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 $P$ defined by

$P\psi (q)=\psi (-q)$

It is easily verified that $P$ is Hermitean. Moreover, $P^{2}=1$ and, consequently, the only possible eigenvalues of $P$ are $+1$ and $-1$; even functions are associated with $+1$, and odd functions with $-1$.

When the Hamiltonian is invariant under the substitution of $-q$ for $q$, we obviously have

$[P,H]=0$

Indeed, if

$H\left({\frac {\hbar }{i}}{\frac {d}{dq}},q\right)=H\left(-{\frac {\hbar }{i}}{\frac {d}{dq}},-q\right)$

one has, for any $\psi (q)$,

$PH\psi =H\left(-{\frac {\hbar }{i}}{\frac {d}{dq}},-q\right)\psi (-q)=H\left({\frac {\hbar }{i}}{\frac {d}{dq}},q\right)\psi (-q)=HP\psi$

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 $(\mathbf {r} _{i}\rightarrow -\mathbf {r} _{i})$ and for which the observable parity is defined by

$P\Psi (\mathbf {r} _{1},\mathbf {r} _{2}.\dots )=\Psi (-\mathbf {r} _{1},-\mathbf {r} _{2},\dots )$

References

[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].