State de Broglie's hypothesis and derive an expression for the wavelength of an electron accelerated through a potential difference V.

1. 1 · State the hypothesis. Moving material particles show wave nature; the wavelength is λ = h/p = h/mv.
2. 2 · Give the reason. Light has a dual nature, and nature loves symmetry, so matter should have it too.
3. 3 · Write the energy gained. An electron accelerated from rest through V gains ½mv² = eV.
4. 4 · Find the momentum. p = mv = √(2meV).
5. 5 · Get the wavelength. λ = h/√(2meV). With h = 6.63 × 10⁻³⁴ J s, m = 9.1 × 10⁻³¹ kg, e = 1.6 × 10⁻¹⁹ C: λ ≈ 12.3/√V Å.
6. 6 · Real life. Fast electrons have tiny wavelengths, so electron beams can show the spacing of atoms in crystals.
7. 7 · Close in one line. So, λ ∝ 1/√V: a higher voltage gives a shorter wavelength.

Scoring tip: Show every step from ½mv² = eV to λ = h/√(2meV) before putting in numbers. Write the unit Å with the final answer.