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In physics, an '''operator''' is a [[Function (mathematics)|function]] acting on the space of physical states. As a result
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of its application on a physical state, another physical state is obtained, very often along with
some extra relevant information.


The simplest example of the utility of operators is the study of [[symmetry]]. Because of this, they
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are a very useful tool in [[classical mechanics]]. In [[quantum mechanics]], on the other hand, they
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are an intrinsic part of the formulation of the theory.
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==Operators in classical mechanics==
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In classical mechanics, the dynamics of a particle (or system of particles) are completely determined by the [[Lagrangian]] ''L''(''q, q̇, t'') or equivalently the [[Hamiltonian mechanics|Hamiltonian]] ''H''(''q, p, t''), a function of the [[generalized coordinates]] ''q'', generalized velocities ''q̇'' = d''q''/d''t'' and its [[conjugate momenta]]:
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


:<math>p = \frac{\partial L}{\partial \dot{q}}</math>
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:<math forcemathmode="png">E=mc^2</math>


If either ''L'' or ''H'' are independent of a generalized coordinate ''q'', meaning the ''L'' and ''H'' so not change when ''q'' is changed, which in turn means the dynamics of the particle are still the same even when ''q'' changes, the corresponding momenta conjugate to those coordinates will be conserved (this is part of [[Noether's theorem]], and the invariance of motion with respect to the coordinate ''q'' is a [[symmetry (physics)|symmetry]]). Operators in classical mechanics are related to these symmetries.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


More technically, when ''H'' is invariant under the action of a certain [[group (mathematics)|group]] of transformations ''G'':
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


:<math>S\in G, H(S(q,p))=H(q,p)</math>.
==Demos==


the elements of ''G'' are physical operators, which map physical states among themselves.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


===Table of classical mechanics operators===


:{| class="wikitable"
* accessibility:
|-
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
! Transformation
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
! Operator
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
! Position
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
! Momentum
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
|-
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
| [[Translational symmetry]]  
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
| <math>X(\bold{a})</math>
| <math>\bold{r}\rightarrow \bold{r} + \bold{a}</math>
| <math>\bold{p}\rightarrow \bold{p}</math>
|-
| [[Time evolution|Time translations]]
| <math>U(t_0)</math>
| <math>\bold{r}(t)\rightarrow \bold{r}(t+t_0)</math>
| <math>\bold{p}(t)\rightarrow \bold{p}(t+t_0)</math>
|-
| [[Rotational invariance]]  
| <math>R(\bold{\hat{n}},\theta)</math>
| <math>\bold{r}\rightarrow R(\bold{\hat{n}},\theta)\bold{r}</math>
| <math>\bold{p}\rightarrow R(\bold{\hat{n}},\theta)\bold{p}</math>
|-  
| [[Galilean transformation]]s
| <math>G(\bold{v})</math>
| <math>\bold{r}\rightarrow \bold{r} + \bold{v}t</math>
| <math>\bold{p}\rightarrow \bold{p} + m\bold{v}</math>
|-
| [[Parity (physics)|Parity]]
| <math>P</math>
| <math>\bold{r}\rightarrow -\bold{r}</math>
| <math>\bold{p}\rightarrow -\bold{p}</math>
|-
| [[T-symmetry]]
| <math>T</math>
| <math>\bold{r}\rightarrow \bold{r}(-t)</math>
| <math>\bold{p}\rightarrow -\bold{p}(-t)</math>
|-
|}


where ''R''('''n̂''', θ) is the [[rotation matrix]] about an axis defined by the [[unit vector]] '''n̂''' and angle θ.
==Test pages ==


==Concept of generator==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
If the transformation is infinitesimal, the operator action should be of the form
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


: <math> I + \epsilon A </math>
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
where <math>I</math> is the identity operator, <math>\epsilon</math> is a small parameter, and <math>A</math> will depend on the transformation at hand, and is called a generator of the group. Again, as a simple example, we will derive the generator of the space translations on 1D functions.
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
As it was stated, <math>T_a f(x)=f(x-a)</math>. If <math>a=\epsilon</math> is infinitesimal, then we may  write
 
: <math>T_\epsilon f(x)=f(x-\epsilon)\approx f(x) - \epsilon f'(x).</math>
 
This formula may be rewritten as
 
: <math>T_\epsilon f(x) = (I-\epsilon D) f(x)</math>
 
where <math>D</math> is the generator of the translation group, which in this case happens to be the ''derivative'' operator. Thus, it is said that the generator of translations is the derivative.
 
