Lets try to get a feel for Neother’s theorem, which says that symmetries and observation laws are directly connected, using an example
Rotational invariance: All directions in space are physically indistinguishable
💼 Case: consider the rotation of a vector $\vec r$ to $\vec r'$ via an angle $\theta$ around the $z$ axis
The relation between the coordinates is
$$ \begin{aligned} x'&=x\cos\theta -y \sin \theta \\ y'&=y\sin\theta + y \cos \theta \\ z'&=z \end{aligned} $$
Now looking at an infinitesimal rotation $\delta \theta$ we get
$$ \begin{aligned} x'&=x-y\delta \theta \\ y'&=x\delta \theta +y \\ z'&=z
\end{aligned} $$
where we used small angle approximation
Now if we apply these coordinate changes to a wave function we get
$$ \begin{aligned} \psi (\vec r')&=\psi(x-y\delta \theta ,\;x\delta \theta+y,\; z) \\ &=\psi(\vec r)-\delta \theta \left ( y \frac{\partial \psi}{\partial x}-x\frac{\partial \psi}{\partial y} \right ) \\ &=(1+i\delta \theta \hat L_z)\psi(\vec r)
\end{aligned} $$
where $\hat L_z=-i\left ( y \frac{\partial \psi}{\partial x} - x \frac{\partial \psi}{\partial y} \right )$ is the x-component of the angular momentum operator $\hat L$
We can define the rotational operator $\hat R_z (\delta \theta)=1+i\delta \theta \hat L_z$ such that $\hat R_z(\delta \theta )\psi(\vec r)=\psi(\vec r')$
We can generalize this equation to any axis $\vec n$ as
$$ \boxed{\hat R_n(\delta \theta )=(1+i\delta \theta \,\hat{\bold L}\cdot \vec n)} $$
where $\vec n$ is a unit vector
Applying this to the Hamiltonian
Applying the $\hat H = \hat T+ \hat V$
$$ \psi'(\vec r)=H(\vec r) \psi(\vec r) $$
Applying $\hat R_n(\delta \theta)$ gives
$$ \begin{aligned} \hat R_n \psi'(\vec r)&=\hat R_n H(\vec r)\psi(\vec r) \\ \psi '(\vec r')&=H(\vec r')\psi(\vec r') \end{aligned} $$
Which is rotational invariance, ie $\boxed{H(\vec r)=H(\vec r')}$
We can now use this to compute the commutability
$$ \begin{aligned} H(\vec r')\psi(\vec r')&=H(\vec r)\psi(\vec r') \\ &=H(\vec r) \hat R_n\psi (\vec r) \\ (\hat R_n H(\vec r)- H(\vec r) \hat R_n)\psi(\vec r)&=0 \\ [H, \hat R_n]&=0
\end{aligned} $$
and since $\hat R_n (\delta \theta) \propto \hat {\bold L}$ it means $[\hat L,H]=0$ which is conservation of angular momentum
If there is a spin $\vec s$ then the total angular momentum $\vec J = \vec L + \vec s$
Angular momentum can be added by vector addition $\vec J = \vec J_1 + \vec J_2$
General properties of $\vec J^2$ and $J_z$
$$ \begin{aligned} |\vec J|^2 &=J(J+1)\hbar^2 \\ J&=|J_1-J_2|,\ldots , |J_1+J_2|\\ J_z&=\hbar M_J \\ M_J&=-J,\ldots,+J
\end{aligned} $$
$L$ is an integer and $s$ can be half-integers
We can also show the following invariances