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. They play a central role in describing observables (measurable quantities like energy, momentum, etc.).

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 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 Quantum Mechanics operators

The operators used in quantum mechanics are collected in the table below. 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 ${\displaystyle \hat{T}=\frac{\hat{\mathbf{J}}\cdot \hat{\mathbf{J}}}{2I}}$[citation needed]	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:

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

Event horizon

From Wikipedia, the free encyclopedia

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

Diagram of the event horizon of a Schwarzschild black hole

In astrophysics, an event horizon is a boundary in spacetime beyond which no signal can ever reach a given observer. Wolfgang Rindler coined the term in the 1950s.

In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact objects that even light cannot escape. At that time, the Newtonian theory of gravitation and the so-called corpuscular theory of light were dominant. In these theories, if the escape velocity of the gravitational influence of a massive object exceeds the speed of light, then light originating inside or from it can escape temporarily but will return. In 1958, David Finkelstein used general relativity to introduce a stricter definition of a local black hole event horizon as a boundary beyond which events of any kind cannot affect an outside observer, leading to information and firewall paradoxes, encouraging the re-examination of the concept of local event horizons and the notion of black holes. Several theories were subsequently developed, some with and some without event horizons. One of the leading developers of theories to describe black holes, Stephen Hawking, suggested that an apparent horizon should be used instead of an event horizon, saying, "Gravitational collapse produces apparent horizons but no event horizons." He eventually concluded that "the absence of event horizons means that there are no black holes – in the sense of regimes from which light can't escape to infinity."

Any object approaching the horizon from the observer's side appears to slow down, never quite crossing the horizon. Due to gravitational redshift, its image reddens over time as the object moves closer to the horizon.

In an expanding universe, the speed of expansion reaches – and even exceeds – the speed of light, preventing signals from traveling to some regions. A cosmic event horizon is a real event horizon because it affects all kinds of signals, including gravitational waves, which travel at the speed of light.

More specific horizon types include the related but distinct absolute and apparent horizons found around a black hole. Other distinct types include:

• The Cauchy and Killing horizons.
• The photon spheres and ergospheres of the Kerr solution.
• Particle and cosmological horizons relevant to cosmology.
• Isolated and dynamical horizons, which are important in current black hole research.

Cosmic event horizon

Main article: Cosmological horizon

The reachable universe as a function of time and distance, in context of the expanding Universe

In cosmology, the event horizon of the observable universe is the largest comoving distance from which light emitted now can ever reach the observer in the future. This differs from the concept of the particle horizon, which represents the largest comoving distance from which light emitted in the past could reach the observer at a given time. For events that occur beyond that distance, light has not had enough time to reach our location, even if it was emitted at the time the universe began. The evolution of the particle horizon with time depends on the nature of the expansion of the universe. If the expansion has certain characteristics, parts of the universe will never be observable, no matter how long the observer waits for the light from those regions to arrive. The boundary beyond which events cannot ever be observed is an event horizon, and it represents the maximum extent of the particle horizon.

The criterion for determining whether a particle horizon for the universe exists is as follows. Define a comoving distance d_p as

${\displaystyle d_p={\int }_0^{t_0}\frac{c}{a(t)}dt.}$

In this equation, a is the scale factor, c is the speed of light, and t₀ is the age of the Universe. If d_p → ∞ (i.e., points arbitrarily as far away as can be observed), then no event horizon exists. If d_p ≠ ∞, a horizon is present.

Examples of cosmological models without an event horizon are universes dominated by matter or by radiation. An example of a cosmological model with an event horizon is a universe dominated by the cosmological constant (a de Sitter universe).

A calculation of the speeds of the cosmological event and particle horizons was given in a paper on the FLRW cosmological model, approximating the Universe as composed of non-interacting constituents, each one being a perfect fluid.

Apparent horizon of an accelerated particle

See also: Hyperbolic motion (relativity)

Spacetime diagram showing a uniformly accelerated particle, P, and an event E that is outside the particle's apparent horizon. The event's forward light cone never intersects the particle's world line.

If a particle is moving at a constant velocity in a non-expanding universe free of gravitational fields, any event that occurs in that Universe will eventually be observable by the particle, because the forward light cones from these events intersect the particle's world line. On the other hand, if the particle is accelerating, in some situations light cones from some events never intersect the particle's world line. Under these conditions, an apparent horizon is present in the particle's (accelerating) reference frame, representing a boundary beyond which events are unobservable.

