Operator (physics)

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https://en.wikipedia.org/wiki/Operator_(physics) 

An operator is a function over a space of physical states onto another space of states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

Operators in classical mechanics

In classical mechanics, the movement of a particle (or system of particles) is completely determined by the Lagrangian ${\displaystyle L(q,\dot{q},t)}$ or equivalently the Hamiltonian ${\displaystyle H(q,p,t)}$, a function of the generalized coordinates q, generalized velocities ${\displaystyle \dot{q}=\mathrm{d}q/\mathrm{d}t}$ and its conjugate momenta:

${\displaystyle p=\frac{\mathrm{\partial }L}{\mathrm{\partial }\dot{q}}}$

If either L or H is independent of a generalized coordinate q, meaning the L and H do 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). Operators in classical mechanics are related to these symmetries.

More technically, when H is invariant under the action of a certain group of transformations G:

${\displaystyle S\in G,H(S(q,p))=H(q,p)}$.

The elements of G are physical operators, which map physical states among themselves.

Table of classical mechanics operators

Transformation	Operator	Position	Momentum
Translational symmetry	${\displaystyle X(\mathbf{a})}$	${\displaystyle \mathbf{r}\to \mathbf{r}+\mathbf{a}}$	${\displaystyle \mathbf{p}\to \mathbf{p}}$
Time translation symmetry	${\displaystyle U(t_0)}$	${\displaystyle \mathbf{r}(t)\to \mathbf{r}(t+t_0)}$	${\displaystyle \mathbf{p}(t)\to \mathbf{p}(t+t_0)}$
Rotational invariance	${\displaystyle R(\hat{\mathbf{n}},\theta )}$	${\displaystyle \mathbf{r}\to R(\hat{\mathbf{n}},\theta )\mathbf{r}}$	${\displaystyle \mathbf{p}\to R(\hat{\mathbf{n}},\theta )\mathbf{p}}$
Galilean transformations	${\displaystyle G(\mathbf{v})}$	${\displaystyle \mathbf{r}\to \mathbf{r}+\mathbf{v}t}$	${\displaystyle \mathbf{p}\to \mathbf{p}+m\mathbf{v}}$
Parity	${\displaystyle P}$	${\displaystyle \mathbf{r}\to -\mathbf{r}}$	${\displaystyle \mathbf{p}\to -\mathbf{p}}$
T-symmetry	${\displaystyle T}$	${\displaystyle \mathbf{r}\to \mathbf{r}(-t)}$	${\displaystyle \mathbf{p}\to -\mathbf{p}(-t)}$

where ${\displaystyle R(\hat{\mathit{n}},\theta )}$ is the rotation matrix about an axis defined by the unit vector ${\displaystyle \hat{\mathit{n}}}$ and angle θ.

Generators

If the transformation is infinitesimal, the operator action should be of the form

${\displaystyle I+ϵA,}$

where ${\displaystyle I}$ is the identity operator, ${\displaystyle ϵ}$ is a parameter with a small value, and ${\displaystyle A}$ 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.

As it was stated, ${\displaystyle T_{a}f(x)=f(x-a)}$. If ${\displaystyle a=ϵ}$ is infinitesimal, then we may write

${\displaystyle T_{ϵ}f(x)=f(x-ϵ)\approx f(x)-ϵf^{\prime }(x).}$

This formula may be rewritten as

${\displaystyle T_{ϵ}f(x)=(I-ϵD)f(x)}$

where ${\displaystyle D}$ 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 ${\displaystyle a}$ may be obtained by repeated application of the infinitesimal translation:

${\displaystyle T_{a}f(x)=\underset{N\to \mathrm{\infty }}{lim}{T}_{a/N}\cdots T_{a/N}f(x)}$

with the ${\displaystyle \cdots }$ standing for the application ${\displaystyle N}$ times. If ${\displaystyle N}$ is large, each of the factors may be considered to be infinitesimal:

${\displaystyle T_{a}f(x)=\underset{N\to \mathrm{\infty }}{lim}{\left(I-\frac{a}{N}D\right)}^{N}f(x).}$

But this limit may be rewritten as an exponential:

${\displaystyle T_{a}f(x)=\exp (-aD)f(x).}$

To be convinced of the validity of this formal expression, we may expand the exponential in a power series:

${\displaystyle T_{a}f(x)=\left(I-aD+\frac{a^2{D}^2}{2!}-\frac{a^3{D}^3}{3!}+\cdots \right)f(x).}$

The right-hand side may be rewritten as

${\displaystyle f(x)-a{f}^{\prime }(x)+\frac{a^2}{2!}{f}^{″}(x)-\frac{a^3}{3!}{f}^{(3)}(x)+\cdots }$

which is just the Taylor expansion of ${\displaystyle f(x-a)}$, which was our original value for ${\displaystyle T_{a}f(x)}$.

