PlanetPhysics/Example of Quantum Commutator Algebra

Here we illustrate a simple example of quantum commutator algebra using a one-dimensional quantum [[../GenericityInOpenSystems/|system]]. Let $f(q)$ be a [[../Bijective/|function]] of $q$. The three [[../Commutator/|commutators]] of $q$ and of each of the functions $p^{2}f(q)$, $pf(q)p$, and $f(q)p^2$ may all be identified (to within the factor $i\hslash$) with the derivative with respect to $p$ of these functions, but they are not the same [[../QuantumOperatorAlgebra4/|operators]]. Indeed, by repeated application of the [[../CommutatorAlgebra/|commutator algebra]] rule

$[q_i,G(p_1,\dots ,p_R)]=i\hslash \frac{\partial G}{\partial p_i}$

we get

$[q,p^{2}f(q)]=2i\hslash pf(q)$ $[q,pfp]=i\hslash (fp+pf)$ $[q,f{p}^2]=2i\hslash fp$

In the same way

$[p,p^{2}f]=\frac{\hslash }{i}{p}^2{f}^{\prime }$ $[p,pfp]=\frac{\hslash }{i}p{f}^{\prime }p$ $[p,f{p}^2]=\frac{\hslash }{i}{f}^{\prime }{p}^2$

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].