Playground
Log-log plot of de Broglie wavelength vs momentum with a sliding dot; shows baseball vs electron scale.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Wavelengthoutput de Broglie wavelength of the particle | m | L | 1e-35 – 0.000001 | |
| Planck's constant Fundamental quantum of action | J*s | M*L^2*T^-1 | 6.62607015e-34 – 6.62607015e-34 | |
| Momentum Linear momentum of the particle | kg*m/s | M*L*T^-1 | 1e-30 – 10000000000 |
Deep dive
Derivation
Starting from Einstein's photon relation E = hf and the photon momentum p = E/c = hf/c = h/lambda. De Broglie postulated the same relation p = h/lambda holds for matter, giving lambda = h/p. For a non-relativistic particle of mass m and speed v, p = mv so lambda = h/(mv).
Experimental verification
Davisson–Germer (1927) observed electron diffraction from nickel crystals matching lambda = h/p. Thomson's electron diffraction (1927), neutron diffraction (1936), and C60 fullerene interferometry (1999) all confirm the relation.
Common misconceptions
- lambda is not a 'size' of the particle — it is a length scale of interference
- Macroscopic objects do have wavelengths, but they are unmeasurably small
- The wave is not a physical oscillation of the particle, but of the probability amplitude
Real-world applications
- Electron microscopes (lambda ≈ 0.004 nm at 100 kV)
- Neutron scattering for material science
- Atom interferometry for precision gravimetry
- Low-energy electron diffraction (LEED) for surface studies
Worked examples
Baseball wavelength
Given:
- m:
- 0.145
- v:
- 40
Find: lambda
Solution
lambda = h/(mv) = 6.62607015e-34 / (0.145 * 40) ≈ 1.14e-34 m
Electron at 1% c
Given:
- m:
- 9.1093837015e-31
- v:
- 2997924.58
Find: lambda
Solution
lambda ≈ 2.43e-10 m ≈ 0.24 nm
Scenarios
What if…
- scenario:
- What if h were zero?
- answer:
- All matter would be purely classical — no wave behavior, no quantum mechanics, no stable atoms.
- scenario:
- What if you slowed an electron to 1 mm/s?
- answer:
- lambda ≈ 0.73 m — a macroscopic matter wave, achievable in ultracold atom experiments.
- scenario:
- What if the particle is a photon?
- answer:
- p = E/c = h/lambda recovers the original Einstein relation — de Broglie's law reduces to it.
Limiting cases
- condition:
- p → 0
- result:
- lambda → ∞
- explanation:
- Slow, low-momentum particles become highly delocalized waves.
- condition:
- p → large (macroscopic)
- result:
- lambda → 0
- explanation:
- Baseball at 40 m/s has lambda ≈ 1.14e-34 m — 19 orders of magnitude below a proton.
- condition:
- p = m_e * 1e6 m/s (fast electron)
- result:
- lambda ≈ 0.73 nm
- explanation:
- Comparable to atomic spacings — basis of electron diffraction.
Context
Louis de Broglie · 1924
In his PhD thesis, de Broglie proposed that if light could be particle-like, particles could be wave-like — confirmed by Davisson–Germer in 1927.
Hook
Does a thrown baseball have a wavelength?
Every moving particle behaves like a wave. What is the wavelength of a 0.145 kg baseball thrown at 40 m/s?
Dimensions:
- lhs:
- lambda → [L]
- rhs:
- [M*L^2*T^-1] / [M*L*T^-1] → [L]
- check:
- Both sides are length. ✓
Validity: Valid for non-interacting free particles. Requires p > 0. Relativistic correction p = gamma*m*v for v comparable to c.