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.

Physical pure states in quantum mechanics are represented as unit-norm vectors (probabilities are normalized to one) in a special complex 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 eigenvalues, since they are values which may come up as the result of the experiment. Mathematically this means the operators must be Hermitian. 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 about Hermitian operators.

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 operators.

In the matrix mechanics formulation, the norm of the physical state should stay fixed, so the evolution operator should be unitary, and the operators can be represented as matrices. Any other symmetry, mapping a physical state into another, should keep this restriction.

Wavefunction

Main article: wavefunction

The wavefunction must be square-integrable (see L^p spaces), meaning:

${\displaystyle {\iiint }_{{\mathbb{R}}^3}|\psi (\mathbf{r}){|}^2{d}^3\mathbf{r}={\iiint }_{{\mathbb{R}}^3}\psi (\mathbf{r}{)}^{\ast }\psi (\mathbf{r})d^3\mathbf{r}<\mathrm{\infty }}$

and normalizable, so that:

${\displaystyle {\iiint }_{{\mathbb{R}}^3}|\psi (\mathbf{r}){|}^2{d}^3\mathbf{r}=1}$

Two cases of eigenstates (and eigenvalues) are:

• for discrete eigenstates ${\displaystyle |{\psi }_i⟩}$ forming a discrete basis, so any state is a sum $${\displaystyle |\psi ⟩=\sum _i{c}_i|{\varphi }_i⟩}$$ where c_i are complex numbers such that |c_i|² = c_i^*c_i is the probability of measuring the state ${\displaystyle |{\varphi }_i⟩}$, and the corresponding set of eigenvalues a_i is also discrete - either finite or countably infinite. In this case, the inner product of two eigenstates is given by ${\displaystyle ⟨{\varphi }_i|{\varphi }_j⟩={\delta }_{ij}}$, where ${\displaystyle {\delta }_{mn}}$ denotes the Kronecker Delta. However,
• for a continuum of eigenstates ${\displaystyle |{\psi }_i⟩}$ forming a continuous basis, any state is an integral $${\displaystyle |\psi ⟩=\int c(\varphi )d\varphi |\varphi ⟩}$$ where c(φ) is a complex function such that |c(φ)|² = c(φ)^*c(φ) is the probability of measuring the state ${\displaystyle |\varphi ⟩}$, and there is an uncountably infinite set of eigenvalues a. In this case, the inner product of two eigenstates is defined as ${\displaystyle ⟨{\varphi }^{\prime }|\varphi ⟩=\delta (\varphi -{\varphi }^{\prime })}$, where here ${\displaystyle \delta (x-y)}$ denotes the Dirac Delta.

Linear operators in wave mechanics

Main articles: Wave function and Bra–ket notation

Let ψ be the wavefunction for a quantum system, and ${\displaystyle \hat{A}}$ be any linear operator for some observable A (such as position, momentum, energy, angular momentum etc.). If ψ is an eigenfunction of the operator ${\displaystyle \hat{A}}$, then

${\displaystyle \hat{A}\psi =a\psi ,}$

where a is the eigenvalue of the operator, corresponding to the measured value of the observable, i.e. observable A has a measured value a.

If ψ is an eigenfunction of a given operator ${\displaystyle \hat{A}}$, then a definite quantity (the eigenvalue a) will be observed if a measurement of the observable A is made on the state ψ. Conversely, if ψ is not an eigenfunction of ${\displaystyle \hat{A}}$, then it has no eigenvalue for ${\displaystyle \hat{A}}$, and the observable does not have a single definite value in that case. Instead, measurements of the observable A will yield each eigenvalue with a certain probability (related to the decomposition of ψ relative to the orthonormal eigenbasis of ${\displaystyle \hat{A}}$).

