PlanetPhysics/Examples of Constants of the Motion

There exists an [[../QuantumSpinNetworkFunctor2/|observable]] which always [[../Commutator/|commutes]] with the [[../Hamiltonian2/|Hamiltonian]]: the Hamiltonian itself. The [[../CosmologicalConstant/|energy]] is therefore a constant of the [[../CosmologicalConstant/|motion]] of all [[../GenericityInOpenSystems/|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 [[../Bijective/|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

$H(\frac{\hslash }{i}\frac{d}{dq},q)=H(-\frac{\hslash }{i}\frac{d}{dq},-q)$

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

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

Under these conditions, if the [[../CosmologicalConstant2/|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 [[../Particle/|particles]] for which the parity [[../Cod/|operation]] amounts to a [[../FluorescenceCrossCorrelationSpectroscopy/|reflection]] in space $({𝐫}_i\to -{𝐫}_i)$ and for which the observable parity is defined by

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

References

[1] Messiah, Albert. "[[../QuantumParadox/|Quantum mechanics]]: [[../Volume/|volume]] I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.

This entry is a derivative of the Public [[../Bijective/|domain]] [[../Work/|work]] [1].