Let \(\Psi(\vec{r}, t)\) be a wave function which depends on position \(\vec{r}\) and time \(t\).
Consider the following expressions:
\(\hat{x}\hat{p}_x\Psi(\vec{r}, t)\)
\(\hat{x}\hat{p}_y\Psi(\vec{r}, t)\)
\(\hat{x}\hat{y}\Psi(\vec{r}, t)\)
\(\hat{p}_x\hat{x}\Psi(\vec{r}, t)\)
\(\hat{p}_y\hat{x}\Psi(\vec{r}, t)\)
\(\hat{y}\hat{x}\Psi(\vec{r}, t)\)
\(\hat{p}_x\hat{p}_y\Psi(\vec{r}, t)\)
\(\hat{p}_y\hat{p}_x\Psi(\vec{r}, t)\)
(2)
where \(\hat{x}, \hat{y}\) are the position operators and \(\hat{p}_x, \hat{p}_y\) are the momentum operators and
their action on function \(\Psi(\vec{r}, t)\) are defined as:
\(\hat{x}\Psi(\vec{r}, t) = x\Psi(\vec{r}, t)\)
\(\hat{y}\Psi(\vec{r}, t) = y\Psi(\vec{r}, t)\)
\(\hat{p}_x\Psi(\vec{r}, t) = -ih\frac{\partial}{\partial x}\Psi(\vec{r}, t)\)
\(\hat{p}_y\Psi(\vec{r}, t) = -ih\frac{\partial}{\partial y}\Psi(\vec{r}, t)\)
Do the values of expressions in (2) depend on the order of operations?
Evaluate
\((\hat{x}\hat{p}_x - \hat{p}_x\hat{x})\Psi(\vec{r}, t),\
and
\((\hat{x}\hat{p}_y - \hat{p}_y\hat{x})\Psi(\vec{r}, t).\)