In bra–ket notation the above can be written;

${\displaystyle \begin{array}{rl}\hat{A}\psi & =\hat{A}\psi (\mathbf{r})=\hat{A}⟨\mathbf{r}\mid \psi ⟩=⟨\mathbf{r}\left|\hat{A}\right|\psi ⟩\\ a\psi & =a\psi (\mathbf{r})=a⟨\mathbf{r}\mid \psi ⟩=⟨\mathbf{r}\mid a\mid \psi ⟩\end{array}}$

that are equal if ${\displaystyle |\psi ⟩}$ is an eigenvector, or eigenket of the observable A.

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:

${\displaystyle \hat{\mathbf{A}}=\sum _{j=1}^n{\mathbf{e}}_j{\hat{A}}_j}$

where e_j are basis vectors corresponding to each component operator A_j. Each component will yield a corresponding eigenvalue ${\displaystyle a_j}$. Acting this on the wave function ψ:

${\displaystyle \hat{\mathbf{A}}\psi =\left(\sum _{j=1}^n{\mathbf{e}}_j{\hat{A}}_j\right)\psi =\sum _{j=1}^n\left({\mathbf{e}}_j{\hat{A}}_j\psi \right)=\sum _{j=1}^n\left({\mathbf{e}}_j{a}_j\psi \right)}$

in which we have used ${\displaystyle {\hat{A}}_j\psi =a_j\psi .}$

In bra–ket notation:

${\displaystyle \begin{array}{rl}\hat{\mathbf{A}}\psi =\hat{\mathbf{A}}\psi (\mathbf{r})=\hat{\mathbf{A}}⟨\mathbf{r}\mid \psi ⟩& =⟨\mathbf{r}\left|\hat{\mathbf{A}}\right|\psi ⟩\\ \left(\sum _{j=1}^n{\mathbf{e}}_j{\hat{A}}_j\right)\psi =\left(\sum _{j=1}^n{\mathbf{e}}_j{\hat{A}}_j\right)\psi (\mathbf{r})=\left(\sum _{j=1}^n{\mathbf{e}}_j{\hat{A}}_j\right)⟨\mathbf{r}\mid \psi ⟩& =⟨\mathbf{r}\left|\sum _{j=1}^n{\mathbf{e}}_j{\hat{A}}_j\right|\psi ⟩\end{array}}$

Commutation of operators on Ψ

Main article: Commutator

If two observables A and B have linear operators ${\displaystyle \hat{A}}$ and ${\displaystyle \hat{B}}$, the commutator is defined by,

${\displaystyle \left[\hat{A},\hat{B}\right]=\hat{A}\hat{B}-\hat{B}\hat{A}}$

The commutator is itself a (composite) operator. Acting the commutator on ψ gives:

${\displaystyle \left[\hat{A},\hat{B}\right]\psi =\hat{A}\hat{B}\psi -\hat{B}\hat{A}\psi .}$

If ψ is an eigenfunction with eigenvalues a and b for observables A and B respectively, and if the operators commute:

${\displaystyle \left[\hat{A},\hat{B}\right]\psi =0,}$

then the observables A and B can be measured simultaneously with infinite precision, i.e., uncertainties ${\displaystyle \mathrm{\Delta }A=0}$, ${\displaystyle \mathrm{\Delta }B=0}$ simultaneously. ψ is then said to be the simultaneous eigenfunction of A and B. To illustrate this:

${\displaystyle \begin{array}{rl}\left[\hat{A},\hat{B}\right]\psi & =\hat{A}\hat{B}\psi -\hat{B}\hat{A}\psi \\ & =a(b\psi )-b(a\psi )\\ & =0.\end{array}}$

It shows that measurement of A and B does not cause any shift of state, i.e., initial and final states are same (no disturbance due to measurement). Suppose we measure A to get value a. We then measure B to get the value b. We measure A again. We still get the same value a. Clearly the state (ψ) of the system is not destroyed and so we are able to measure A and B simultaneously with infinite precision.

