}_i} </math>
| style="width:300px;"| '''Hamilton's equations'''
: <math> \dfrac{\partial \mathbf{p}}{\partial t} = -\dfrac{\partial H}{\partial \mathbf{q}} </math><br /><math> \dfrac{\partial \mathbf{q}}{\partial t} = \dfrac{\partial H}{\partial \mathbf{p}} </math>

The Hamiltonian as a function of generalized coordinates and momenta has the general form: <br />
: <math>H (\mathbf{q}, \mathbf{p}, t) = \mathbf{p}\cdot\mathbf{\dot{q}}-L</math>
|-
|[[Hamilton–Jacobi equation]]
: <math>H \left(\mathbf{q}, \frac{\partial S}{\partial\mathbf{q}}, t\right) = -\frac{\partial S}{\partial t}</math>
|- style="border-top: 3px solid;"
| colspan="2" scope="col" style="width:600px;"| '''Newton's laws'''

'''[[Newton's laws of motion]]'''

They are low-limit solutions to [[theory of relativity|relativity]]. Alternative formulations of Newtonian mechanics are [[Lagrangian mechanics|Lagrangian]] and [[Hamiltonian mechanics|Hamiltonian]] mechanics.

The laws can be summarized by two equations (since the 1st is a special case of the 2nd, zero resultant acceleration):
: <math> \mathbf{F} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}, \quad \mathbf{F}_{ij}=-\mathbf{F}_{ji} </math>
where '''p''' = momentum of body, '''F'''<sub>''ij''</sub> = force ''on'' body ''i'' ''by'' body ''j'', '''F'''<sub>''ji''</sub> = force ''on'' body ''j'' ''by'' body ''i''.

For a [[dynamical system]] the two equations (effectively) combine into one:
: <math> \frac{\mathrm{d}\mathbf{p}_\mathrm{i}}{\mathrm{d}t} = \mathbf{F}_\text{E} + \sum_{\mathrm{i} \neq \mathrm{j}} \mathbf{F}_\mathrm{ij} </math>
in which '''F'''<sub>E</sub> = resultant external force (due to any agent not part of system). Body ''i'' does not exert a force on itself. 
|}

From the above, any equation of motion in classical mechanics can be derived.

'''Corollaries in mechanics :'''
* [[Euler's laws of motion]]
* [[Euler's equations (rigid body dynamics)]]

'''Corollaries in [[fluid mechanics]] :'''<br />

Equations describing fluid flow in various situations can be derived, using the above classical equations of motion and often conservation of mass, energy and momentum. Some elementary examples follow.
* [[Archimedes' principle]]
* [[Bernoulli's principle]]
* [[Poiseuille's law]]
* [[Stokes' law]]
* [[Navier–Stokes equations]]
* [[Faxén's law]]

=== Laws of gravitation and relativity ===

Some of the more famous laws of nature are found in [[Isaac Newton]]'s theories of (now) [[classical mechanics]], presented in his ''[[Philosophiae Naturalis Principia Mathematica]]'', and in [[Albert Einstein]]'s [[theory of relativity]].

==== Modern laws ====

'''[[Special relativity]] :'''<br />

The two postulates of special relativity are not "laws" in themselves, but assumptions of their nature in terms of ''relative motion''.

They can be stated as "the laws of physics are the same in all [[inertial frames]]" and "the [[speed of light]] is constant and has the same value in all inertial frames".

The said postulates lead to the [[Lorentz transformations]] – the transformation law between two [[frame of reference]]s moving relative to each other. For any [[4-vector]]
: <math>A' =\Lambda A </math>
this replaces the [[Galilean transformation]] law from classical mechanics. The Lorentz transformations reduce to the Galilean transformations for low velocities much less than the speed of light ''c''.

The magnitudes of 4-vectors are invariants – ''not'' "conserved", but the same for all inertial frames (i.e. every observer in an inertial frame will agree on the same value), in particular if ''A'' is the [[four-momentum]], the magnitude can derive the famous invariant equation for mass–energy and momentum conservation (see [[invariant mass]]):
: <math> E^2 = (pc)^2 + (mc^2)^2 </math>
in which the (more famous) [[mass–energy equivalence]] {{nowrap|1=''E'' = ''mc''<sup>2</sup>}} is a special case.

