Editing Operator (physics) (section)

==Operators in classical mechanics==

In classical mechanics, the movement of a particle (or system of particles) is completely determined by the [[Lagrangian mechanics|Lagrangian]] <math>L(q, \dot{q}, t)</math> or equivalently the [[Hamiltonian mechanics|Hamiltonian]] <math>H(q, p, t)</math>, a function of the [[generalized coordinates]] ''q'', generalized velocities <math>\dot{q} = \mathrm{d} q / \mathrm{d} t</math> and its [[conjugate momenta]]:

:<math>p = \frac{\partial L}{\partial \dot{q}}</math>

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 (physics)|symmetry]]). Operators in classical mechanics are related to these symmetries.

More technically, when ''H'' is invariant under the action of a certain [[group (mathematics)|group]] of transformations ''G'':

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

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

===Table of classical mechanics operators===
{| class="wikitable"
|-
! Transformation 
! Operator
! Position
! Momentum
|-
| [[Translational symmetry]] 
| <math>X(\mathbf{a})</math>
| <math>\mathbf{r}\rightarrow \mathbf{r} + \mathbf{a}</math>
| <math>\mathbf{p}\rightarrow \mathbf{p}</math>
|-
| [[Time translation symmetry]]
| <math>U(t_0)</math>
| <math>\mathbf{r}(t)\rightarrow \mathbf{r}(t+t_0)</math>
| <math>\mathbf{p}(t)\rightarrow \mathbf{p}(t+t_0)</math>
|-
| [[Rotational invariance]] 
| <math>R(\mathbf{\hat{n}},\theta)</math>
| <math>\mathbf{r}\rightarrow R(\mathbf{\hat{n}},\theta)\mathbf{r}</math>
| <math>\mathbf{p}\rightarrow R(\mathbf{\hat{n}},\theta)\mathbf{p}</math>
|- 
| [[Galilean transformation]]s
| <math>G(\mathbf{v})</math>
| <math>\mathbf{r}\rightarrow \mathbf{r} + \mathbf{v}t</math>
| <math>\mathbf{p}\rightarrow \mathbf{p} + m\mathbf{v}</math>
|-
| [[Parity (physics)|Parity]]
| <math>P</math>
| <math>\mathbf{r}\rightarrow -\mathbf{r}</math>
| <math>\mathbf{p}\rightarrow -\mathbf{p}</math>
|-
| [[T-symmetry]]
| <math>T</math>
| <math>\mathbf{r}\rightarrow \mathbf{r}(-t)</math>
| <math>\mathbf{p}\rightarrow -\mathbf{p}(-t)</math>
|-
|}

where <math>R(\hat{\boldsymbol{n}}, \theta)</math> is the [[rotation matrix]] about an axis defined by the [[unit vector]] <math>\hat{\boldsymbol{n}}</math> and angle ''θ''.