If the operators do not commute:

${\displaystyle \left[\hat{A},\hat{B}\right]\psi \ne 0,}$

they cannot be prepared simultaneously to arbitrary precision, and there is an uncertainty relation between the observables

${\displaystyle \mathrm{\Delta }A\mathrm{\Delta }B\ge \left|\frac{1}{2}⟨[A,B]⟩\right|}$

even if ψ is an eigenfunction the above relation holds. Notable pairs are position-and-momentum and energy-and-time uncertainty relations, and the angular momenta (spin, orbital and total) about any two orthogonal axes (such as L_x and L_y, or s_y and s_z, 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 ${\displaystyle ⟨\hat{A}⟩}$ of the operator ${\displaystyle \hat{A}}$ is calculated from:

${\displaystyle ⟨\hat{A}⟩={\int }_R{\psi }^{\ast }\left(\mathbf{r}\right)\hat{A}\psi \left(\mathbf{r}\right){\mathrm{d}}^3\mathbf{r}=⟨\psi \left|\hat{A}\right|\psi ⟩.}$

This can be generalized to any function F of an operator:

${\displaystyle ⟨F\left(\hat{A}\right)⟩={\int }_R\psi (\mathbf{r}{)}^{\ast }\left[F\left(\hat{A}\right)\psi (\mathbf{r})\right]{\mathrm{d}}^3\mathbf{r}=⟨\psi \left|F\left(\hat{A}\right)\right|\psi ⟩,}$

An example of F is the 2-fold action of A on ψ, i.e. squaring an operator or doing it twice:

${\displaystyle \begin{array}{rl}F\left(\hat{A}\right)& ={\hat{A}}^2\\ \Rightarrow ⟨{\hat{A}}^2⟩& ={\int }_R{\psi }^{\ast }\left(\mathbf{r}\right){\hat{A}}^2\psi \left(\mathbf{r}\right){\mathrm{d}}^3\mathbf{r}=⟨\psi \left|{\hat{A}}^2\right|\psi ⟩\end{array}}$

Hermitian operators

Main article: Self-adjoint operator

The definition of a Hermitian operator is:

${\displaystyle \hat{A}={\hat{A}}^{†}}$

Following from this, in bra–ket notation:

${\displaystyle ⟨{\varphi }_i\left|\hat{A}\right|{\varphi }_j⟩={⟨{\varphi }_j\left|\hat{A}\right|{\varphi }_i⟩}^{\ast }.}$

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 are linear, the matrix is a linear transformation (aka transition matrix) between bases. Each basis element ${\displaystyle {\varphi }_j}$ can be connected to another, by the expression:

${\displaystyle A_{ij}=⟨{\varphi }_i\left|\hat{A}\right|{\varphi }_j⟩,}$

which is a matrix element:

${\displaystyle \hat{A}=\left(\begin{array}{cccc}{A}_{11}& A_{12}& \cdots & A_{1n}\\ A_{21}& A_{22}& \cdots & A_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ A_{n1}& A_{n2}& \cdots & A_{nn}\end{array}\right)}$

A further property of a Hermitian operator is that eigenfunctions corresponding to different eigenvalues are orthogonal. 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:

${\displaystyle det\left(\hat{A}-a\hat{I}\right)=0,}$

where I is the n × n identity matrix, as an operator it corresponds to the identity operator. For a discrete basis:

${\displaystyle \hat{I}=\sum _i|{\varphi }_i⟩⟨{\varphi }_i|}$

while for a continuous basis:

${\displaystyle \hat{I}=\int |\varphi ⟩⟨\varphi |\mathrm{d}\varphi }$

Inverse of an operator

A non-singular operator ${\displaystyle \hat{A}}$ has an inverse ${\displaystyle {\hat{A}}^{-1}}$ defined by:

${\displaystyle \hat{A}{\hat{A}}^{-1}={\hat{A}}^{-1}\hat{A}=\hat{I}}$

If an operator has no inverse, it is a singular operator. In a finite-dimensional space, an operator is non-singular if and only if its determinant is nonzero:

${\displaystyle det\left(\hat{A}\right)\ne 0}$

and hence the determinant 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). The bold-face vectors with circumflexes are not unit vectors, they are 3-vector operators; all three spatial components taken together.

