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The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum problems involving rotational symmetry. Being an observable, its eigenfunctions represent the distinguishable physical states of a system's angular momentum, and the corresponding eigenvalues the observable experimental values.
In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics , an observable is a real -valued "function" on the set of all possible system states, e.g., position and momentum .
Such an observable is itself a self-sufficient CSCO. However, if some of the eigenvalues of are degenerate (such as having degenerate energy levels), then the above result no longer holds. In such a case, we need to distinguish between the eigenfunctions corresponding to the same eigenvalue.
The approach codified by John von Neumann represents a measurement upon a physical system by a self-adjoint operator on that Hilbert space termed an "observable". [1]: 17 These observables play the role of measurable quantities familiar from classical physics: position, momentum, energy, angular momentum and so on.
Examples of integrals of motion are the angular momentum vector, =, or a Hamiltonian without time dependence, such as (,) = + (). An example of a function that is a constant of motion but not an integral of motion would be the function C ( x , v , t ) = x − v t {\displaystyle C(x,v,t)=x-vt} for an object moving at a constant speed in one ...
Quantum mechanics enters the picture when observed quantities are measured and found to be discrete rather than continuous. The allowed observable values are determined by the eigenvalues of the operators associated with the observable. In the case angular momentum, for instance, the allowed observable values are the eigenvalues of the spin ...
Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs. In control theory, the observability and controllability of a linear system are mathematical duals.
These galaxies are observable above and below the Zone of Avoidance; all are redshifted in accordance with the Hubble flow, indicating that they are receding relative to the Milky Way and to each other, but the variations in their redshifts are large enough and regular enough to reveal that they are slightly drawn towards the attraction.