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
![]() |
(1) |
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
[1] Messiah, Albert. "Quantum mechanics: volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.
This entry is a derivative of the Public domain work [1].
As of this snapshot date, this entry was owned by bloftin.