Operator (common name/s)	Cartesian component	General definition	SI unit	Dimension
Position	${\displaystyle \begin{array}{rlrlrl}\hat{x}& =x,& \hat{y}& =y,& \hat{z}& =z\end{array}}$	${\displaystyle \hat{\mathbf{r}}=\mathbf{r}}$	m	[L]
Momentum	General ${\displaystyle \begin{array}{rlrlrl}{\hat{p}}_x& =-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }x},& {\hat{p}}_y& =-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }y},& {\hat{p}}_z& =-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }z}\end{array}}$	General ${\displaystyle \hat{\mathbf{p}}=-i\hslash \mathrm{\nabla }}$	J s m−1 = N s	[M] [L] [T]−1
	Electromagnetic field ${\displaystyle \begin{array}{r}{\hat{p}}_x=-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }x}-q{A}_x\\ {\hat{p}}_y=-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }y}-q{A}_y\\ {\hat{p}}_z=-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }z}-q{A}_z\end{array}}$	Electromagnetic field (uses kinetic momentum; A, vector potential) ${\displaystyle \begin{array}{rl}\hat{\mathbf{p}}& =\hat{\mathbf{P}}-q\mathbf{A}\\ & =-i\hslash \mathrm{\nabla }-q\mathbf{A}\end{array}}$	J s m−1 = N s	[M] [L] [T]−1
Kinetic energy	Translation ${\displaystyle \begin{array}{rl}{\hat{T}}_x& =-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\\ {\hat{T}}_y& =-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}\\ {\hat{T}}_z& =-\frac{{\hslash }^2}{2m}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}\end{array}}$	${\displaystyle \begin{array}{rl}\hat{T}& =\frac{1}{2m}\hat{\mathbf{p}}\cdot \hat{\mathbf{p}}\\ & =\frac{1}{2m}(-i\hslash \mathrm{\nabla })\cdot (-i\hslash \mathrm{\nabla })\\ & =\frac{-{\hslash }^2}{2m}{\mathrm{\nabla }}^2\end{array}}$	J	[M] [L]2 [T]−2
	Electromagnetic field ${\displaystyle \begin{array}{rl}{\hat{T}}_x& =\frac{1}{2m}{\left(-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }x}-q{A}_x\right)}^2\\ {\hat{T}}_y& =\frac{1}{2m}{\left(-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }y}-q{A}_y\right)}^2\\ {\hat{T}}_z& =\frac{1}{2m}{\left(-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }z}-q{A}_z\right)}^2\end{array}}$	Electromagnetic field (A, vector potential) ${\displaystyle \begin{array}{rl}\hat{T}& =\frac{1}{2m}\hat{\mathbf{p}}\cdot \hat{\mathbf{p}}\\ & =\frac{1}{2m}(-i\hslash \mathrm{\nabla }-q\mathbf{A})\cdot (-i\hslash \mathrm{\nabla }-q\mathbf{A})\\ & =\frac{1}{2m}(-i\hslash \mathrm{\nabla }-q\mathbf{A}{)}^2\end{array}}$	J	[M] [L]2 [T]−2
	Rotation (I, moment of inertia) ${\displaystyle \begin{array}{rl}{\hat{T}}_{xx}& =\frac{{\hat{J}}_x^2}{2I_{xx}}\\ {\hat{T}}_{yy}& =\frac{{\hat{J}}_y^2}{2I_{yy}}\\ {\hat{T}}_{zz}& =\frac{{\hat{J}}_z^2}{2I_{zz}}\end{array}}$	Rotation ${\displaystyle \hat{T}=\frac{\hat{\mathbf{J}}\cdot \hat{\mathbf{J}}}{2I}}$	J	[M] [L]2 [T]−2
Potential energy	N/A	${\displaystyle \hat{V}=V\left(\mathbf{r},t\right)=V}$	J	[M] [L]2 [T]−2
Total energy	N/A	Time-dependent potential: ${\displaystyle \hat{E}=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}}$ Time-independent: ${\displaystyle \hat{E}=E}$	J	[M] [L]2 [T]−2
Hamiltonian		${\displaystyle \begin{array}{rl}\hat{H}& =\hat{T}+\hat{V}\\ & =\frac{1}{2m}\hat{\mathbf{p}}\cdot \hat{\mathbf{p}}+V\\ & =\frac{1}{2m}{\hat{p}}^2+V\end{array}}$	J	[M] [L]2 [T]−2
Angular momentum operator	${\displaystyle \begin{array}{rl}{\hat{L}}_x& =-i\hslash \left(y\frac{\mathrm{\partial }}{\mathrm{\partial }z}-z\frac{\mathrm{\partial }}{\mathrm{\partial }y}\right)\\ {\hat{L}}_y& =-i\hslash \left(z\frac{\mathrm{\partial }}{\mathrm{\partial }x}-x\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)\\ {\hat{L}}_z& =-i\hslash \left(x\frac{\mathrm{\partial }}{\mathrm{\partial }y}-y\frac{\mathrm{\partial }}{\mathrm{\partial }x}\right)\end{array}}$	${\displaystyle \hat{\mathbf{L}}=\mathbf{r}\times -i\hslash \mathrm{\nabla }}$	J s = N s m	[M] [L]2 [T]−1
Spin angular momentum	${\displaystyle \begin{array}{rlrlrl}{\hat{S}}_x& =\frac{\hslash }{2}{\sigma }_x& {\hat{S}}_y& =\frac{\hslash }{2}{\sigma }_y& {\hat{S}}_z& =\frac{\hslash }{2}{\sigma }_z\end{array}}$ where ${\displaystyle \begin{array}{rl}{\sigma }_x& =\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\\ {\sigma }_y& =\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)\\ {\sigma }_z& =\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\end{array}}$ are the Pauli matrices for spin-1/2 particles.	${\displaystyle \hat{\mathbf{S}}=\frac{\hslash }{2}\mathit{\sigma }}$ where σ is the vector whose components are the Pauli matrices.	J s = N s m	[M] [L]2 [T]−1
Total angular momentum	${\displaystyle \begin{array}{rl}{\hat{J}}_x& ={\hat{L}}_x+{\hat{S}}_x\\ {\hat{J}}_y& ={\hat{L}}_y+{\hat{S}}_y\\ {\hat{J}}_z& ={\hat{L}}_z+{\hat{S}}_z\end{array}}$	${\displaystyle \begin{array}{rl}\hat{\mathbf{J}}& =\hat{\mathbf{L}}+\hat{\mathbf{S}}\\ & =-i\hslash \mathbf{r}\times \mathrm{\nabla }+\frac{\hslash }{2}\mathit{\sigma }\end{array}}$	J s = N s m	[M] [L]2 [T]−1
Transition dipole moment (electric)	${\displaystyle \begin{array}{rlrlrl}{\hat{d}}_x& =q\hat{x},& {\hat{d}}_y& =q\hat{y},& {\hat{d}}_z& =q\hat{z}\end{array}}$	${\displaystyle \hat{\mathbf{d}}=q\hat{\mathbf{r}}}$	C m	[I] [T] [L]