'''[[General relativity]] :'''<br />

General relativity is governed by the [[Einstein field equations]], which describe the curvature of space-time due to mass–energy equivalent to the gravitational field. Solving the equation for the geometry of space warped due to the mass distribution gives the [[metric tensor]]. Using the geodesic equation, the motion of masses falling along the geodesics can be calculated.

'''[[Gravitoelectromagnetism]] :'''<br />

In a relatively flat spacetime due to weak gravitational fields, gravitational analogues of Maxwell's equations can be found; the '''GEM equations''', to describe an analogous ''[[Gravitoelectromagnetism|gravitomagnetic field]]''. They are well established by the theory, and experimental tests form ongoing research.<ref>{{cite book | last1=Ciufolini | first1=Ignazio | last2=Wheeler | first2=John Archibald | title=Gravitation and Inertia |series=Princeton Physics |publisher=Princeton University Press | publication-place=Princeton, N.J | date=1995-08-13 | isbn=0-691-03323-4}}</ref>

: {| class="wikitable" align="center"
|- valign="top"
| scope="col" style="width:300px;"|'''[[Einstein field equations]] (EFE):'''
: <math>R_{\mu \nu} + \left ( \Lambda - \frac{R}{2} \right ) g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}\,\!</math>
where Λ = [[cosmological constant]], ''R<sub>μν</sub>'' = [[Ricci curvature tensor]], ''T<sub>μν</sub>'' = [[stress–energy tensor]], ''g<sub>μν</sub>'' = [[metric tensor]]
| scope="col" style="width:300px;"|'''[[Geodesic equation]]:'''
: <math>\frac{{\rm d}^2x^\lambda }{{\rm d}t^2} + \Gamma^{\lambda}_{\mu \nu }\frac{{\rm d}x^\mu }{{\rm d}t}\frac{{\rm d}x^\nu }{{\rm d}t} = 0\ ,</math>
where Γ is a [[Christoffel symbol]] of the [[Christoffel symbols#Christoffel symbols of the second kind (symmetric definition)|second kind]], containing the metric.
|- style="border-top: 3px solid;"
|colspan="2"| '''GEM Equations'''

If '''g''' the gravitational field and '''H''' the gravitomagnetic field, the solutions in these limits are:
: <math> \nabla \cdot \mathbf{g} = -4 \pi G \rho \,\!</math>
: <math> \nabla \cdot \mathbf{H} = \mathbf{0} \,\!</math>
: <math> \nabla \times \mathbf{g} = -\frac{\partial \mathbf{H}} {\partial t} \,\!</math>
: <math> \nabla \times \mathbf{H} =  \frac{4}{c^2}\left( - 4 \pi G\mathbf{J} + \frac{\partial \mathbf{g}} {\partial t} \right) \,\!</math>
where ''ρ'' is the [[density|mass density]] and '''J''' is the mass current density or [[mass flux]].
|-
|colspan="2"| In addition there is the '''gravitomagnetic Lorentz force''':
: <math>\mathbf{F} = \gamma(\mathbf{v}) m \left( \mathbf{g} + \mathbf{v} \times \mathbf{H} \right) </math>
where ''m'' is the [[rest mass]] of the particlce and γ is the [[Lorentz factor]].
|}

==== Classical laws ====

{{Main|Kepler's laws of planetary motion|Newton's law of gravitation}}

Kepler's laws, though originally discovered from planetary observations (also due to [[Tycho Brahe]]), are true for any ''[[central force]]s''.<ref>{{cite book | last=Kibble | first=T. W. B. | title=Classical Mechanics |series=European Physics| publisher=McGraw Hill | publication-place=London; New York | date=1973 | isbn=0-07-084018-0 }}</ref>

