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

$\displaystyle [q_i,G(p_1,\dots,p_R)] = i\hbar \frac{\partial G}{\partial p_i}$ (1)

we get

$\displaystyle [q,p^2 f(q)] = 2 i \hbar p f(q)$

$\displaystyle [q,pfp] = i \hbar(fp+pf)$

$\displaystyle [q,fp^2] = 2 i \hbar f p$

In the same way

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

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

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

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].



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