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 position basis in one dimension is:

${\displaystyle \hat{p}=-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }x}}$

Letting this act on ψ we obtain:

${\displaystyle \hat{p}\psi =-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }x}\psi ,}$

if ψ is an eigenfunction of ${\displaystyle \hat{p}}$, then the momentum eigenvalue p is the value of the particle's momentum, found by:

${\displaystyle -i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }x}\psi =p\psi .}$

For three dimensions the momentum operator uses the nabla operator to become:

${\displaystyle \hat{\mathbf{p}}=-i\hslash \mathrm{\nabla }.}$

In Cartesian coordinates (using the standard Cartesian basis vectors e_x, e_y, e_z) this can be written;

${\displaystyle {\mathbf{e}}_{\mathrm{x}}{\hat{p}}_x+{\mathbf{e}}_{\mathrm{y}}{\hat{p}}_y+{\mathbf{e}}_{\mathrm{z}}{\hat{p}}_z=-i\hslash \left({\mathbf{e}}_{\mathrm{x}}\frac{\mathrm{\partial }}{\mathrm{\partial }x}+{\mathbf{e}}_{\mathrm{y}}\frac{\mathrm{\partial }}{\mathrm{\partial }y}+{\mathbf{e}}_{\mathrm{z}}\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right),}$

that is:

${\displaystyle {\hat{p}}_x=-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }x},{\hat{p}}_y=-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }y},{\hat{p}}_z=-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }z}}$

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 ${\displaystyle \hat{\mathbf{p}}}$ on ψ obtains:

${\displaystyle \begin{array}{rl}{\hat{p}}_x\psi & =-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }x}\psi =p_x\psi \\ {\hat{p}}_y\psi & =-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }y}\psi =p_y\psi \\ {\hat{p}}_z\psi & =-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }z}\psi =p_z\psi \end{array}}$

Invariant mass

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Invariant_mass

The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, it is a characteristic of the system's total energy and momentum that is the same in all frames of reference related by Lorentz transformations. If a center-of-momentum frame exists for the system, then the invariant mass of a system is equal to its total mass in that "rest frame". In other reference frames, where the system's momentum is nonzero, the total mass (a.k.a. relativistic mass) of the system is greater than the invariant mass, but the invariant mass remains unchanged.