: {| class="wikitable" align="center"
|- valign="top"
| scope="col" style="width:300px;"|'''[[Newton's law of universal gravitation]]:''' 
For two point masses:
: <math>\mathbf{F} = \frac{G m_1 m_2}{\left | \mathbf{r} \right |^2} \mathbf{\hat{r}} \,\!</math>
For a non uniform mass distribution of local mass density ''ρ'' ('''r''') of body of Volume ''V'', this becomes:
: <math> \mathbf{g} = G \int_V \frac{\mathbf{r} \rho  \, \mathrm{d}{V}}{\left | \mathbf{r} \right |^3}\,\!</math>
| scope="col" style="width:300px;"| '''[[Gauss's law for gravity]]:'''

An equivalent statement to Newton's law is:
: <math>\nabla\cdot\mathbf{g} = 4\pi G\rho \,\!</math>
|- style="border-top: 3px solid;"
| colspan="2" scope="col" style="width:600px;"|'''Kepler's 1st law:''' Planets move in an ellipse, with the star at a focus
: <math>r = \frac\ell{1+e \cos\theta}  \,\!</math>
where 
: <math> e = \sqrt{1- (b/a)^2} </math>
is the [[Eccentricity (mathematics)|eccentricity]] of the elliptic orbit, of  semi-major axis ''a'' and semi-minor axis ''b'', and ''ℓ'' is the semi-latus rectum. This equation in itself is nothing physically fundamental; simply the [[Polar coordinate system|polar equation]] of an [[ellipse]] in which the pole (origin of polar coordinate system) is positioned at a focus of the ellipse, where the orbited star is.
|-
| colspan="2" style="width:600px;"|'''Kepler's 2nd law:''' equal areas are swept out in equal times (area bounded by two radial distances and the orbital circumference):
: <math>\frac{\mathrm{d}A}{\mathrm{d}t} = \frac{\left | \mathbf{L} \right |}{2m} \,\!</math>
where '''L''' is the orbital angular momentum of the particle (i.e. planet) of mass ''m'' about the focus of orbit, 
|-
|colspan="2"|'''Kepler's 3rd law:''' The square of the orbital time period ''T'' is proportional to the cube of the semi-major axis ''a'':
: <math>T^2 =  \frac{4\pi^2}{G \left ( m + M \right ) } a^3\,\!</math>
where ''M'' is the mass of the central body (i.e. star).
|}

=== Thermodynamics ===

: {| class="wikitable" align="center"
|- 
!colspan="2"|'''[[Laws of thermodynamics]]'''
|- valign="top"
| scope="col" style="width:150px;"|'''[[First law of thermodynamics]]:''' The change in internal energy d''U'' in a closed system is accounted for entirely by the heat &delta;''Q'' absorbed by the system and the work &delta;''W'' done by the system:
: <math>\mathrm{d}U=\delta Q-\delta W\,</math>
'''[[Second law of thermodynamics]]:''' There are many statements of this law, perhaps the simplest is "the entropy of isolated systems never decreases",
: <math>\Delta S \ge 0</math>
meaning reversible changes have zero entropy change, irreversible process are positive, and impossible process are negative.
| rowspan="2" style="width:150px;"| '''[[Zeroth law of thermodynamics]]:''' If two systems are in [[thermal equilibrium]] with a third system, then they are in thermal equilibrium with one another.
: <math>T_A = T_B \,, T_B=T_C \Rightarrow T_A=T_C\,\!</math>

'''[[Third law of thermodynamics]]:'''
: As the temperature ''T'' of a system approaches absolute zero, the entropy ''S'' approaches a minimum value ''C'': as ''T''&nbsp;&rarr;&nbsp;0, ''S''&nbsp;&rarr;&nbsp;''C''.
|-
| For homogeneous systems the first and second law can be combined into the '''[[Fundamental thermodynamic relation]]''':
: <math>\mathrm{d} U = T \, \mathrm{d} S - P \, \mathrm{d} V + \sum_i \mu_i \, \mathrm{d}N_i \,\!</math>
|- style="border-top: 3px solid;"
| colspan="2" style="width:500px;"|'''[[Onsager reciprocal relations]]:''' sometimes called the ''fourth law of thermodynamics''
: <math> \mathbf{J}_u = L_{uu}\, \nabla(1/T) - L_{ur}\, \nabla(m/T);</math>
: <math> \mathbf{J}_r = L_{ru}\, \nabla(1/T) - L_{rr}\, \nabla(m/T).</math>
|}

