If light has both wave and particle properties, matter should also have wave properties—this is de Broglie’s hypothesis and wave-particle duality.
Basic Knowledge
de Broglie’s hypothesis（1924）: A particle with momentum p has an associated matter wave (de Broglie wave) with wavelength λ:λ = h/p = h/(mv)
Wave interpretation of Bohr’s quantum condition: A standing wave forms when the circumference of an electron’s circular orbit is an integer multiple of the matter wavelength:2πr = nλ = nh/(mv) → mvr = nh/(2π)（agrees with Bohr’s quantum condition）
Experimental confirmation:
・Davisson–Germer experiment（1927）: Confirmed that diffraction occurs when an electron beam strikes a nickel crystal
・Electron diffraction experiment by G. P. Thomson
・Today: Electron microscopes use de Broglie waves (shorter wavelengths than visible light provide high resolution)
Matter wave and de Broglie wave（a wave associated with a material particle. λ = h/p）
Wave-particle duality（both light and matter have wave and particle properties）
Electron diffraction（the diffraction of an electron beam by a crystal lattice. Evidence of wave nature）
Electron microscope（a high-resolution microscope using the de Broglie waves of electrons）
Uncertainty principle（Heisenberg: position and momentum cannot both be determined simultaneously. Δx・Δp ≥ ℏ/2）
Further Exploration (Background and Significance)
de Broglie wavelengths are extremely short for everyday objects and therefore cannot be observed: for a baseball（0.14 kg, 40 m/s）, λ ≈ 10⁻³⁴ m. For an electron（m = 9.1 × 10⁻³¹ kg, v = 10⁶ m/s）, λ ≈ 10⁻⁹ m（about 1/500 of visible light’s wavelength, comparable to the distance between atoms）. This is why electron microscopes have high resolution.
de Broglie’s hypothesis led to Schrödinger’s wave equation（1926）. The square of the absolute value of the wave function ψ represents the probability of finding a particle (Copenhagen interpretation).
The uncertainty principle Δx・Δp ≥ ℏ/2 is fundamental to quantum mechanics: it is impossible in principle to measure a particle’s position and momentum with unlimited precision simultaneously (this is not a problem with measurement instruments but a fundamental property of nature).
- λ = h/p = h/(mv)（de Broglie wavelength）
- The quantum condition can be derived from matter waves
- Electron diffraction experimentally confirms wave nature
- Electron microscope: shorter λ than light → high resolution
- Wave-particle duality is common to both light and matter
- Uncertainty principle: Δx・Δp ≥ ℏ/2
- Everyday objects have extremely small λ → wave nature is not visible
Notes (Easily Confused Points)
① The de Broglie wavelength is inversely proportional to momentum p = mv (the faster the particle, the shorter the wavelength). ② Electron diffraction is “evidence of wave nature.” ③ The uncertainty principle is a principle of nature, not a problem with measuring instruments. ④ de Broglie（1924）and Schrödinger’s equation（1926）are different but deeply related.
Practice
- Find the de Broglie wavelength of an electron moving at 2.0 × 10⁶ m/s（m = 9.1 × 10⁻³¹ kg、h = 6.6 × 10⁻³⁴ J·s）.
- Use matter waves to derive Bohr’s quantum condition mvr = nℏ.
- Explain why an electron microscope has higher resolution than an optical microscope.