Question

(a) Work out all of the canonical commutation relations for components of the operators $\mathbf{r}$ and $\mathbf{p}:[x, y],\left[x, p_{y}\right]\left[x, p_{x}\right],\left[p_{y}, p_{z}\right],$ and so on. (b) Confirm the three-dimensional version of Ehrenfest's theorem, $$\frac{d}{d t}\langle\mathbf{r}\rangle=\frac{1}{m}\langle\mathbf{p}\rangle, \quad \text { and } \quad \frac{d}{d t}\langle\mathbf{p}\rangle=\langle-\nabla V\rangle$$ (Each of these, of course, stands for three equations- one for each component.) Hint: First check that the "generalized" Ehrenfest theorem, Equation $3.73,$ is valid in three dimensions. (c) Formulate Heisenberg's uncertainty principle in three dimensions.

          (a) Work out all of the canonical commutation relations for components of the operators $\mathbf{r}$ and $\mathbf{p}:[x, y],\left[x, p_{y}\right]\left[x, p_{x}\right],\left[p_{y}, p_{z}\right],$ and so on.
(b) Confirm the three-dimensional version of Ehrenfest's theorem,
$$\frac{d}{d t}\langle\mathbf{r}\rangle=\frac{1}{m}\langle\mathbf{p}\rangle, \quad \text { and } \quad \frac{d}{d t}\langle\mathbf{p}\rangle=\langle-\nabla V\rangle$$
(Each of these, of course, stands for three equations- one for each component.) Hint: First check that the "generalized" Ehrenfest theorem, Equation $3.73,$ is valid in three dimensions.
(c) Formulate Heisenberg's uncertainty principle in three dimensions.

Added by Keith T.

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