Derivation of Squared Angular Momentum in Spherical Coordinates

Question

While reading my textbook, I found the following:

$$ \vec{L}^2=-\hbar^2r^2(\hat{r}\times\vec{\nabla})\cdot(\hat{r}\times\vec{\nabla})=-\hbar^2r^2\left[\nabla^2-\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right)\right]$$

I tried to prove the above equation by doing the following.

Knowing that : $$(\vec{A}\times\vec{B}).(\vec{C}\times\vec{D})=(\vec{A}.\vec{C})(\vec{B}.\vec{D})-(\vec{A}.\vec{D})(\vec{B}.\vec{C})$$ And making the paropiate substitutions I'm left with this: $$\nabla^2-(\hat{r}.\vec{\nabla})(\vec{\nabla}.\hat{r})$$ I am working in spherical coordinates so we have: $$\hat{r}.\vec{\nabla}=\frac{\partial}{\partial r}$$ $$\vec{\nabla}.\hat{r}=?$$ where in place of the question mark I found $\frac{2}{r}$ but I know it must be wrong. I used the following partial derivatives of unit vector properties in spherical coordiantes:

\begin{aligned} \frac{\partial \hat{r}}{\partial r}&=0, & \frac{\partial\hat{\theta}}{\partial r}&=0, & \frac{\partial\hat{\varphi}}{\partial r}&=0,\\[6pt] \frac{\partial\hat{r}}{\partial\theta}&=\hat{\theta}, & \frac{\partial\hat{\theta}}{\partial\theta}&=-\hat{r}, & \frac{\partial\hat{\varphi}}{\partial\theta}&=0,\\[6pt] \frac{\partial\hat{r}}{\partial\varphi}&=\hat{\varphi}\sin\theta, & \frac{\partial\hat{\theta}}{\partial\varphi}&=\hat{\varphi}\cos\theta, & \frac{\partial\hat{\varphi}}{\partial\varphi}&=-\hat{r}\sin\theta-\hat{\theta}\cos\theta, \end{aligned}

Also Knowing the gradient for spherical coordinates: $$\vec{\nabla} = \hat{x}\frac{\partial}{\partial x}+\hat{y}\frac{\partial}{\partial y}+\hat{z}\frac{\partial}{\partial z}=\hat{r}\frac{\partial}{\partial r}+\hat{\theta}\frac{1}{r}\frac{\partial}{\partial \theta}+\hat{\varphi}\frac{1}{r\sin\theta}\frac{\partial}{\partial \varphi}, $$

I know I must be wrong because the quatruple product stands for vectors, but it might not be the case for operators. I tried taking in consideration the order of the products, as a last resource i tried to take into account the commutator of the unit vector and the gradient but failed to find an expression because i'm still green in this area. I would greatly apreciate if someone could clear my doubts, I also know I could've taken the cross products directly but is there a way of doing this by taking this path? It's always enlightning to be able to do computations in different ways.

Answer 1

You must remember that $\textbf{r}$ is an operator and to compute $\nabla\cdot\hat r$ you must act it on a function of coordinates. Here is how I derived it.\begin{equation} \textbf{L}^2=(\textbf{r}\times\textbf{p})\cdot(\textbf{r}\times\textbf{p}) \end{equation} Using the formula $\textbf{A}\cdot(\textbf{B}\times\textbf{C})=\textbf{C}\cdot(\textbf{A}\times\textbf{B})$ twice, we get, \begin{equation} \textbf{L}^2=\textbf{r}\cdot(\textbf{p}\times(\textbf{r}\times\textbf{p})) \end{equation} Using the formula for vector triple product we get, \begin{equation} \textbf{L}^2=\textbf{r}\cdot(p^2\textbf{r}-\textbf{p}(\textbf{p}\cdot\textbf{r})) \end{equation} Using $[\textbf{r},p^2]=2i\hbar\textbf{p}$ and $\textbf{r}\cdot\textbf{p}-\textbf{p}\cdot\textbf{r}=3i\hbar 1$, (to prove these commutation relations just use $[x_i,p_j]=i\hbar\delta_{ij}$) we get \begin{equation} \textbf{L}^2=r^2p^2+i\hbar\textbf{r}\cdot\textbf{p}-(\textbf{r}\cdot\textbf{p})(\textbf{r}\cdot\textbf{p}) \end{equation} Now use the fact that $\hat{r}\cdot\nabla f(\textbf{r})=\frac{\partial f}{\partial r}$ to obtain\begin{equation} \textbf{L}^2=r^2p^2+\hbar^2\frac{\partial }{\partial r}(r^2\frac{\partial}{\partial r})=-\hbar^2r^2[\nabla^2-\frac{1}{r^2}\frac{\partial }{\partial r}(r^2\frac{\partial}{\partial r})] \end{equation}

Answer 2

I worked on your line of reasoning and was able to obtain the answer. I use $\hat{r}$ to mean an operator (not unit vector), which to compute, you have to act on a function (from right to left as operators always do), as SRS have said.

After applying the quadruple product for vectors, you obtained: $$ \hat{L}^2 f = -\hbar^2 r^2 \nabla^2 f + \hbar^2\left(r\frac{\partial}{\partial r}\right)\left(\nabla\cdot(\hat{r}f)\right) $$ Using the identity $$ \frac{1}{r} \frac{\partial^2}{\partial r^2}(rf) \equiv \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial f}{\partial r} \right) $$ I was able to tame the second term on the RHS and obtain the desired form.

Disclaimer: I am unsure if my resolution of the divergence is legitimate though.