* [[Newton's law of cooling]]
* [[Conduction (heat)|Fourier's law]]
* [[Ideal gas law]], combines a number of separately developed gas laws;
** [[Boyle's law]]
** [[Charles's law]]
** [[Gay-Lussac's law]]
** [[Avogadro's law]], into one
: now improved by other [[equations of state]]
* [[Dalton's law]] (of partial pressures)
* [[Boltzmann equation]]
* [[Carnot's theorem (thermodynamics)|Carnot's theorem]]
* [[Kopp's law]]

=== Electromagnetism ===

[[Maxwell's equations]] give the time-evolution of the [[electric field|electric]] and [[magnetic field|magnetic]] fields due to [[electric charge]] and [[Electric current|current]] distributions. Given the fields, the [[Lorentz force]] law is the [[equation of motion]] for charges in the fields.

: {| class="wikitable" align="center"
|- valign="top"
| scope="col" style="width:300px;"|'''[[Maxwell's equations]]'''

'''[[Gauss's law]] for electricity'''
: <math> \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} </math>
'''[[Gauss's law for magnetism]]'''
: <math>\nabla \cdot \mathbf{B} = 0 </math>
'''[[Faraday's law of induction|Faraday's law]]'''
: <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
'''[[Ampère's circuital law]] (with Maxwell's correction)'''
: <math>\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \frac{1}{c^2}  \frac{\partial \mathbf{E}}{\partial t} \ </math>
| scope="col" style="width:300px;"| '''[[Lorentz force]] law:'''
: <math>\mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)</math>
|- style="border-top: 3px solid;"
| colspan="2" scope="col" style="width:600px;"| '''[[Quantum electrodynamics]] (QED):''' Maxwell's equations are generally true and consistent with relativity - but they do not predict some observed quantum phenomena (e.g. light propagation as [[EM wave]]s, rather than [[photons]], see [[Maxwell's equations]] for details). They are modified in QED theory.
|}

These equations can be modified to include [[magnetic monopole]]s, and are consistent with our observations of monopoles either existing or not existing; if they do not exist, the generalized equations reduce to the ones above, if they do, the equations become fully symmetric in electric and magnetic charges and currents. Indeed, there is a duality transformation where electric and magnetic charges can be "rotated into one another", and still satisfy Maxwell's equations.

'''Pre-Maxwell laws :'''<br />

These laws were found before the formulation of Maxwell's equations. They are not fundamental, since they can be derived from Maxwell's equations. Coulomb's law can be found from Gauss's law (electrostatic form) and the Biot–Savart law can be deduced from Ampere's law (magnetostatic form). Lenz's law and Faraday's law can be incorporated into the Maxwell–Faraday equation. Nonetheless, they are still very effective for simple calculations.
* [[Lenz's law]]
* [[Coulomb's law]]
* [[Biot–Savart law]]

'''Other laws :'''
* [[Ohm's law]]
* [[Kirchhoff's circuit laws|Kirchhoff's laws]]
* [[Joule's first law|Joule's law]]

=== Photonics ===

Classically, [[optics]] is based on a [[variational principle]]: light travels from one point in space to another in the shortest time.
* [[Fermat's principle]]

In [[geometric optics]] laws are based on approximations in Euclidean geometry (such as the [[paraxial approximation]]).
* [[Law of reflection]]
* [[Law of refraction]], [[Snell's law]]

In [[physical optics]], laws are based on physical properties of materials.
* [[Brewster's law|Brewster's angle]]
* [[Malus's law]]
* [[Beer–Lambert law]]

In actuality, optical properties of matter are significantly more complex and require quantum mechanics.

