Quantum Mechanics Course

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← Part IV/Spherical Harmonics

4. Spherical Harmonics

Reading time: ~35 minutes | Pages: 10

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Spherical harmonics are the eigenfunctions of angular momentum operators—the mathematical foundation for understanding atomic orbitals, molecular structure, and anything with spherical symmetry. They appear everywhere from quantum mechanics to cosmology.

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Video Lecture

Visualizing Spherical Harmonics - 3D Animations

Beautiful 3D visualizations of spherical harmonics showing the angular structure of atomic orbitals and their quantum numbers

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

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Development of Spherical Harmonics

1782

Legendre Polynomials

— Adrien-Marie Legendre

Introduced Legendre polynomials Pₗ(cos θ) while studying gravitational attraction of ellipsoids

Significance: Fundamental building blocks for spherical harmonics, describing θ-dependence

1839

Associated Legendre Functions

— Gabriel Lamé

Generalized Legendre polynomials to associated Legendre functions Pₗᵐ(cos θ) for solving Laplace equation in spherical coordinates

Significance: Completed the mathematical framework needed for spherical harmonics

1926

Connection to Quantum Angular Momentum

— Erwin Schrödinger & Wolfgang Pauli

Recognized that spherical harmonics Yₗᵐ(θ,φ) are eigenfunctions of L² and Lz operators in quantum mechanics

Significance: Explained the angular structure of atomic orbitals and selection rules

1992

CMB Angular Power Spectrum

— COBE Satellite Team

Measured cosmic microwave background temperature fluctuations using spherical harmonic decomposition

Significance: Spherical harmonics became essential tool in cosmology for analyzing the universe's baby picture

💡 Understanding the historical development helps contextualize why certain concepts emerged and how they fit into the broader quantum revolution.

Definition & Eigenvalue Equations

Spherical harmonics $Y_\ell^m(\theta,\phi)$ are simultaneous eigenfunctions of $\hat{L}^2$ and $\hat{L}_z$:

$$\hat{L}^2 Y_\ell^m = \hbar^2\ell(\ell+1) Y_\ell^m$$
$$\hat{L}_z Y_\ell^m = \hbar m Y_\ell^m$$

where $\ell = 0, 1, 2, \ldots$ and $m = -\ell, -\ell+1, \ldots, \ell$

Physical meaning: ℓ determines total angular momentum magnitude; m determines z-component.

💡 Quantum Numbers as Labels

Spectroscopic notation for atomic orbitals:

  • ℓ = 0 → s orbitals (sharp) - spherical
  • ℓ = 1 → p orbitals (principal) - dumbbell shaped
  • ℓ = 2 → d orbitals (diffuse) - cloverleaf/donut
  • ℓ = 3 → f orbitals (fundamental) - complex lobes

Explicit Form

General formula:

$$Y_\ell^m(\theta,\phi) = \sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-|m|)!}{(\ell+|m|)!}}P_\ell^{|m|}(\cos\theta)e^{im\phi}$$

where $P_\ell^m(x)$ are associated Legendre polynomials

Structure:

  • Normalization constant (ensures $\int |Y_\ell^m|^2 d\Omega = 1$)
  • θ-dependence: $P_\ell^{|m|}(\cos\theta)$ (associated Legendre polynomial)
  • φ-dependence: $e^{im\phi}$ (azimuthal phase factor)

First Few Spherical Harmonics

ℓ = 0 (s-orbital):

$$Y_0^0 = \frac{1}{\sqrt{4\pi}}$$

Spherically symmetric—no angular dependence

ℓ = 1 (p-orbitals):

$$Y_1^0 = \sqrt{\frac{3}{4\pi}}\cos\theta \quad \text{(p}_z\text{)}$$

$$Y_1^{\pm 1} = \mp\sqrt{\frac{3}{8\pi}}\sin\theta e^{\pm i\phi} \quad \text{(p}_x, \text{p}_y\text{)}$$

Dumbbell shapes along different axes

ℓ = 2 (d-orbitals):

$$Y_2^0 = \sqrt{\frac{5}{16\pi}}(3\cos^2\theta - 1) \quad \text{(d}_{z^2}\text{)}$$

$$Y_2^{\pm 1} = \mp\sqrt{\frac{15}{8\pi}}\sin\theta\cos\theta e^{\pm i\phi} \quad \text{(d}_{xz}, \text{d}_{yz}\text{)}$$

$$Y_2^{\pm 2} = \sqrt{\frac{15}{32\pi}}\sin^2\theta e^{\pm 2i\phi} \quad \text{(d}_{xy}, \text{d}_{x^2-y^2}\text{)}$$

Cloverleaf and donut shapes

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Video Lecture

Atomic Orbitals and Quantum Numbers

Comprehensive explanation of how quantum numbers (n, ℓ, m) determine orbital shapes and energies in hydrogen

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

📝 Worked Example: Verifying Orthonormality

Problem: Verify that $Y_0^0$ and $Y_1^0$ are orthonormal.

