Overview
This lecture covers the Planck constant, its origin, fundamental role in quantum mechanics, applications (such as the photoelectric effect, uncertainty principle, and de Broglie wavelength), and its precise value and units.
Planck Constant Basics
- The Planck constant (h) is a fundamental physical constant in quantum mechanics.
- Symbol: h; SI unit: joule-second (Js); alternative: electronvolt-second (eVs).
- Exact value: h = 6.626 070 15 × 10⁻³⁴ J·s (or J/Hz).
Origin and Historical Context
- Max Planck introduced h in 1900 to explain black-body radiation.
- Planck's law describes the spectral distribution of thermal radiation using h.
- The quantization of energy (energy elements) was first introduced by Planck.
- Einstein used the concept to explain the photoelectric effect, earning the Nobel Prize.
Key Quantum Formulas
- Photon energy: E = h·f, where f is frequency.
- Planck's law for black-body radiation includes h in its formula.
- de Broglie wavelength: λ = h/p, where p is momentum.
Applications and Quantum Phenomena
- Photoelectric effect: Electron emission depends on light frequency, not intensity, showing energy is delivered in quanta (photons).
- Bohr model: Electron energies in atoms quantized using h and the reduced Planck constant (ħ).
- Uncertainty principle: Δx·Δp ≥ ħ/2, linking position (x) and momentum (p) uncertainties.
- Commutator relation: [p̂, x̂] = -iħδᵢⱼ.
Reduced Planck Constant (ħ)
- ħ = h/(2π), also called h-bar or Dirac constant.
- Value: ħ = 1.054 571 817 × 10⁻³⁴ J·s.
- Used in equations involving angular frequency or angular momentum.
Modern Significance and Units
- The Planck constant defines the kilogram in the SI system.
- ħ and h have the same dimensions as action and angular momentum (Js).
- Quantum effects become significant when energy × time approaches h.
Key Terms & Definitions
- Planck constant (h) — Proportionality constant linking energy and frequency in quantum mechanics.
- Reduced Planck constant (ħ) — h divided by 2π, used in angular momentum and wave equations.
- Photon — Quantum of light energy, with energy E = h·f.
- de Broglie wavelength — Wavelength associated with a particle: λ = h/p.
- Quantum — Smallest discrete unit of a physical property (e.g., energy).
- Uncertainty principle — Limits precision of simultaneous measurements of conjugate variables.
- Photoelectric effect — Emission of electrons from a surface by light, explained by quantized energy.
Action Items / Next Steps
- Review Planck's law, the formulas for photon energy and de Broglie wavelength.
- Practice problems involving the photoelectric effect and uncertainty principle.
- Read about the role of h in defining SI units and its historical context in modern physics.