Vector calculus: angular momentum operator in spherical coordinates

[MisterX]

Note: physics conventions, $\theta$ is measured from z-axis
We have a vector operator

$\vec{L} = -i \vec{r} \times \vec{\nabla} = -i\left(\hat{\phi} \frac{\partial}{\partial \theta} - \hat{\theta} \frac{1}{\sin\theta} \frac{\partial}{\partial \phi} \right)$
And apparently
$\vec{L}\cdot\vec{L}= -\left(\frac{\partial^2}{\partial \theta^2} + \cot \theta\frac{\partial}{\partial \theta}+ \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial \phi^2} \right)$
I am wondering about a way to obtain the second expression (the $\cot \theta$ term in particular) from the first expression without taking the circuitous route followed in my references. I realize the unit vectors aren't constant.
$\frac{\partial \boldsymbol{\hat{\theta}}} {\partial \phi} = \cos \theta\boldsymbol{\hat \phi}$
$\frac{\partial \boldsymbol{\hat{\phi}}} {\partial \theta} = 0$

The basis vectors are orthonormal however, and for the norm-squared of a regular vector in spherical coordinates, we can just square each of the components and add.

 

[member 428835]

while I'm not quite sure what you're asking for (not a physicist), i can say the del operator is not explicitly defined in spherical coordinates. thus, if you are looking for a way to literally take $\vec{L}\cdot \vec{L}$ perhaps the only way is the "circuitous" route from your references.

maybe this helps a little?

 

[member 428835]

while I'm not quite sure what you're asking for (not a physicist), i can say the del operator is not explicitly defined in spherical coordinates. thus, if you are looking for a way to literally take $\vec{L}\cdot \vec{L}$ perhaps the only way is the "circuitous" route from your references.

maybe this helps a little?

 

[member 428835]

while I'm not quite sure what you're asking for (not a physicist), i can say the del operator is not explicitly defined in spherical coordinates. thus, if you are looking for a way to literally take $\vec{L}\cdot \vec{L}$ perhaps the only way is the "circuitous" route from your references.

maybe this helps a little?

 

[vanhees71]

The gradient operator is defined in spherical coordinates (except along the polar axis, where spherical coordinates are singular).

To evaluate $\hat{\vec{L}}^2$, it's easier to go back to Cartesian coordinates first.
$$\hat{\vec{L}}^2 \psi =-\epsilon_{ijk}x_j \partial_k (\epsilon_{ilm} x_l \partial_m) \psi.$$
First we use the product rule and then to get
$$\epsilon_{ijk} \epsilon_{ilm}=\delta_{jl} \delta_{km}-\delta_{jm} \delta_{kl}$$
$$\hat{\vec{L}}^2 \psi =-(delta_{jl} \delta_{km}-\delta_{jm} \delta_{kl}) x_j (\delta_{kl} \partial_m+x_j x_l \partial_k \partial_m)=-(r^2 \Delta - 2 (\vec{x} \cdot \vec{\nabla}) \psi - x_j (\vec{x}\cdot \vec{\nabla}) \partial_j \psi).$$
The latter expression is not nice, but we can use
$$(\vec{x} \cdot \vec{\nabla})(\vec{x} \cdot \vec{\nabla})=x_j \partial_j (x_k \partial_k \psi) = (\vec{x} \cdot \vec{\nabla}) \psi + x_j (\vec{x} \cdot \vec{\nabla}) \partial_j \psi$$
to finally write
$$\hat{\vec{L}}^2 \psi = - (r^2 \Delta - (\vec{x} \cdot \vec{\nabla}) \psi - (\vec{x} \cdot \vec{\nabla})(\vec{x} \cdot \vec{\nabla}) \psi.$$
Now you use the Laplacian in spherical coordinates and
$$\vec{x} \cdot \vec{ \nabla} \psi = r \hat{r} \cdot \vec{\nabla} \psi = r \partial_r \psi$$
to finally get
$$\hat{\vec{L}}^2 \psi=-\frac{1}{\sin \vartheta} \left [\frac{\partial}{\partial \vartheta} \left (\sin \vartheta \frac{\partial \psi}{\partial \vartheta} \right ) + \frac{1}{\sin \vartheta}\frac{\partial^2 \psi}{\partial \varphi^2} \right ].$$

 

[MisterX]

To evaluate $\hat{\vec{L}}^2$, it's easier to go back to Cartesian coordinates first.

I guess what I'm wondering is if there is a way without going back to Cartesian coordinates.

 

[vanhees71]

It's a bit tricky in non-Cartesian coordinates as soon as vector (or higher-rank tensor) fields are involved, and $\hat{\vec{L}} \psi$ already is a vector field. That's why it is much simpler to first derive a scalar expression in Cartesian coordinates, which you already now how to express in spherical coordinates.