==The exponential map==
The whole group may be recovered, under normal circumstances, from the generators, via the [[exponential map]]. In the case of the translations the idea works like this.
 
The translation for a finite value of <math>a</math> may be obtained by repeated application of the infinitesimal translation:
 
: <math>T_a f(x) = \lim_{N\to\infty} T_{a/N} \cdots T_{a/N} f(x)</math>
 
with the <math>\cdots</math> standing for the application <math>N</math> times. If <math>N</math> is large, each of the factors may be considered to be infinitesimal:
 
: <math>T_a f(x) = \lim_{N\to\infty} (I -(a/N) D)^N f(x).</math>
 
But this limit may be rewritten as an exponential:
 
: <math>T_a f(x)= \exp(-aD) f(x).</math>
 
To be convinced of the validity of this formal expression, we may expand the exponential in a power series:
 
: <math>T_a f(x) = \left( I - aD + {a^2D^2\over 2!} - {a^3D^3\over 3!} + \cdots \right) f(x).</math>
 
The right-hand side may be rewritten as
 
: <math>f(x) - a f'(x) + {a^2\over 2!} f''(x) - {a^3\over 3!} f'''(x) + \cdots</math>
 
which is just the Taylor expansion of <math>f(x-a)</math>, which was our original value for <math>T_a f(x)</math>.
 
The mathematical properties of physical operators are a topic of great importance in itself. For further information, see [[C*-algebra]] and [[Gelfand-Naimark theorem]].
 
==Operators in quantum mechanics==
 
The [[mathematical formulation of quantum mechanics]] (QM) is built upon the concept of an operator.
 
The wavefunction represents the [[probability amplitude]] of finding the system in that state. The terms "wavefunction" and "state" in QM context are usually used interchangeably.
 
Physical [[pure state]]s in quantum mechanics are represented as [[unit-norm vector]]s (probabilities are normalized to one) in a special [[complex number|complex]] [[vector space]]: a [[Hilbert space]]. [[Time evolution]] in this vector space is given by the application of the [[evolution operator]].
 
Any [[observable]], i.e., any quantity which can be measured in a physical experiment, should be associated with a [[self-adjoint]] [[linear operator]]. The operators must yield real [[eigenvalue]]s, since they are values which may come up as the result of the experiment. Mathematically this means the operators must be [[Hermitian matrix|Hermitian]].<ref>Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISRTY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0</ref> The probability of each eigenvalue is related to the projection of the physical state on the subspace related to that eigenvalue. See below for mathematical details.
 
In the [[wave mechanics]] formulation of QM, the wavefunction varies with space and time, or equivalently momentum and time (see [[position and momentum space]] for details), so observables are [[differential operator]]s.
 
In the [[matrix mechanics]] formulation, the [[Norm (mathematics)|norm]] of the physical state should stay fixed, so the evolution operator should be [[unitary transformation|unitary]], and the operators can be represented as matrices. Any other symmetry, mapping a physical state into another, should keep this restriction.
 
===Wavefunction ===
 
{{Main|wavefunction}}
 
The wavefunction must be [[square-integrable]] on the Hilbert space (see [[Lp spaces]]) meaning:
 
:<math>\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty |\psi(\bold{r})|^2 {\rm d}^3\bold{r} = \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty \psi(\bold{r})^*\psi(\bold{r}){\rm d}^3\bold{r} < \infty </math>
 
and normalizable, so that:
 
:<math>\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty |\psi(\bold{r})|^2 {\rm d}^3\bold{r} = 1 </math>
 
Two cases of eigenstates (and eigenvalues) are:
*for '''discrete''' eigenstates <math> | \psi_i \rangle </math> forming a discrete basis, so the state is a [[sum]]
::<math>|\psi\rangle = \sum_i c_i|\phi_i\rangle</math>
:where ''c<sub>i</sub>'' are complex numbers such that |''c<sub>i</sub>''|<sup>2</sup> = ''c<sub>i</sub>''<sup>*</sup>''c<sub>i</sub>'' = probability of measuring the state <math>|\phi_i\rangle</math>, and has the corresponding set of eigenvalues ''a<sub>i</sub>'' is also discrete - either [[finite]] or [[countably infinite]],
*for a '''continuum''' of eigenstates <math> | \psi \rangle </math> forming a continuous basis, so the state is an [[integral]]
::<math>|\psi\rangle = \int c(\phi){\rm d}\phi|\phi_i\rangle </math>
:where ''c''(φ) is a complex function such that |''c''(φ)|<sup>2</sup> = ''c''(φ)<sup>*</sup>''c''(φ) = probability of measuring the state <math>|\phi\rangle</math>, there is an [[uncountably infinite]] set of eigenvalues ''a''.
 
