**Problem 3.** Calculate the following commutator:\[[\hat{L}, f(r)],\]where $\hat{L}$ is the operator of orbital momentum, and $f(r)$ is a function of the radial coordinate $r = \sqrt{x^2 + y^2 + z^2}$.

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**Problem 3.** Calculate the following commutator:

\[
[\hat{L}, f(r)],
\]

where $\hat{L}$ is the operator of orbital momentum, and $f(r)$ is a function of the radial coordinate $r = \sqrt{x^2 + y^2 + z^2}$.

Expert Solution

Step 1

Note that the orbital angular momentum operator (L) may be expressed in spherical polar form from the vector product of the radius vector (r) and the linear momentum operator (p) as follows:

$\begin{array}{rcl} \hat{L} & = & r \times \hat{p} \\ = & (r \hat{r}) \times (- i ℏ \nabla) \\ = & - i ℏ \hat{r} \times (\hat{r} \frac{\partial}{\partial r} + \frac{\hat{θ}}{r} \frac{\partial}{\partial θ} + \frac{\hat{φ}}{r \sin θ} \frac{\partial}{\partial φ}) \\ = & - i ℏ (\hat{φ} \frac{\partial}{\partial θ} - \frac{\hat{θ}}{\sin θ} \frac{\partial}{\partial φ}) (∵ \hat{r} \times \hat{r} = 0, \hat{r} \times \hat{θ} = \hat{φ}, \hat{r} \times \hat{φ} = - \hat{θ}) \end{array}$

Note that there is no r dependence in L.

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PROBLEM 3. Calculate the following commutator: [L, f(r)], where L is the operator of orbital momentum, and f(r) is a function of the radial coordinate r = Vr2 + y? + z?.