De Broglie's hypothesis
Since light shows both wave and particle properties, de Broglie suggested that moving material particles should also show wave properties. The wavelength associated with a particle of momentum p is: λ = h/p = h/(mv) where h = 6.63 × 10⁻³⁴ J s is Planck's constant. This is the de Broglie wavelength. It is inversely proportional to momentum: doubling the momentum halves the wavelength.
Why we do not notice matter waves in daily life
For everyday objects, m is large, so λ is extremely small. A 0.12 kg ball moving at 20 m/s has λ = 6.63 × 10⁻³⁴ / (0.12 × 20) ≈ 2.8 × 10⁻³⁴ m, far too small to detect. For electrons, m is tiny and λ is comparable to the spacing of atoms in crystals, so wave effects can be observed.
Charged particle accelerated through a potential
A particle of charge q and mass m accelerated from rest through potential difference V gains kinetic energy K = qV. Since p = √(2mK): λ = h/√(2mqV) For an electron this becomes λ ≈ 1.227/√V nm (V in volts). An electron accelerated through 100 V has λ ≈ 0.123 nm. For particles with the same kinetic energy, λ ∝ 1/√m, so the lighter particle has the longer wavelength. For a proton and an alpha particle accelerated through the same V, λ_p/λ_α = √(m_α q_α / m_p q_p) = √(4 × 2) = 2√2.
Experimental confirmation
Davisson and Germer directed a beam of electrons at a nickel crystal and found strong scattering at particular angles, exactly as expected for diffraction of waves. For electrons accelerated through 54 V, the measured wavelength agreed with the de Broglie value. Electron diffraction is the basis of the electron microscope.
Key points
- λ = h/p = h/mv.
- For an electron: λ ≈ 1.227/√V nm.
- Same KE: λ ∝ 1/√m.
- Davisson–Germer confirmed the wave nature of electrons.
Exam tips
- When comparing particles, write λ ∝ 1/√(mqV) and substitute ratios rather than full values.