The angular momentum in classical mechanics is defined as the vector (outer) product of \(r\) and \(p\), \[\hat{L}=-i \hbar\left(y \frac{\partial}{\partial z}-z \frac{\partial}{\partial y}, z \frac{\p...The angular momentum in classical mechanics is defined as the vector (outer) product of \(r\) and \(p\), \[\hat{L}=-i \hbar\left(y \frac{\partial}{\partial z}-z \frac{\partial}{\partial y}, z \frac{\partial}{\partial x}-x \frac{\partial}{\partial z}, x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}\right)\] Since \(L\) commutes with \(H\) we can diagonalise one of the components of \(L\) at the same time as \(H\). \[\hat{L}_z Y_{L M}(\theta, \phi)=\hbar M Y_{L M}(\theta, \phi) .\]
The simultaneous eigenstates, \(Y_{l,m}(\theta,\phi)\), of \(L^2\) and \(L_z\) are known as the spherical harmonics . Let us investigate their functional form.