Because of mass–energy equivalence, the rest energy of the system is simply the invariant mass times the speed of light squared. Similarly, the total energy of the system is its total (relativistic) mass times the speed of light squared.

Systems whose four-momentum is a null vector (for example, a single photon or many photons moving in exactly the same direction) have zero invariant mass and are referred to as massless. A physical object or particle moving faster than the speed of light would have space-like four-momenta (such as the hypothesized tachyon), and these do not appear to exist. Any time-like four-momentum possesses a reference frame where the momentum (3-dimensional) is zero, which is a center of momentum frame. In this case, invariant mass is positive and is referred to as the rest mass.

If objects within a system are in relative motion, then the invariant mass of the whole system will differ from the sum of the objects' rest masses. This is also equal to the total energy of the system divided by c². See mass–energy equivalence for a discussion of definitions of mass. Since the mass of systems must be measured with a weight or mass scale in a center of momentum frame in which the entire system has zero momentum, such a scale always measures the system's invariant mass. For example, a scale would measure the kinetic energy of the molecules in a bottle of gas to be part of invariant mass of the bottle, and thus also its rest mass. The same is true for massless particles in such system, which add invariant mass and also rest mass to systems, according to their energy.

For an isolated massive system, the center of mass of the system moves in a straight line with a steady subluminal velocity (with a velocity depending on the reference frame used to view it). Thus, an observer can always be placed to move along with it. In this frame, which is the center-of-momentum frame, the total momentum is zero, and the system as a whole may be thought of as being "at rest" if it is a bound system (like a bottle of gas). In this frame, which exists under these assumptions, the invariant mass of the system is equal to the total system energy (in the zero-momentum frame) divided by c². This total energy in the center of momentum frame, is the minimum energy which the system may be observed to have, when seen by various observers from various inertial frames.

Note that for reasons above, such a rest frame does not exist for single photons, or rays of light moving in one direction. When two or more photons move in different directions, however, a center of mass frame (or "rest frame" if the system is bound) exists. Thus, the mass of a system of several photons moving in different directions is positive, which means that an invariant mass exists for this system even though it does not exist for each photon.

Possible 4-momenta of particles. One has zero invariant mass, the other is massive

Sum of rest masses

The invariant mass of a system includes the mass of any kinetic energy of the system constituents that remains in the center of momentum frame, so the invariant mass of a system may be greater than sum of the invariant masses (rest masses) of its separate constituents. For example, rest mass and invariant mass are zero for individual photons even though they may add mass to the invariant mass of systems. For this reason, invariant mass is in general not an additive quantity (although there are a few rare situations where it may be, as is the case when massive particles in a system without potential or kinetic energy can be added to a total mass).

Consider the simple case of two-body system, where object A is moving towards another object B which is initially at rest (in any particular frame of reference). The magnitude of invariant mass of this two-body system (see definition below) is different from the sum of rest mass (i.e. their respective mass when stationary). Even if we consider the same system from center-of-momentum frame, where net momentum is zero, the magnitude of the system's invariant mass is not equal to the sum of the rest masses of the particles within it.

The kinetic energy of such particles and the potential energy of the force fields increase the total energy above the sum of the particle rest masses, and both terms contribute to the invariant mass of the system. The sum of the particle kinetic energies as calculated by an observer is smallest in the center of momentum frame (again, called the "rest frame" if the system is bound).

They will often also interact through one or more of the fundamental forces, giving them a potential energy of interaction, possibly negative.