===Linear operators in wave mechanics===
 
{{Main|Wave function|Bra-ket notation}}
 
Let ''ψ'' be the wavefunction for a quantum system, and <math>\hat{A}</math> be any [[linear operator]] for some observable ''A'' (such as position, momentum, energy, angular momentum etc.), then
 
:<math>\hat{A} \psi = a \psi ,</math>
 
where:
 
* ''a'' is the [[Eigenvalues and eigenvectors|eigenvalue]] of the operator, corresponding to the measured value of the observable, i.e. observable ''A'' has a measured value ''a''
*''ψ'' is the [[eigenfunction]] of <math>\hat{A}</math> if this relation holds.
 
If ''ψ''  is an eigenfunction of an operator, it means the eigenvalue can be found and so the observable can be measured, conversely if is not an eigenfunction then the eigenvalue can't be found and the observable can't be measured for that case.
 
In bra-ket notation the above can be written;
 
:<math>\begin{align} & \hat{A} \psi = \hat{A} \psi ( \mathbf{r} ) = \hat{A} \langle \mathbf{r} | \psi \rangle = \langle \mathbf{r} | \hat {A} | \psi \rangle \\
& a \psi = a \psi ( \mathbf{r} ) = a \langle \mathbf{r} | \psi \rangle = \langle \mathbf{r} | a | \psi \rangle \\
\end{align} </math>
 
in which case <math> | \psi \rangle </math> is an [[eigenvector]], or [[eigenket]].
 
Due to linearity, vectors can be defined in any number of dimensions, as each component of the vector acts on the function separately. One mathematical example is the [[del operator]], which is itself a vector (useful in momentum-related quantum operators, in the table below).
 
An operator in ''n''-dimensional space can be written:
 
:<math> \mathbf{\hat{A}} = \sum_{j=1}^n \mathbf{e}_\mathrm{j} \hat{A}_j </math>
 
where '''e'''<sub>''j''</sub> are basis vectors corresponding to each component operator ''A<sub>j</sub>''. Each component will yield a corresponding eigenvalue. Acting this on the wave function ''ψ'':
 
:<math> \mathbf{\hat{A}} \psi = \left ( \sum_{j=1}^n \mathbf{e}_\mathrm{j} \hat{A}_j \right ) \psi = \sum_{j=1}^n \left ( \mathbf{e}_\mathrm{j} \hat{A}_j \psi \right ) = \sum_{j=1}^n \left ( \mathbf{e}_\mathrm{j} a_j \psi \right ) </math>
 
in which
 
:<math> \hat{A}_j \psi = a_j \psi .</math>
 
In bra-ket notation:
 
:<math>\begin{align} & \mathbf{\hat{A}} \psi = \mathbf{\hat{A}} \psi ( \mathbf{r} ) = \mathbf{\hat{A}} \langle \mathbf{r} | \psi \rangle = \langle \mathbf{r} | \mathbf{\hat{A}} | \psi \rangle \\
 
& \left ( \sum_{j=1}^n \mathbf{e}_\mathrm{j} \hat{A}_j \right ) \psi = \left ( \sum_{j=1}^n \mathbf{e}_\mathrm{j} \hat{A}_j \right ) \psi ( \mathbf{r} ) = \left ( \sum_{j=1}^n \mathbf{e}_\mathrm{j} \hat{A}_j \right ) \langle \mathbf{r} | \psi \rangle = \left \langle \mathbf{r} \Bigg | \sum_{j=1}^n \mathbf{e}_\mathrm{j} \hat{A}_j \Bigg | \psi \right \rangle \\
 
\end{align} \,\!</math>
 
===Commutation of operators on ''Ψ''===
 
{{main|Commutator}}
 
If two observables ''A'' and ''B'' have linear operators <math> \hat{A} </math> and <math> \hat{B} </math>, the commutator is defined by,
 
