PlanetPhysics/Example of Quantum Commutator Algebra

Here we illustrate a simple example of quantum commutator algebra using a one-dimensional quantum system. Let $$f(q)$$ be a function of $$q$$. The three 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 $$i \hbar$$) with the derivative with respect to $$p$$ of these functions, but they are not the same operators. Indeed, by repeated application of the commutator algebra rule

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

we get

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

In the same way

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