As defined in particle physics

In particle physics, the invariant mass m₀ is equal to the mass in the rest frame of the particle, and can be calculated by the particle's energy E and its momentum p as measured in any frame, by the energy–momentum relation: $${\displaystyle m_0^2{c}^2={\left(\frac{E}{c}\right)}^2-{\Vert \mathbf{p}\Vert }^2}$$ or in natural units where c = 1, $${\displaystyle m_0^2=E^2-{\Vert \mathbf{p}\Vert }^2.}$$

This invariant mass is the same in all frames of reference (see also special relativity). This equation says that the invariant mass is the pseudo-Euclidean length of the four-vector (E, p), calculated using the relativistic version of the Pythagorean theorem which has a different sign for the space and time dimensions. This length is preserved under any Lorentz boost or rotation in four dimensions, just like the ordinary length of a vector is preserved under rotations. In quantum theory the invariant mass is a parameter in the relativistic Dirac equation for an elementary particle. The Dirac quantum operator corresponds to the particle four-momentum vector.

Since the invariant mass is determined from quantities which are conserved during a decay, the invariant mass calculated using the energy and momentum of the decay products of a single particle is equal to the mass of the particle that decayed. The mass of a system of particles can be calculated from the general formula: $${\displaystyle {\left(W{c}^2\right)}^2={\left(\sum E\right)}^2-{\Vert \sum \mathbf{p}c\Vert }^2,}$$ where

• ${\displaystyle W}$ is the invariant mass of the system of particles, equal to the mass of the decay particle.
• $\sum E$ is the sum of the energies of the particles
• $\sum \mathbf{p}$ is the vector sum of the momentum of the particles (includes both magnitude and direction of the momenta)

The term invariant mass is also used in inelastic scattering experiments. Given an inelastic reaction with total incoming energy larger than the total detected energy (i.e. not all outgoing particles are detected in the experiment), the invariant mass (also known as the "missing mass") W of the reaction is defined as follows (in natural units): $${\displaystyle W^2={\left(\sum E_{\text{in}}-\sum E_{\text{out}}\right)}^2-{\Vert \sum {\mathbf{p}}_{\text{in}}-\sum {\mathbf{p}}_{\text{out}}\Vert }^2.}$$

If there is one dominant particle which was not detected during an experiment, a plot of the invariant mass will show a sharp peak at the mass of the missing particle.

In those cases when the momentum along one direction cannot be measured (i.e. in the case of a neutrino, whose presence is only inferred from the missing energy) the transverse mass is used.

Example: two-particle collision

In a two-particle collision (or a two-particle decay) the square of the invariant mass (in natural units) is $${\displaystyle \begin{array}{rl}{M}^2& =(E_1+E_2{)}^2-{\Vert {\mathbf{p}}_1+{\mathbf{p}}_2\Vert }^2\\ & =m_1^2+m_2^2+2\left(E_1{E}_2-{\mathbf{p}}_1\cdot {\mathbf{p}}_2\right).\end{array}}$$

Massless particles

The invariant mass of a system made of two massless particles whose momenta form an angle ${\displaystyle \theta }$ has a convenient expression: $${\displaystyle \begin{array}{rl}{M}^2& =(E_1+E_2{)}^2-{\Vert {\mathbf{\text{p}}}_1+{\mathbf{\text{p}}}_2\Vert }^2\\ & =[(p_1,0,0,p_1)+(p_2,0,p_2\sin \theta ,p_2\cos \theta ){]}^2\\ & =(p_1+p_2{)}^2-p_2^2{\sin }^2\theta -(p_1+p_2\cos \theta {)}^2\\ & =2p_1{p}_2(1-\cos \theta ).\end{array}}$$

Collider experiments

In particle collider experiments, one often defines the angular position of a particle in terms of an azimuthal angle ${\displaystyle \varphi }$ and pseudorapidity ${\displaystyle \eta }$. Additionally the transverse momentum, ${\displaystyle p_T}$, is usually measured. In this case if the particles are massless, or highly relativistic (${\displaystyle E\gg m}$) then the invariant mass becomes: $${\displaystyle M^2=2p_{T1}{p}_{T2}(\cosh ({\eta }_1-{\eta }_2)-\cos ({\varphi }_1-{\varphi }_2)).}$$

Rest energy

Rest energy (also called rest mass energy) is the energy associated with a particle's invariant mass.

The rest energy ${\displaystyle E_0}$ of a particle is defined as: $${\displaystyle E_0=m_0{c}^2,}$$ where ${\displaystyle c}$ is the speed of light in vacuum. In general, only differences in energy have physical significance.

The concept of rest energy follows from the special theory of relativity that leads to Einstein's famous conclusion about equivalence of energy and mass. See Special relativity § Relativistic dynamics and invariance.