Q1(a). Free-particle waves and de Broglie relation

Answer in brief

The free-particle solutions are plane waves ψ = Ae^±ikx, with momentum p = ±ħk. Since k = 2π/λ, |p| = h/λ.

Key points

• Choose the constant potential as zero. The stationary equation becomes −ħ²ψ″/(2m) = Eψ, or ψ″ + k²ψ = 0, with k² = 2mE/ħ².
• Its general solution is Ae^ikx + Be^−ikx. The two terms describe opposite propagation directions.
• Acting with p̂ = −iħ d/dx on Ae^ikx gives p̂ψ = ħkψ. Hence E = p²/(2m) and p = ħk for that momentum eigenstate.
• A spatial period satisfies kλ = 2π. Therefore λ = 2πħ/|p| = h/|p|, the de Broglie relation.