:<math> \left [ \hat{A}, \hat{B} \right ] = \hat{A} \hat{B} - \hat{B} \hat{A} </math>
 
The commutator is itself a (composite) operator. Acting the commutator on ''ψ'' gives:
 
:<math> \left [ \hat{A}, \hat{B} \right ] \psi = \hat{A} \hat{B} \psi - \hat{B} \hat{A} \psi . </math>
 
If ''ψ'' is an eigenfunction with eigenvalues ''a'' and ''b'' for observables ''A'' and ''B'' respectively, and if the operators commute:
 
:<math> \left [ \hat{A}, \hat{B} \right ] \psi = 0, </math>
 
then the observables ''A'' and ''B'' can be measured at the same time with measurable eigenvalues ''a'' and ''b'' respectively. To illustrate this:
 
:<math> \begin{align}\left [ \hat{A}, \hat{B} \right ] \psi & = \hat{A} \hat{B} \psi - \hat{B} \hat{A} \psi \\
& = a(b \psi) - b(a \psi) \\
& = 0 .\\
\end{align} </math>
 
If the operators do not commute:
 
:<math> \left [ \hat{A}, \hat{B} \right ] \psi \neq 0, </math>
 
they can't be measured simultaneously to arbitrary precision, and there is an uncertainty relation between the observables, even if ''ψ''  is an eigenfunction. Notable pairs are position and momentum, and energy and time - [[Uncertainty principle|Hiesenberg's uncertainty relations]], and the angular momenta (spin, orbital and total) about any two orthogonal axes (such as ''L<sub>x</sub>'' and ''L<sub>y</sub>'', or ''s<sub>y</sub>'' and ''s<sub>z</sub>'' etc.).
 
===Expectation values of operators on ''Ψ''===
 
The [[expectation value]] (equivalently the average or mean value) is the average measurement of an observable, for particle in region ''R''. The expectation value <math>\langle \hat{A} \rangle </math> of the operator <math> \hat{A} </math> is calculated from<ref>Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10) 0 07 145546 9</ref>:
 
:<math>\langle \hat{A} \rangle = \int_R \psi^{*}\left( \mathbf{r} \right ) \hat{A} \psi \left( \mathbf{r} \right ) \mathrm{d}^3\mathbf{r} = \langle \psi | \hat{A} | \psi \rangle .</math>
 
This can be generalized to any function ''F'' of an operator:
 
:<math> \langle F ( \hat{A} ) \rangle = \int_R \psi(\mathbf{r})^{*} \left [ F ( \hat{A} ) \psi(\mathbf{r}) \right ] \mathrm{d}^3 \mathbf{r} = \langle \psi | F ( \hat{A} ) | \psi \rangle , </math>
 
An example of ''F'' is the 2-fold action of ''A'' on ''ψ'', i.e. squaring an operator or doing it twice:
 
:<math>\begin{align}
& F(\hat{A}) = \hat{A}^2 \\
& \Rightarrow \langle \hat{A}^2 \rangle = \int_R \psi^{*} \left( \mathbf{r} \right ) \hat{A}^2 \psi \left( \mathbf{r} \right ) \mathrm{d}^3\mathbf{r} = \langle \psi \vert \hat{A}^2 \vert \psi \rangle \\
\end{align}\,\!</math>
 
===Hermitian operators===
 
{{Main|Self-adjoint operator}}
 
The definition of a [[Hermitian operator]] is <ref>Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISRTY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0</ref>:
 
:<math>\hat{A} = \hat{A}^\dagger</math>
 
Following from this, in bra-ket notation:
 
:<math>\langle \phi_i | \hat{A} | \phi_j \rangle = \langle \phi_j | \hat{A} | \phi_i \rangle^*.</math>
 
Important properties of Hermitian operators include:
 
*real eigenvalues,
*eigenvectors with different eigenvalues are [[orthogonal]],
*eigenvectors can be chosen to be a complete [[orthonormal basis]],
 
===Operators in Matrix mechanics ===
 
An operator can be written in matrix form to map one basis vector to another. Since the operators and basis vectors are linear, the matrix is a [[linear transformation]] (aka transition matrix) between bases. Each basis element <math>\phi_j </math> can be connected to another<ref>Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10) 0 07 145546 9</ref>, by the expression:
 
:<math>A_{ij} = \langle \phi_i | \hat{A} | \phi_j \rangle,</math>
 
which is a matrix element:
 
