1.16: The Commutator

One must be careful to observe the correct order of operators. For example,

\[ \hat{x}\hat{k} \neq \hat{k}\hat{x} \nonumber \]

but

\[ \hat{x}\hat{\omega} = \hat{\omega}\hat{x} \nonumber \]

In quantum mechanics we define the commutator:

\[ [\hat{q},\hat{r}]=\hat{q}\hat{r}-\hat{r}\hat{q} \nonumber \]

We find that the operators \(\hat{r}\) and \(\hat{\omega}\) commute because \([\hat{x},\hat{\omega}]=0\).

Considering the operators \(\hat{x}\) and \(\hat{k}\):

\[ [\hat{x},\hat{k}]=-ix\frac{d}{dx}+i\frac{d}{dx}x \nonumber \]

To simplify this further we need to operate on some function, f(x):

\[\begin{align*} [\hat{x},\hat{k}]f(x) &=-ix\frac{df}{dx}+i\frac{d}{dx}(xf) \\[4pt] &= -ix\frac{df}{dx}+if\frac{dx}{dx}+ix\frac{df}{dx} \\[4pt] &=if \end{align*} \nonumber \]

Thus, the operators \(\hat{x}\) and \(\hat{k}\) do not commute, i.e.

\[ [\hat{x},\hat{k}] = i \nonumber \]

Although we used Fourier transforms, Equation (1.10.13) can also be derived from the relation (1.16.5) for the non-commuting operators operators \(\hat{x}\) and \(\hat{k}\). It follows that all operators that do not commute are subject to a similar limit on the product of their uncertainties. We shall see in the next section that this limit is known as "the uncertainty principle‟.

\(^{†}\)We have applied Parseval's theorem; see the Problem Sets.