Step 1: Check normalization of Y₀⁰

$$\int_0^{2\pi}\int_0^\pi |Y_0^0|^2 \sin\theta\,d\theta\,d\phi = \int_0^{2\pi}d\phi \int_0^\pi \frac{1}{4\pi}\sin\theta\,d\theta$$
$$= \frac{1}{4\pi}(2\pi)[-\cos\theta]_0^\pi = \frac{1}{4\pi}(2\pi)(2) = 1 \quad \checkmark$$

Step 2: Check orthogonality between Y₀⁰ and Y₁⁰

$$\int Y_0^{0*} Y_1^0 d\Omega = \frac{1}{\sqrt{4\pi}}\sqrt{\frac{3}{4\pi}}\int_0^{2\pi}d\phi\int_0^\pi\cos\theta\sin\theta\,d\theta$$
$$= \sqrt{\frac{3}{16\pi^2}}(2\pi)\left[\frac{\sin^2\theta}{2}\right]_0^\pi = 0 \quad \checkmark$$

💡 Result: Y₀⁰ and Y₁⁰ are orthonormal, as expected for eigenfunctions of Hermitian operators with different eigenvalues.

Key Properties

1. Orthonormality

$$\int_0^{2\pi}\int_0^\pi Y_{\ell'}^{m'*}(\theta,\phi)Y_\ell^m(\theta,\phi)\sin\theta\,d\theta\,d\phi = \delta_{\ell\ell'}\delta_{mm'}$$

Different (ℓ, m) states are orthogonal; same states are normalized to 1

2. Completeness

Any function on the unit sphere can be expanded:

$$f(\theta,\phi) = \sum_{\ell=0}^\infty\sum_{m=-\ell}^\ell c_{\ell m}Y_\ell^m(\theta,\phi)$$

Coefficients: $c_{\ell m} = \int Y_{\ell}^{m*}(\theta,\phi)f(\theta,\phi)\,d\Omega$

This is the angular analog of Fourier series—decomposing functions into "angular frequencies"

3. Parity

Under spatial inversion $\vec{r} \to -\vec{r}$ (i.e., $\theta \to \pi - \theta, \phi \to \phi + \pi$):

$$Y_\ell^m(\pi-\theta, \phi+\pi) = (-1)^\ell Y_\ell^m(\theta,\phi)$$
  • Even ℓ (s, d, g, ...): even parity (+1)
  • Odd ℓ (p, f, h, ...): odd parity (-1)

Selection rules: Electric dipole transitions require Δℓ = ±1 (parity must change)

4. Complex Conjugation

$$Y_\ell^{m*}(\theta,\phi) = (-1)^m Y_\ell^{-m}(\theta,\phi)$$

Reflects the relationship between states with opposite m (mirror images about z-axis)

🤔 Self-Check Question

Question: How many independent spherical harmonics are there for a given ℓ? Why does this match the degeneracy of energy levels in the hydrogen atom?

Show Answer

Answer: For each ℓ, there are 2ℓ + 1 independent spherical harmonics corresponding to m = -ℓ, -ℓ+1, ..., +ℓ.

In hydrogen, the energy depends only on n (principal quantum number), not on ℓ or m. For a given n, ℓ ranges from 0 to n-1, giving total degeneracy:

$$g_n = \sum_{\ell=0}^{n-1}(2\ell + 1) = n^2$$

Physical meaning: The 2ℓ+1 states differ only in orientation of angular momentum vector—all have same magnitude |L|² = ℏ²ℓ(ℓ+1) but different z-components.

Real Forms (for Visualization)

For m ≠ 0, the complex exponentials $e^{im\phi}$ make visualization difficult. We often use real combinations:

$$Y_{\ell,m}^{\text{cos}} = \frac{1}{\sqrt{2}}\left(Y_\ell^m + (-1)^m Y_\ell^{-m}\right) \propto P_\ell^m(\cos\theta)\cos(m\phi)$$
$$Y_{\ell,m}^{\text{sin}} = \frac{1}{i\sqrt{2}}\left(Y_\ell^m - (-1)^m Y_\ell^{-m}\right) \propto P_\ell^m(\cos\theta)\sin(m\phi)$$

These real forms give the familiar px, py, pz and dxy, dxz, etc. orbitals seen in chemistry textbooks.

▶️

Video Lecture

Legendre Polynomials and Spherical Harmonics

Mathematical derivation showing how Legendre polynomials arise from solving Laplace's equation in spherical coordinates

💡 Tip: Watch at 1.25x or 1.5x speed for efficient learning. Use YouTube's subtitle feature if available.