:<math>\hat{A} = \begin{pmatrix}
A_{11} & A_{12} & \cdots & A_{1n} \\
A_{21} & A_{22} & \cdots & A_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n1} & A_{n2} & \cdots & A_{nn} \\
\end{pmatrix}
</math>
 
A further property of a Hermitian operator is that eigenfunctions corresponding to different eigenvalues are orthogonal.<ref>Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISRTY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0</ref> In matrix form, operators allow real eigenvalues to be found, corresponding to measurements. Orthogonality allows a suitable basis set of vectors to represent the state of the quantum system. The eigenvalues of the operator are also evaluated in the same way as for the square matrix, by solving the [[characteristic polynomial]]:
 
:<math> \det\left ( \hat{A} - a \hat{I} \right ) = 0 ,</math>
 
where ''I'' is the ''n'' × ''n'' [[identity matrix]], as an operator it corresponds to the identity operator. For a discrete basis:
 
:<math> \hat{I} = \sum_i |\phi_i\rangle\langle\phi_i|</math>
 
while for a continuous basis:
 
:<math> \hat{I} = \int |\phi\rangle\langle\phi|d\phi</math>
 
=== Inverse of an operator ===
 
A non-singular operator <math>\hat{A}</math> has an inverse <math> \hat{A}^{-1} </math> defined by:
 
:<math> \hat{A}\hat{A}^{-1} = \hat{A}^{-1}\hat{A} = \hat{I} </math>
 
If an operator has no inverse, it is a singular operator. In a finite-dimensional space, the determinant of a non-singular operator is non-zero:
 
:<math> \det(\hat{A}) \neq 0</math>
 
and hence it is zero for a singular operator.
 
===Table of QM operators===
 
The operators used in quantum mechanics are collected in the table below (see for example,<ref>Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISRTY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0</ref><ref>Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, ISBN 0-19-855493-1</ref>). The bold-face vectors with circumflexes are not [[unit vector]]s, they are 3-vector operators; all three spatial components taken together.
 
:{| class="wikitable"
|-valign="top"
 
! scope="col" width="200" | Operator (common name/s)
! scope="col" width="200" | Cartesian component
! scope="col" width="200" | General definition
! scope="col" width="100" | SI unit
! scope="col" width="100" | Dimension
|-valign="top"
! [[Position operator|Position]]
|<math>\begin{align} \hat{x} = x \\
\hat{y} = y \\
\hat{z} = z
\end{align}</math>
|<math> \mathbf{\hat{r}} = \mathbf{r} \,\!</math>
| m
| [L]
|-valign="top"
!rowspan="2"| [[Momentum operator|Momentum]]
|General
 
<math> \begin{align}
\hat{p}_x & = -i \hbar \frac{\partial }{\partial x} \\
\hat{p}_y & = -i \hbar \frac{\partial }{\partial y} \\
\hat{p}_z & = -i \hbar \frac{\partial }{\partial z}
\end{align}</math>
|General
 
<math> \mathbf{\hat{p}} = -i \hbar \nabla \,\!</math>
| J s m<sup>−1</sup> = N s
| [M] [L] [T]<sup>−1</sup>
|-valign="top"
|Electromagnetic field
 
<math> \begin{align}
\hat{p}_x = -i \hbar \frac{\partial }{\partial x} - qA_x \\
\hat{p}_y = -i \hbar \frac{\partial }{\partial y} - qA_y \\
\hat{p}_z = -i \hbar \frac{\partial }{\partial z} - qA_z
\end{align}</math>
|Electromagnetic field (uses [[kinetic momentum]], '''A''' = vector potential)
 
<math> \begin{align}
\mathbf{\hat{p}} & = \bold{\hat{P}} - q\bold{A} \\
& = -i \hbar \nabla - q\bold{A} \\
\end{align}\,\!</math>
| J s m<sup>−1</sup> = N s
| [M] [L] [T]<sup>−1</sup>
|-valign="top"
!rowspan="3"| [[Kinetic energy]]
| Translation
<math> \begin{align} \hat{T}_x & = -\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial x^2} \\
\hat{T}_y & = -\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial y^2} \\
\hat{T}_z & = -\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial z^2} \\
\end{align} </math>
|
<math> \begin{align} \hat{T} & = \frac{\mathbf{\hat{p}}\cdot\mathbf{\hat{p}}}{2m} \\
& = \frac{(-i \hbar \nabla)\cdot(-i \hbar \nabla)}{2m} \\
& = \frac{-\hbar^2 }{2m}\nabla^2
\end{align}\,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-valign="top"
|Electromagnetic field
 
