Operator (physics)

Operator (physics)

In physics, an operator is a function acting on the space of physical states. As a result 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 are a very useful tool in classical mechanics. In quantum mechanics, on the other hand, they are an intrinsic part of the formulation of the theory.

Contents

Operators in classical mechanics

Let us consider a classical mechanics system led by a certain Hamiltonian H(q,p), function of the generalized coordinates q and its conjugate momenta. Let us consider this function to be invariant under the action of a certain group of transformations G, i.e., if S\in G, H(S(q,p))=H(q,p). The elements of G are physical operators, which map physical states among themselves.

An easy example is given by space translations. The hamiltonian of a translationally invariant problem does not change under the transformation q\to T_a q=q+a. Other straightforward symmetry operators are the ones implementing rotations.

If the physical system is described by a function, as in classical field theories, the translation operator is generalized in a straightforward way:

f(x) \to T_a f(x)=f(x-a).

Notice that the transformation inside the parenthesis should be the inverse of the transformation done on the coordinates.

Concept of generator

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

 I + \epsilon A

where I is the identity operator, \epsilon is a small parameter, and 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, Taf(x) = f(xa). If a=\epsilon is infinitesimal, then we may write

T_\epsilon f(x)=f(x-\epsilon)\approx f(x) - \epsilon f'(x).

This formula may be rewritten as

T_\epsilon f(x) = (I-\epsilon D) f(x)

where D is the generator of the translation group, which happens to be just 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 a may be obtained by repeated application of the infinitesimal translation:

T_a f(x) = \lim_{N\to\infty} T_{a/N} \cdots T_{a/N} f(x)

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

T_a f(x) = \lim_{N\to\infty} (I -(a/N) D)^N f(x).

But this limit may be rewritten as an exponential:

Taf(x) = exp( − aD)f(x).

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

T_a f(x) = \left( I - aD + {a^2D^2\over 2!} - {a^3D^3\over 3!} + \cdots \right) f(x).

The right-hand side may be rewritten as

f(x) - a f'(x) + {a^2\over 2!} f''(x) - {a^3\over 3!} f'''(x) + \cdots

which is just the Taylor expansion of f(xa), which was our original value for Taf(x).

Operators in quantum mechanics

The mathematical description of quantum mechanics is built upon the concept of an operator.

Physical pure states in quantum mechanics are unit-norm vectors in a certain vector space (a Hilbert space). Time evolution in this vector space is given by the application of a certain operator, called the evolution operator. Since the norm of the physical state should stay fixed, the evolution operator should be unitary. Any other symmetry, mapping a physical state into another, should keep this restriction.

Any observable, i.e., any quantity which can be measured in a physical experiment, should be associated with a self-adjoint linear operator. The values which may come up as the result of the experiment are the eigenvalues of the operator. The probability of each eigenvalue is related to the projection of the physical state on the subspace related to that eigenvalue.

Table of QM operators

The operators used in quantum mechanics are collected in the table below (see for example [1], [2]). 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) Component definitions General definition SI unit Dimension
Position  \hat{x} = x \,\!

 \hat{y} = y \,\!
 \hat{z} = z \,\!

 \mathbf{\hat{r}} = \mathbf{r} \,\! m [L]
Momentum  \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} \,\!

 \mathbf{\hat{p}} = -i \hbar \nabla \,\! J s m-1 = N s [M] [L] [T]-1
Potential energy  \hat{V}_x = V(x) \,\!

 \hat{V}_y = V(y) \,\!
 \hat{V}_z = V(z) \,\!

 \hat{V} = V\left ( \mathbf{r}, t \right ) = V \,\! J [M] [L]2 [T]-2
Energy Time-independent:

 \hat{E}_x = E(x) \,\!
 \hat{E}_y = E(y) \,\!
 \hat{E}_z = E(z) \,\!

Time-Independent:

 \hat{E} = E(\mathbf{r}) \,\!
Time-Dependent:
 \hat{E} = i \hbar \frac{\partial }{\partial t} \,\!

J [M] [L]2 [T]-2
Hamiltonian  \begin{align} & \hat{H} = \hat{T} + \hat{V} \\
& = \frac{\hat{p}^2}{2m} + V \\
& = -\frac{\hbar^2}{2m}\nabla^2 + V \\
\end{align} \,\! J [M] [L]2 [T]-2
Angular momentum operator \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)

\mathbf{\hat{L}} = -i\hbar \mathbf{r} \times \nabla J s = N s m-1 [M] [L]2 [T]-1
Spin angular momentum  \hat{S}_x = {\hbar \over 2} \sigma_x

 \hat{S}_y = {\hbar \over 2} \sigma_y
 \hat{S}_z = {\hbar \over 2} \sigma_z

where


\sigma_x = \begin{pmatrix}
0&1\\
1&0
\end{pmatrix}


\sigma_y = \begin{pmatrix}
0&-i\\
i&0
\end{pmatrix}


\sigma_z = \begin{pmatrix}
1&0\\
0&-1
\end{pmatrix}

are the pauli matrices for spin-½ particles.

\mathbf{\hat{S}} = {\hbar \over 2} \boldsymbol{\sigma} \,\!

where σ is the vector whose components are the pauli matricies.

J s = N s m-1 [M] [L]2 [T]-1

General mathematical properties of quantum operators

The mathematical properties of physical operators are a topic of great importance in itself. For further information, see C*-algebra and Gelfand-Naimark theorem.

See also

References

  1. ^ 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
  2. ^ Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, ISBN 0-19-855493-1

Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Operator — may refer to: Contents 1 Music 2 Computers 3 Science and mathematics …   Wikipedia

  • Operator (disambiguation) — Operator can mean:* Operator, a type of mathematical function * Operator (biology), a segment of DNA regulating the activity of genes * Operator (extension), an extension for the Firefox web browser, for reading microformats * Operator (IRC), a… …   Wikipedia

  • Operator (mathematics) — This article is about operators in mathematics. For other uses, see Operator (disambiguation). In basic mathematics, an operator is a symbol or function representing a mathematical operation. In terms of vector spaces, an operator is a mapping… …   Wikipedia

  • Operator product expansion — Contents 1 2D Euclidean quantum field theory 2 General 3 See also 4 External links 2D Euclidean quantum field theory …   Wikipedia

  • Operator (profession) — For other uses, see operator (disambiguation). Statue of a boom operator on the Avenue of Stars in Hong Kong. An operator is a professional designation used in various industries, including broadcasting (in television and radio …   Wikipedia

  • Parity (physics) — Flavour in particle physics Flavour quantum numbers: Isospin: I or I3 Charm: C Strangeness: S Topness: T Bottomness: B′ Related quantum numbers: Baryon number: B Lepton number: L Weak isospin: T or T3 Electric charge: Q …   Wikipedia

  • Spin (physics) — This article is about spin in quantum mechanics. For rotation in classical mechanics, see angular momentum. In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles… …   Wikipedia

  • Self-adjoint operator — In mathematics, on a finite dimensional inner product space, a self adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose.… …   Wikipedia

  • Laplace operator — This article is about the mathematical operator. For the Laplace probability distribution, see Laplace distribution. For graph theoretical notion, see Laplacian matrix. Del Squared redirects here. For other uses, see Del Squared (disambiguation) …   Wikipedia

  • Coherent states in mathematical physics — Coherent states have been introduced in a physical context, first as quasi classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states (see also [1]). However …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”