Playground
Log-log plot of de Broglie wavelength vs momentum with reference markers.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Wavelengthoutput De Broglie wavelength of the particle | m | L | 1e-15 – 0.000001 | |
| Momentum Linear momentum of the particle | kg·m/s | M·L·T⁻¹ | 1e-30 – 1e-20 |
Deep dive
Derivation
De Broglie generalized the photon relation p = h/λ (which Compton had verified for X-rays) to all matter. Combined with Bohr's quantization condition for atomic orbits, requiring an integer number of wavelengths around an orbit (nλ = 2πr) reproduces angular momentum quantization L = nℏ — beautiful internal consistency.
Experimental verification
Davisson and Germer (1927) accidentally diffracted electrons off a nickel crystal and observed the predicted angular pattern. G.P. Thomson independently confirmed it the same year. Since then: neutron diffraction (1940s), atom interferometry, and even C₆₀ buckyball interference (Zeilinger, 1999) confirm matter waves.
Common misconceptions
- λ is not the size of the particle — it's the spacing of probability oscillations of its wavefunction.
- The wave is not a 'physical' wave in 3D space; it's a probability amplitude (|ψ|² gives probability density).
- Matter waves don't violate locality — interference patterns build up statistically over many particles.
Real-world applications
- Electron microscopes — λ ~ 0.005 nm gives nanometer resolution, far better than light.
- Neutron diffraction for crystallography (sensitive to light atoms like H).
- Atom interferometers for ultra-precise gravity and inertial sensing.
- Scanning tunneling microscopy relies on the wave nature of electrons.
Worked examples
Electron accelerated through 100 V
Given:
- V:
- 100
Find: λ
Solution
λ = h/√(2m_e·eV) ≈ 1.23×10⁻¹⁰ m = 0.123 nm
1 kg baseball at 40 m/s
Given:
- m:
- 1
- v:
- 40
Find: λ
Solution
λ = h/(mv) = 6.626×10⁻³⁴ / 40 ≈ 1.66×10⁻³⁵ m
Scenarios
What if…
- scenario:
- What if h were a million times larger?
- answer:
- Macroscopic matter waves would have λ ~ 10⁻²⁹ m — still tiny, but quantum effects would creep into chemistry and biology in dramatic ways.
- scenario:
- What if you slow an electron down to 1 m/s?
- answer:
- λ = h/(m_e·1) ≈ 0.7 mm — nearly visible! Ultra-cold neutrons and atoms exploit this.
- scenario:
- What if you tried to diffract a proton with the same energy as an electron?
- answer:
- Same kinetic energy → larger momentum (because m_p ≫ m_e) → smaller λ by factor √(m_p/m_e) ≈ 43. Proton beams give shorter-wavelength probes.
Limiting cases
- condition:
- p → 0
- result:
- λ → ∞
- explanation:
- A particle at rest has infinite wavelength — its position is maximally uncertain.
- condition:
- p → ∞
- result:
- λ → 0
- explanation:
- Highly energetic particles behave like point particles; wave nature becomes invisible.
- condition:
- macroscopic m, ordinary v
- result:
- λ ≪ atom
- explanation:
- A 1 kg ball at 1 m/s has λ ≈ 6.6×10⁻³⁴ m — far below any measurable scale, hence no diffraction.
Context
Louis de Broglie · 1924
In his PhD thesis, de Broglie proposed wave-particle duality for matter; confirmed by Davisson-Germer electron diffraction in 1927.
Hook
If electrons are waves, why don't baseballs diffract?
Find the de Broglie wavelength of an electron accelerated through 100 V.
Dimensions: [λ] = [h]/[p] = (M·L²·T⁻¹)/(M·L·T⁻¹) = L ✓
Validity: Valid for all matter, from electrons to bowling balls. Use the relativistic momentum p = γmv when v approaches c. For non-localized states use wave-packet treatment.