📝 Worked Example: Computing Expectation Value

Problem: Calculate $\langle L_z \rangle$ for the state $|\psi\rangle = \frac{1}{\sqrt{2}}(Y_1^0 + Y_1^1)$.

Step 1: Recall eigenvalue equation

Each Yₗᵐ is an eigenstate of L_z with eigenvalue mℏ:

$$\hat{L}_z Y_1^0 = 0 \cdot \hbar Y_1^0 = 0, \quad \hat{L}_z Y_1^1 = 1 \cdot \hbar Y_1^1 = \hbar Y_1^1$$

Step 2: Compute expectation value

$$\langle L_z \rangle = \langle\psi|\hat{L}_z|\psi\rangle = \frac{1}{2}\left(\langle Y_1^0| + \langle Y_1^1|\right)\hat{L}_z\left(|Y_1^0\rangle + |Y_1^1\rangle\right)$$
$$= \frac{1}{2}\left(\langle Y_1^0|0 + \langle Y_1^1|\hbar Y_1^1\right) = \frac{1}{2}(\hbar) = \frac{\hbar}{2}$$

Used orthonormality: ⟨Y₁⁰|Y₁¹⟩ = 0

💡 Result: The average z-component of angular momentum is ℏ/2—halfway between the two eigenstates.

Addition Theorem

Relates Legendre polynomials to spherical harmonics:

$$P_\ell(\cos\gamma) = \frac{4\pi}{2\ell+1}\sum_{m=-\ell}^\ell Y_\ell^{m*}(\theta_1,\phi_1)Y_\ell^m(\theta_2,\phi_2)$$

where γ is the angle between directions (θ₁,φ₁) and (θ₂,φ₂)

Application: Multipole expansions in electromagnetism and gravity

💡 Application: Cosmic Microwave Background

The CMB temperature distribution T(θ,φ) across the sky is expanded in spherical harmonics:

$$\frac{\Delta T}{T}(\theta,\phi) = \sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell} a_{\ell m} Y_\ell^m(\theta,\phi)$$

The angular power spectrum Cₗ = ⟨|aₗₘ|²⟩ encodes information about dark matter, dark energy, and the geometry of the universe. Planck satellite measured this to incredible precision!

🤔 Self-Check Question

Question: Why do spherical harmonics with m ≠ 0 have complex values, and how does this relate to their physical meaning?

Show Answer

The factor $e^{im\phi}$ is complex because it represents rotation about the z-axis. For m > 0, the phase increases with φ (counterclockwise rotation); for m < 0, it decreases (clockwise).

Physical interpretation: States with m ≠ 0 have angular momentum circulating around the z-axis. The complex exponential captures this rotational character.

For real-valued functions (like electron density in stationary states), we take linear combinations to get real spherical harmonics—this is why chemistry textbooks show px and py orbitals instead of p+1 and p-1.

Applications Across Physics

Atomic & Molecular Physics

  • Electron orbitals in atoms
  • Molecular orbital theory (LCAO)
  • Selection rules for transitions
  • Atomic scattering cross sections

Nuclear Physics

  • Nuclear shell model
  • Nuclear magnetic moments
  • Gamma decay angular distributions
  • Nucleon-nucleon scattering

Cosmology

  • CMB temperature anisotropies
  • Gravitational lensing maps
  • Large-scale structure analysis
  • Primordial fluctuation power spectrum

Electromagnetism & Gravity

  • Multipole expansion of potentials
  • Antenna radiation patterns
  • Gravitational wave modes
  • Planetary gravity fields

📝 Chapter Summary

Key Equations

Eigenvalue: L²Yₗᵐ = ℏ²ℓ(ℓ+1)Yₗᵐ
z-component: LzYₗᵐ = mℏYₗᵐ
Orthonormality: ⟨Yₗ'ᵐ'|Yₗᵐ⟩ = δₗₗ'δₘₘ'
Parity: Yₗᵐ(π-θ,φ+π) = (-1)ˡYₗᵐ(θ,φ)

Key Concepts

  • Eigenfunctions of L² and Lz simultaneously
  • 2ℓ+1 states for each ℓ (different m values)
  • Form complete orthonormal basis on sphere
  • Describe angular structure of atomic orbitals
  • Complex Yₗᵐ → real linear combinations
  • Parity determines selection rules
  • Universal in spherically symmetric problems

Spherical harmonics are the language of angular structure in quantum mechanics. From atomic orbitals to cosmic microwave background, they appear wherever spherical symmetry exists. Mastering spherical harmonics is essential for three-dimensional quantum problems.

🔗 Related Topics:Angular Momentum • Hydrogen Atom • Addition of Angular Momentum

← Angular MomentumHydrogen Atom →