=== Laws of quantum mechanics ===

Quantum mechanics has its roots in [[postulates of quantum mechanics|postulates]]. This leads to results which are not usually called "laws", but hold the same status, in that all of quantum mechanics follows from them. These postulates can be summarized as follows:
* The state of a physical system, be it a particle or a system of many particles, is described by a [[wavefunction]].
* Every physical quantity is described by an [[operators (physics)|operator]] acting on the system; the measured quantity has a [[Born rule|probabilistic nature]].
* The [[wavefunction]] obeys the [[Schrödinger equation]]. Solving this wave equation predicts the time-evolution of the system's behavior, analogous to solving Newton's laws in classical mechanics.
* Two [[identical particles]], such as two electrons, cannot be distinguished from one another by any means. Physical systems are classified by their symmetry properties.

These postulates in turn imply many other phenomena, e.g., [[uncertainty principle]]s and the [[Pauli exclusion principle]].

: {| class="wikitable" align="center"
|- valign="top"
| style="width:300px;"| '''[[Quantum mechanics]], [[Quantum field theory]]'''

'''[[Schrödinger equation]] (general form):''' Describes the time dependence of a quantum mechanical system.
: <math> i\hbar \frac{d}{dt} \left| \psi \right\rangle = \hat{H} \left| \psi \right\rangle </math>

The [[Hamiltonian quaternions|Hamiltonian]] (in quantum mechanics) ''H'' is a [[self-adjoint operator]] acting on the state space, <math>| \psi \rangle </math> (see [[Dirac notation]]) is the instantaneous [[quantum state vector]] at time ''t'', position '''r''', ''i'' is the unit [[imaginary number]], {{nowrap|1=''ħ'' = ''h''/2π}} is the [[reduced Planck constant]].
| rowspan="2" scope="col" style="width:300px;"|'''[[Wave–particle duality]]'''

'''[[Planck constant|Planck–Einstein law]]:''' the [[energy]] of [[photon]]s is proportional to the [[frequency]] of the light (the constant is the [[Planck constant]], ''h'').  
: <math> E = h\nu = \hbar \omega </math>

'''[[Matter wave|De Broglie wave]]length:''' this laid the foundations of wave–particle duality, and was the key concept in the [[Schrödinger equation]],
: <math> \mathbf{p} = \frac{h}{\lambda}\mathbf{\hat{k}} = \hbar \mathbf{k}</math>

'''[[Heisenberg uncertainty principle]]:''' [[Uncertainty]] in position multiplied by uncertainty in [[momentum]] is at least half of the [[reduced Planck constant]], similarly for time and [[energy]];
: <math>\Delta x \, \Delta p \ge \frac{\hbar}{2},\, \Delta E \, \Delta t \ge \frac{\hbar}{2} </math>

The uncertainty principle can be generalized to any pair of observables – see main article.
|-
| '''Wave mechanics'''

'''[[Schrödinger equation]] (original form):''' 
: <math> i\hbar \frac{\partial}{\partial t}\psi = -\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi </math>
|- style="border-top: 3px solid;"
| colspan="2" style="width:600px;"| '''[[Pauli exclusion principle]]:''' No two identical [[fermion]]s can occupy the same quantum state ([[boson]]s can). Mathematically, if two particles are interchanged, fermionic wavefunctions are anti-symmetric, while bosonic wavefunctions are symmetric:
: <math>\psi(\cdots\mathbf{r}_i\cdots\mathbf{r}_j\cdots) = (-1)^{2s}\psi(\cdots\mathbf{r}_j\cdots\mathbf{r}_i\cdots)</math>
where '''r'''<sub>''i''</sub> is the position of particle ''i'', and ''s'' is the [[Spin (physics)|spin]] of the particle. There is no way to keep track of particles physically, labels are only used mathematically to prevent confusion.
|}

=== Radiation laws ===

Applying electromagnetism, thermodynamics, and quantum mechanics, to atoms and molecules, some laws of [[electromagnetic radiation]] and light are as follows.
* [[Stefan–Boltzmann law]]
* [[Planck's law]] of black-body radiation
* [[Wien's displacement law]]
* [[Radioactive decay law]]