Angular Momentum

Chem 3240 · Lecture 4.4

Davit Potoyan

Why angular momentum?

  • Rotation is to angular momentum what straight-line motion is to linear momentum.

  • Central to atomic structure and any system with rotational symmetry.

  • In QM it is a vector operator, like \(\vec{p}\).

  • Key twist: its three components do not commute.

Conservation from symmetry

Noether’s theorem

Every continuous symmetry of the laws of physics corresponds to a conserved quantity.

  • Translation \(\to\) momentum
  • Rotation \(\to\) angular momentum
  • Time shift \(\to\) energy

Emmy Noether

Classical definition

\[\vec{L}=\vec{r}\times\vec{p}\]

  • A vector: direction by the right-hand rule.
  • Components: \[L_x = yp_z - zp_y\] \[L_y = zp_x - xp_z\] \[L_z = xp_y - yp_x\]

Right-hand rule for \(\vec{L}\)

Conservation in action

  • No external torque \(\Rightarrow\) \(\vec{L}\) conserved.
  • Rotational energy: \[E_\text{rot}=\frac{L^2}{2I}\]
  • Pull mass inward \(\to\) smaller \(I\) \(\to\) faster spin.

Lower \(I\) speeds up rotation

Spherical coordinates

  • Natural choice for rotation: \((r,\theta,\phi)\).

  • Fixed orbit \(r=\text{const}\) removes the radial degree of freedom.

\[x=r\sin\theta\cos\phi\] \[y=r\sin\theta\sin\phi\] \[z=r\cos\theta\]

Spherical coordinates

Quantum operators

  • Promote classical \(\vec{L}\) to operators, e.g.

\[\hat{L}_z = -i\hbar\frac{\partial}{\partial\phi}\]

  • Eigenvalue problem \(\hat{L}_z f = a f\) gives \[f(\phi)=e^{im\phi},\qquad a=\hbar m\]
  • The \(z\)-axis is a choice (often set by an external field).

Components do not commute

\[\left[\hat{L}_x,\hat{L}_y\right]=i\hbar\hat{L}_z,\;\left[\hat{L}_y,\hat{L}_z\right]=i\hbar\hat{L}_x,\;\left[\hat{L}_z,\hat{L}_x\right]=i\hbar\hat{L}_y\]

\[\left[\hat{L}_x,\vec{\hat{L}}^2\right]=\left[\hat{L}_y,\vec{\hat{L}}^2\right]=\left[\hat{L}_z,\vec{\hat{L}}^2\right]=0\]

  • Only \(\vec{\hat{L}}^2\) and one component are knowable simultaneously.

  • Note the cyclic pattern \(x\to y\to z\to x\).

Quantized eigenvalues

  • Shared eigenfunctions \(Y_l^m(\theta,\phi)\):

\[\hat{L}^2 Y_l^m = \hbar^2 l(l+1)\, Y_l^m\] \[\hat{L}_z Y_l^m = \hbar m\, Y_l^m\]

  • Magnitude quantized: \(L^2 \to \hbar^2 l(l+1)\).

  • Projection quantized: \(L_z \to m\hbar\).

  • \(l=0,1,2,\dots\) and \(|m|\le l\), giving \(2l+1\) values of \(m\).

A vector on a cone

  • Fixed length \(\sqrt{l(l+1)}\,\hbar\).

  • Only \(2l+1\) projections onto \(z\).

  • \(\vec{L}\) never points exactly along \(z\): \(L_x, L_y\) stay uncertain.

Quantized magnitude and projection

Spherical harmonics

  • \(Y_l^m(\theta,\phi)\) or \(|l,m\rangle\): the angular wavefunctions.

  • Orthonormal: \(\langle l',m'|l,m\rangle=\delta_{ll'}\delta_{mm'}\).

  • Total nodes \(= l\) (polar bands from \(l\), longitude lines from \(|m|\)).

Spherical harmonics

Takeaway

Angular momentum is a vector operator with quantized magnitude \(L^2\to\hbar^2 l(l+1)\) and projection \(L_z\to m\hbar\). Its components do not commute, so only \(\vec{L}^2\) and one component are simultaneously sharp, and the spherical harmonics \(Y_l^m\) are their shared eigenfunctions.