For example, this occurs with a uniformly accelerated particle. A spacetime diagram of this situation is shown in the figure to the right. As the particle accelerates, it approaches, but never reaches, the speed of light with respect to its original reference frame. On the spacetime diagram, its path is a hyperbola, which asymptotically approaches a 45-degree line (the path of a light ray). An event whose light cone's edge is this asymptote or is farther away than this asymptote can never be observed by the accelerating particle. In the particle's reference frame, there is a boundary behind it from which no signals can escape (an apparent horizon). The distance to this boundary is given by ${\displaystyle c^2/a}$, where a is the constant proper acceleration of the particle.

While approximations of this type of situation can occur in the real world (in particle accelerators, for example), a true event horizon is never present, as this requires the particle to be accelerated indefinitely (requiring arbitrarily large amounts of energy and an arbitrarily large apparatus).

Interacting with a cosmic horizon

In the case of a horizon perceived by a uniformly accelerating observer in empty space, the horizon seems to remain a fixed distance from the observer no matter how its surroundings move. Varying the observer's acceleration may cause the horizon to appear to move over time or may prevent an event horizon from existing, depending on the acceleration function chosen. The observer never touches the horizon and never passes a location where it appeared to be.

In the case of a horizon perceived by an occupant of a de Sitter universe, the horizon always appears to be a fixed distance away for a non-accelerating observer. It is never contacted, even by an accelerating observer.

Event horizon of a black hole

Main article: Black hole

 

Far away from the black hole, a particle can move in any direction, as illustrated by the set of arrows. It is restricted only by the speed of light.

 

Closer to the black hole, spacetime starts to deform. There are more paths going towards the black hole than paths moving away.

 

Inside of the event horizon, all paths bring the particle closer to the centre of the black hole. It is no longer possible for the particle to escape.

 

NASA simulation (360-degree equirectangular views) of a test camera orbiting a black hole. Due to its speed, the accretion disk appears brighter in the direction of motion, with the black hole shadow below.

 

As the camera approaches the event horizon, the view of the outside universe appears to contract ahead while the shadow grows beneath.

 

Inside of the event horizon the roles of space and time switch. The singularity becomes an inevitable moment in the future and conversely the event horizon becomes a moment in the past. The black hole shadow approaches half the field of view, while simultaneously the aberration of light (due to exponentially-increasing tidal forces) concentrate the view of the outside universe into a thin band. In the final moments, all views of the outside universe eventually pinch off into total darkness.

One of the best-known examples of an event horizon derives from general relativity's description of a black hole, a celestial object so dense that no nearby matter or radiation can escape its gravitational field. Often, this is described as the boundary within which the black hole's escape velocity is greater than the speed of light. However, a more detailed description is that within this horizon, all lightlike paths (paths that light could take) (and hence all paths in the forward light cones of particles within the horizon) are warped so as to fall farther into the hole. Once a particle is inside the horizon, moving into the hole is as inevitable as moving forward in time – no matter in what direction the particle is travelling – and can be thought of as equivalent to doing so, depending on the spacetime coordinate system used.

The surface at the Schwarzschild radius acts as an event horizon in a non-rotating body that fits inside this radius (although a rotating black hole operates slightly differently). The Schwarzschild radius of an object is proportional to its mass. Theoretically, any amount of matter will become a black hole if compressed into a space that fits within its corresponding Schwarzschild radius. For the mass of the Sun, this radius is approximately 3 kilometers (1.9 miles); for Earth, it is about 9 millimeters (0.35 inches). In practice, however, neither Earth nor the Sun have the necessary mass (and, therefore, the necessary gravitational force) to overcome electron and neutron degeneracy pressure. The minimal mass required for a star to collapse beyond these pressures is the Tolman–Oppenheimer–Volkoff limit, which is approximately three solar masses.

According to the fundamental gravitational collapse models, an event horizon forms before the singularity of a black hole. If all the stars in the Milky Way would gradually aggregate towards the galactic center while keeping their proportionate distances from each other, they will all fall within their joint Schwarzschild radius long before they are forced to collide. Up to the collapse in the far future, observers in a galaxy surrounded by an event horizon would proceed with their lives normally.

Black hole event horizons are widely misunderstood. Common, although erroneous, is the notion that black holes "vacuum up" material in their neighborhood, where in fact they are no more capable of seeking out material to consume than any other gravitational attractor. As with any mass in the universe, matter must come within its gravitational scope for the possibility to exist of capture or consolidation with any other mass. Equally common is the idea that matter can be observed falling into a black hole. This is not possible. Astronomers can detect only accretion disks around black holes, where material moves with such speed that friction creates high-energy radiation that can be detected (similarly, some matter from these accretion disks is forced out along the axis of spin of the black hole, creating visible jets when these streams interact with matter such as interstellar gas or when they happen to be aimed directly at Earth). Furthermore, a distant observer will never actually see something reach the horizon. Instead, while approaching the hole, the object will seem to go ever more slowly, while any light it emits will be further and further redshifted.

