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

$$P\Psi(\mathbf{r}_1,\mathbf{r}_2.\dots)=\Psi(-\mathbf{r}_1,-\mathbf{r}_2,\dots)$$