UPSC Combined Geo-Scientist, Mains · 2023 · Mains Paper II
Q1(k). Degeneracy of the level 6h²/8mL² in a cubic box
Question
What is the degree of degeneracy of the energy level of a particle in a cubic box having energy 6h²/8mL²? Write the wave function(s) of the particle in this energy level.
Answer in brief
Triply degenerate (g = 3): (nx, ny, nz) = (2, 1, 1), (1, 2, 1), (1, 1, 2).
Key points
- E = (h²/8mL²)(nx² + ny² + nz²); the only way to make 6 is 4 + 1 + 1: (2,1,1), (1,2,1), (1,1,2).
- Three states, one energy: g = 3 (triply degenerate).
- Wave functions ψ = (2/L)³ᐟ² sin(nxπx/L) sin(nyπy/L) sin(nzπz/L): ψ₂₁₁, ψ₁₂₁, ψ₁₁₂.
- Each has one nodal plane; the symmetry of the cube (Lx = Ly = Lz) makes them equal in energy. A box with unequal edges splits the level.
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 3-D boxdegeneracycubic boxwave functionsquantum numbers
All parts of Question 1
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- 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
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- 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