The event horizon can also be defined by the causal structure of spacetime. Trajectories crossing a point in spacetime can only follow paths in a light cone limited by the speed of light. Curvature of spacetime tips the light cones. At the event horizon of a black hole, curvature becomes so strong that there are no paths that lead away from the black hole.

Topologically, the event horizon is defined from the causal structure as the past null cone of future conformal timelike infinity. A black hole event horizon is teleological in nature, meaning that it is determined by future causes. More precisely, one would need to know the entire history of the universe and all the way into the infinite future to determine the presence of an event horizon, which is not possible for quasilocal observers (not even in principle). In other words, there is no experiment and/or measurement that can be performed within a finite-size region of spacetime and within a finite time interval that answers the question of whether or not an event horizon exists. Because of the purely theoretical nature of the event horizon, the traveling object does not necessarily experience strange effects and does, in fact, pass through the calculated boundary in a finite amount of its proper time.

Interacting with black hole horizons

A misconception concerning event horizons, especially black hole event horizons, is that they represent an immutable surface that destroys objects that approach them. In practice, all event horizons appear to be some distance away from any observer, and objects sent towards an event horizon never appear to cross it from the sending observer's point of view (as the horizon-crossing event's light cone never intersects the observer's world line). Attempting to make an object near the horizon remain stationary with respect to an observer requires applying a force whose magnitude increases unboundedly (becoming infinite) the closer it gets.

In the case of the horizon around a black hole, observers stationary with respect to a distant object will all agree on where the horizon is. While this seems to allow an observer lowered towards the hole on a rope (or rod) to contact the horizon, in practice this cannot be done. The proper distance to the horizon is finite, so the length of rope needed would be finite as well, but if the rope were lowered slowly (so that each point on the rope was approximately at rest in Schwarzschild coordinates), the proper acceleration (G-force) experienced by points on the rope closer and closer to the horizon would approach infinity, so the rope would be torn apart. If the rope is lowered quickly (perhaps even in freefall), then indeed the observer at the bottom of the rope can touch and even cross the event horizon. But once this happens it is impossible to pull the bottom of rope back out of the event horizon, since if the rope is pulled taut, the forces along the rope increase without bound as they approach the event horizon and at some point the rope must break. Furthermore, the break must occur not at the event horizon, but at a point where the second observer can observe it.

Assuming that the possible apparent horizon is far inside the event horizon, or there is none, observers crossing a black hole event horizon would not actually see or feel anything special happen at that moment. In terms of visual appearance, observers who fall into the hole perceive the eventual apparent horizon as a black impermeable area enclosing the singularity. Other objects that had entered the horizon area along the same radial path but at an earlier time would appear below the observer as long as they are not entered inside the apparent horizon, and they could exchange messages. Increasing tidal forces are also locally noticeable effects, as a function of the mass of the black hole. In realistic stellar black holes, spaghettification occurs early: tidal forces tear materials apart well before the event horizon. However, in supermassive black holes, which are found in centers of galaxies, spaghettification occurs inside the event horizon. A human astronaut would survive the fall through an event horizon only in a black hole with a mass of approximately 10,000 solar masses or greater.

Beyond general relativity

A cosmic event horizon is commonly accepted as a real event horizon, whereas the description of a local black hole event horizon given by general relativity is found to be incomplete and controversial. When the conditions under which local event horizons occur are modeled using a more comprehensive picture of the way the Universe works, that includes both relativity and quantum mechanics, local event horizons are expected to have properties that are different from those predicted using general relativity alone.

At present, it is expected by the Hawking radiation mechanism that the primary impact of quantum effects is for event horizons to possess a temperature and so emit radiation. For black holes, this manifests as Hawking radiation, and the larger question of how the black hole possesses a temperature is part of the topic of black hole thermodynamics. For accelerating particles, this manifests as the Unruh effect, which causes space around the particle to appear to be filled with matter and radiation.

According to the controversial black hole firewall hypothesis, matter falling into a black hole would be burned to a crisp by a high energy "firewall" at the event horizon.

An alternative is provided by the complementarity principle, according to which, in the chart of the far observer, infalling matter is thermalized at the horizon and reemitted as Hawking radiation, while in the chart of an infalling observer matter continues undisturbed through the inner region and is destroyed at the singularity. This hypothesis does not violate the no-cloning theorem as there is a single copy of the information according to any given observer. Black hole complementarity is actually suggested by the scaling laws of strings approaching the event horizon, suggesting that in the Schwarzschild chart they stretch to cover the horizon and thermalize into a Planck length-thick membrane.

A complete description of local event horizons generated by gravity is expected to, at minimum, require a theory of quantum gravity. One such candidate theory is M-theory. Another such candidate theory is loop quantum gravity.