UPSC Combined Geo-Scientist, Mains · 2023 · Mains Paper II
Q1(l). Is Peⁱᵏˣ + Qe⁻ⁱᵏˣ an eigenfunction?
Question
Is the function ψ(x) = Peⁱᵏˣ + Qe⁻ⁱᵏˣ an eigenfunction of d/dx and d²/dx²? P, Q and k are real numbers.
Answer in brief
Not an eigenfunction of d/dx (unless P = 0 or Q = 0). An eigenfunction of d²/dx² with eigenvalue −k².
Key points
- dψ/dx = ik(Peⁱᵏˣ − Qe⁻ⁱᵏˣ) ≠ (constant) × ψ for k, P, Q ≠ 0 ⇒ not an eigenfunction of d/dx.
- d²ψ/dx² = (ik)²Peⁱᵏˣ + (−ik)²Qe⁻ⁱᵏˣ = −k²ψ ⇒ an eigenfunction, eigenvalue −k².
- p̂x = −iħ d/dx, so ψ mixes momenta +ħk and −ħk: momentum not sharp; Ĥ gives one value E = ħ²k²/2m: energy sharp.
- P = Q gives 2P cos kx; P = −Q gives 2iP sin kx, the particle-in-a-box form.
The full model answer
As written for the answer booklet: the diagram carries the working, the lines stay short. Read here only; it is not offered as a download.
Coverseigenfunctioneigenvalueoperatorsmomentum operatorquantum mechanics
All parts of Question 1
- 1(a) van der Waals equation: meaning of a and b5 marks
- 1(b) Exceptional behaviour of hydrogen and helium5 marks
- 1(c) Unit cell of solid C₆₀ and its carbon atoms5 marks
- 1(d) Planes that diffract: simple cubic, a = 5 Å5 marks
- 1(e) Chemical potential and its link with Gibbs energy5 marks
- 1(f) Debye–Hückel limiting law for γ±5 marks
- 1(g) Activation energy of 2HI → H₂ + I₂5 marks
- 1(h) Best coagulant for the Sb₂S₃ sol5 marks
- 1(i) E° of Na⁺/Na from ΔrH° and ΔrS°5 marks
- 1(j) pH from a calomel–quinhydrone cell5 marks
- 1(k) Degeneracy of the level 6h²/8mL² in a cubic box5 marks
- 1(l) Is Peⁱᵏˣ + Qe⁻ⁱᵏˣ an eigenfunction?5 marks
- 1(m) Zero-point energy in joule and in cm⁻¹5 marks
- 1(n) Microwave activity of SF₆, CO, H₂, H₂O and CO₂5 marks
- 1(o) Stark–Einstein law and the quantum yield5 marks
- 1(p) Decay routes of a photoexcited molecule5 marks