UPSC Combined Geo-Scientist, Mains · 2018 · Mains Paper II
Q1(n). Particle-in-a-box eigenvalue
Question
Show that ψₙ=Nsin (nπ x/L) satisfies the Schrodinger equation for V(x)=0 on 0≤ x≤ L and infinite potential elsewhere. Find its energy eigenvalue.
Answer in brief
Eₙ=n²π²ħ²/(2mL²)=n²h²/(8mL²), with n=1,2,….
Key points
- Inside the box, Ĥ=-ħ²(d²/dx²)/(2m). Differentiating twice gives ψₙ′′=-(nπ/L)²ψₙ.
- Thus Ĥψₙ=[n²π²ħ²/(2mL²)]ψₙ, proving the eigenvalue equation.
- The wavefunction vanishes at x=0,L and outside the infinite walls.
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.
CoversParticle-in-a-box eigenvalueQuantum mechanics
All parts of Question 1
- 1(a) Equal molecular speeds5 marks
- 1(b) Fugacity under repulsion5 marks
- 1(c) Wetting of glass5 marks
- 1(d) FCC particle count5 marks
- 1(e) Joule–Thomson cooling5 marks
- 1(f) Gibbs and Helmholtz changes5 marks
- 1(g) Equilibrium-constant conventions5 marks
- 1(h) Very dilute hydrochloric acid5 marks
- 1(i) Pressure controlling boiling5 marks
- 1(j) Primary kinetic salt effect5 marks
- 1(k) Heat of spontaneous adsorption5 marks
- 1(l) Conductivity as an intensive property5 marks
- 1(m) Anthracene photodimerization5 marks
- 1(n) Particle-in-a-box eigenvalue5 marks
- 1(o) Second Balmer line5 marks
- 1(p) Normal-mode counts5 marks