<math> \begin{align} \hat{T}_x & = \frac{1}{2m}\left(-i \hbar \frac{\partial }{\partial x } - q A_x \right)^2 \\
\hat{T}_y & = \frac{1}{2m}\left(-i \hbar \frac{\partial }{\partial y} - q A_y \right)^2 \\
\hat{T}_z & = \frac{1}{2m}\left(-i \hbar \frac{\partial }{\partial z} - q A_z \right)^2
\end{align}\,\!</math>
|Electromagnetic field ('''A''' = [[vector potential]])
 
<math> \begin{align} \hat{T} & = \frac{\mathbf{\hat{p}}\cdot\mathbf{\hat{p}}}{2m} \\
& = \frac{1}{2m}(-i \hbar \nabla - q\bold{A})\cdot(-i \hbar \nabla - q\bold{A}) \\
& = \frac{1}{2m}(-i \hbar \nabla - q\bold{A})^2
\end{align}\,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-valign="top"
|Rotation (''I'' = [[moment of inertia]])
 
<math> \begin{align}
\hat{T}_{xx} & = \frac{\hat{J}_x^2}{2I_{xx}} \\
\hat{T}_{yy} & = \frac{\hat{J}_y^2}{2I_{yy}} \\
\hat{T}_{zz} & = \frac{\hat{J}_y^2}{2I_{zz}} \\
\end{align}\,\!</math>
|Rotation
 
<math> \hat{T} = \frac{\bold{\hat{J}}\cdot\bold{\hat{J}}}{2I} \,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-valign="top"
! Potential energy
| N/A
|<math> \hat{V} = V\left ( \mathbf{r}, t \right ) = V \,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-valign="top"
! Total [[Energy operator|energy]]
|N/A
|Time-dependent potential:<br />
<math> \hat{E} = i \hbar \frac{\partial }{\partial t} \,\!</math>
 
Time-independent:<br />
<math> \hat{E} = E \,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-valign="top"
! [[Hamiltonian operator|Hamiltonian]]
|
|<math> \begin{align} \hat{H} & = \hat{T} + \hat{V} \\
& = \frac{\bold{\hat{p}}\cdot\bold{\hat{p}}}{2m} + V \\
& = \frac{\hat{p}^2}{2m} + V \\
\end{align} \,\!</math>
| J
| [M] [L]<sup>2</sup> [T]<sup>−2</sup>
|-valign="top"
! [[Angular momentum operator]]
|<math>\begin{align}
\hat{L}_x & = -i\hbar \left(y {\partial\over \partial z} - z {\partial\over \partial y}\right)\\
\hat{L}_y & = -i\hbar \left(z {\partial\over \partial x} - x {\partial\over \partial z}\right)\\
\hat{L}_z & = -i\hbar \left(x {\partial\over \partial y} - y {\partial\over \partial x}\right)
\end{align}</math>
||<math>\mathbf{\hat{L}} = -i\hbar \mathbf{r} \times \nabla </math>
|| J s = N s m<sup>−1</sup>
|| [M] [L]<sup>2</sup> [T]<sup>−1</sup>
|-valign="top"
! [[Spin (physics)|Spin]] angular momentum
|<math>\begin{align}
\hat{S}_x & = {\hbar \over 2} \sigma_x \\
\hat{S}_y = {\hbar \over 2} \sigma_y \\
\hat{S}_z = {\hbar \over 2} \sigma_z
\end{align}</math>
 
where
 
<math>
\sigma_x = \begin{pmatrix}
0&1\\
1&0
\end{pmatrix}
</math>
 
<math>
\sigma_y = \begin{pmatrix}
0&-i\\
i&0
\end{pmatrix}
</math>
 
<math>
\sigma_z = \begin{pmatrix}
1&0\\
0&-1
\end{pmatrix}
</math>
 
are the [[pauli matrices]] for [[spin-½]] particles.
|<math>\mathbf{\hat{S}} = {\hbar \over 2} \boldsymbol{\sigma} \,\!</math>
 
where '''σ''' is the vector whose components are the pauli matrices.
| J s = N s m<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−1</sup>
|-valign="top"
 
