UPSC Combined Geo-Scientist, Mains · 2016 · Mains Paper II
Q1(n). Momentum in a box
Question
Show that ψ=Asin (nπ x/L) in a one-dimensional box is not an eigenfunction of p̂ₓ=(ħ/i)d/dx, but is an eigenfunction of p̂ₓ². Discuss the significance.
Answer in brief
The state has definite p²=(nπħ/L)², but no definite signed momentum.
Key points
- Acting once gives p̂ₓψ=-iħ A(nπ/L)cos (nπ x/L), which is not a constant multiple of the sine function.
- Acting twice gives p̂ₓ²ψ=-ħ²ψ′′=(nπħ/L)²ψ. Consequently the kinetic energy is definite, Eₙ=n²π²ħ²/(2mL²).
- A stationary box state is a standing wave with oppositely directed travelling-wave components.
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.
CoversMomentum in a boxQuantum mechanics
All parts of Question 1
- 1(a) Virial equation5 marks
- 1(b) Inclined capillary5 marks
- 1(c) Entropy and diffraction5 marks
- 1(d) Equipartition in hydrogen5 marks
- 1(e) Expansion cooling of helium5 marks
- 1(f) Temperature dependence of equilibrium5 marks
- 1(g) Ammonium chloride pH5 marks
- 1(h) Reverse osmosis5 marks
- 1(i) Rate-law determination5 marks
- 1(j) Monolayer surface area5 marks
- 1(k) Electrolysis of copper sulfate5 marks
- 1(l) Ionization of helium ion5 marks
- 1(m) Photochemical quantum yield5 marks
- 1(n) Momentum in a box5 marks
- 1(o) Formaldehyde electronic transitions5 marks
- 1(p) NMR distinction of ethers5 marks