Electrons start from negligible kinetic energy and accelerate through a potential difference 200 V in vacuum. Use |e| = 1.602 × 10⁻¹⁹ C, m_e = 9.11 × 10⁻³¹ kg, h = 6.626 × 10⁻³⁴ J s and c = 3.00 × 10⁸ m s⁻¹. Calculate their momentum and de Broglie wavelength using a nonrelativistic model. Predict the wavelength if the accelerating voltage is quadrupled. Explain why electron diffraction is evidence against a purely classical particle description, and check that the initial voltage justifies the nonrelativistic approximation.

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Momentum 7.640 × 10^-24 kg m s⁻¹; wavelength 8.672 × 10^-11 m; at four times voltage 4.336 × 10^-11 m. Initial speed is 2.796% of c. Electrical work gives each electron kinetic energy K = |e|V = 3.204 × 10^-17 J. For nonrelativistic motion K = p²/(2m), so momentum is p = √(2m|e|V). Substitution gives p = 7.640 × 10^-24 kg m s⁻¹. The de Broglie relation then yields λ = h/p = 8.672 × 10^-11 m. Since momentum scales with √V, wavelength scales with V⁻¹ᐟ². Quadrupling voltage therefore halves the wavelength to 4.336 × 10^-11 m rather than dividing it by four. A diffraction pattern has maxima and minima produced by wave interference. Classical particles following single definite trajectories without wave behaviour do not by themselves account for that pattern; a matter-wave description predicts its wavelength dependence. The initial-voltage speed estimate is p/m = 8.387 × 10^6 m s⁻¹, only 2.796% of c. This small ratio supports the stated nonrelativistic approximation.

Electron acceleration and matter-wave wavelength · set 5

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