Introduction: What Values Can L Take?
Using ladder operators (analogous to the harmonic oscillator), we can derive the allowed eigenvalues of \(\hat{L}^2\) and \(\hat{L}_z\) algebraically.
Ladder Operators for Angular Momentum
\[\hat{L}_{\pm} = \hat{L}_x \pm i\hat{L}_y\]These raise and lower the \(m\) value:
\[\hat{L}_{\pm}|l, m\rangle = \hbar\sqrt{l(l+1) - m(m \pm 1)}|l, m \pm 1\rangle\]The Eigenvalue Equations
\[\hat{L}^2|l, m\rangle = \hbar^2 l(l+1)|l, m\rangle\] \[\hat{L}_z|l, m\rangle = \hbar m|l, m\rangle\]With restrictions:
- \(l = 0, 1, 2, 3, \ldots\) (non-negative integers for orbital)
- \(m = -l, -l+1, \ldots, l-1, l\) (2l + 1 values)
Physical Meaning
- Magnitude of angular momentum: \(|\vec{L}| = \hbar\sqrt{l(l+1)}\)
- z-component: \(L_z = m\hbar\)
- Maximum \(L_z = l\hbar < |\vec{L}|\) (vector can't point exactly along z)
The Quantum Connection
The non-intuitive result \(|\vec{L}| = \hbar\sqrt{l(l+1)} \neq l\hbar\) means angular momentum can't point in a definite direction. For \(l = 1\): \(|\vec{L}| = \sqrt{2}\hbar\) but max \(L_z = \hbar\). The "extra" angular momentum is distributed in \(L_x\) and \(L_y\) through quantum fluctuations.