PlanetPhysics/Examples of Constants of the Motion
There exists an observable which always commutes with the Hamiltonian: the Hamiltonian itself. The energy is therefore a constant of the motion of all systems whose Hamiltonian does not depend explicitly upon the time.
As another possible constant of the motion, let us mention parity . We denote under the name of parity the observable defined by
It is easily verified that is Hermitean. Moreover, and, consequently, the only possible eigenvalues of are and ; even functions are associated with , and odd functions with .
When the Hamiltonian is invariant under the substitution of for , we obviously have
Indeed, if
one has, for any ,
Under these conditions, if the wave function has a definite parity at a given initial instant of time, it conserves the same parity in the course of time.
This property is easily extended to a system having an arbitrary number of dimensions; in particular, it applies to systems of particles for which the parity operation amounts to a reflection in space and for which the observable parity is defined by
References
edit[1] Messiah, Albert. "Quantum mechanics: volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.