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Concept
Operator (physics)
Summary
In physics, an operator is a function over a space of physical states onto another space of physical 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.
In classical mechanics, the movement of a particle (or system of particles) is completely determined by the Lagrangian or equivalently the Hamiltonian , a function of the generalized coordinates q, generalized velocities and its conjugate momenta:
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:
The elements of G are physical operators, which map physical states among themselves.
where is the rotation matrix about an axis defined by the unit vector and angle θ.
If the transformation is infinitesimal, the operator action should be of the form
where is the identity operator, is a parameter with a small value, and 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, . If is infinitesimal, then we may write
This formula may be rewritten as
where 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.
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