! Total angular momentum
||<math>\begin{align}
\hat{J}_x & = \hat{L}_x + \hat{S}_x\\
\hat{J}_y & = \hat{L}_y + \hat{S}_y\\
\hat{J}_z & = \hat{L}_z + \hat{S}_z
\end{align}</math>
||<math>\begin{align}
\mathbf{\hat{J}} & = \mathbf{\hat{L}}+\mathbf{\hat{S}} \\
& = -i\hbar \bold{r}\times\nabla + \frac{\hbar}{2}\boldsymbol{\sigma}
\end{align}</math>
|| C m
|| [I] [T] [L]
|-valign="top"
! [[Transition dipole moment]] (electric)
||<math>\begin{align}
\hat{d}_x & = q\hat{x}\\
\hat{d}_y & = q\hat{y}\\
\hat{d}_z & = q\hat{z}
\end{align}</math>
||<math>\mathbf{\hat{d}} = q \mathbf{\hat{r}} </math>
|| C m
|| [I] [T] [L]
|-valign="top"
|}
 
===Examples of applying quantum operators===
 
The procedure for extracting information from a wave function is as follows. Consider the momentum ''p'' of a particle as an example. The momentum operator in one dimension is:
 
:<math>\hat{p} = -i\hbar\frac{\partial }{\partial x}</math>
 
Letting this act on ''ψ'' we obtain:
 
:<math>\hat{p} \psi = -i\hbar\frac{\partial }{\partial x} \psi ,</math>
 
if ''ψ'' is an eigenfunction of <math>\hat{p}</math>, then the momentum eigenvalue ''p'' is the value of the particle's momentum, found by:
 
:<math> -i\hbar\frac{\partial }{\partial x} \psi = p \psi.</math>
 
For three dimensions the momentum operator uses the [[nabla symbol|nabla]] operator to become:
 
:<math>\mathbf{\hat{p}} = -i\hbar\nabla .</math>
 
In Cartesian coordinates (using the standard Cartesian basis vectors '''e'''<sub>x</sub>, '''e'''<sub>y</sub>, '''e'''<sub>z</sub>) this can be written;
 
:<math>\mathbf{e}_\mathrm{x}\hat{p}_x + \mathbf{e}_\mathrm{y}\hat{p}_y + \mathbf{e}_\mathrm{z}\hat{p}_z = -i\hbar\left ( \mathbf{e}_\mathrm{x} \frac{\partial }{\partial x} + \mathbf{e}_\mathrm{y} \frac{\partial }{\partial y} + \mathbf{e}_\mathrm{z} \frac{\partial }{\partial z} \right ),</math>
 
that is:
 
:<math> \hat{p}_x = -i\hbar \frac{\partial}{\partial x}, \quad \hat{p}_y = -i\hbar \frac{\partial}{\partial y} , \quad \hat{p}_z = -i\hbar \frac{\partial}{\partial z} \,\!</math>
 
The process of finding eigenvalues is the same. Since this is a vector and operator equation, if ''ψ'' is an eigenfunction, then each component of the momentum operator will have an eigenvalue corresponding to that component of momentum. Acting <math> \mathbf{\hat{p}} </math> on ''ψ'' obtains:
 
:<math> \begin{align}
\hat{p}_x \psi & = -i\hbar \frac{\partial}{\partial x} \psi = p_x \psi \\
\hat{p}_y \psi & = -i\hbar \frac{\partial}{\partial y} \psi = p_y \psi \\
\hat{p}_z \psi & = -i\hbar \frac{\partial}{\partial z} \psi = p_z \psi \\
\end{align} \,\!</math>
 
==See also==
<div class="references-small" style="-moz-column-count:3; column-count:3;">
*[[Bounded linear operator]]
*[[Representation theory]]
</div>
 
==References==
 
{{reflist}}
 
{{Physics operator}}
 
{{DEFAULTSORT:Operator (Physics)}}
[[Category:Operator theory]]
[[Category:Theoretical physics]]
 
[[ar:مؤثر (فيزياء)]]
[[de:Operator (Mathematik)#Operatoren der Physik]]
[[fr:Opérateur (physique)]]
[[lt:Operatoriai kvantinėje mechanikoje]]
[[pt:Operador (física)]]
[[ru:Оператор (физика)]]
[[zh:算符 (物理學)]]

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