Introduction: Quantizing Rotation
Angular momentum in quantum mechanics is represented by operators \(\hat{\vec{L}}\). We can't measure all components simultaneously—only \(\hat{L}^2\) and one component (conventionally \(\hat{L}_z\)).
The Angular Momentum Operators
\[\hat{L}_x = \hat{y}\hat{p}_z - \hat{z}\hat{p}_y, \quad \hat{L}_y = \hat{z}\hat{p}_x - \hat{x}\hat{p}_z, \quad \hat{L}_z = \hat{x}\hat{p}_y - \hat{y}\hat{p}_x\] \[\hat{L}^2 = \hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2\]Commutation Relations
\[[\hat{L}_x, \hat{L}_y] = i\hbar\hat{L}_z \quad \text{(cyclic)}\] \[[\hat{L}^2, \hat{L}_i] = 0 \quad \text{for all } i\]In Spherical Coordinates
\[\hat{L}_z = -i\hbar\frac{\partial}{\partial\phi}\] \[\hat{L}^2 = -\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right]\]The Quantum Connection
The commutation relations mean \(L_x\), \(L_y\), \(L_z\) can't all be known precisely. We choose to diagonalize \(\hat{L}^2\) and \(\hat{L}_z\) together, giving quantum numbers \(l\) and \(m\). The uncertainty in \(L_x\) and \(L_y\) is why orbital angular momentum "points in a cone" around the z-axis rather than in a definite direction.