Q1(l). Momentum eigenfunction

Answer in brief

 eⁱᵏˣ=ħ k eⁱᵏˣ, so the eigenvalue is ħ k.

Key points

• An eigenfunction satisfies Ây=ay for a constant eigenvalue a. Differentiate: dy/dx=ik eⁱᵏˣ.
• Thus Ây=(-iħ)(ik)eⁱᵏˣ=ħ k eⁱᵏˣ, since (-i)i=1.
• The operator is the x-momentum operator, so the travelling wave has definite momentum pₓ=ħ k and, for a free particle, energy ħ²k²/(2m).
• The wave number k has unit m⁻¹ and ħ k has unit kg m s⁻¹.