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UPSC Combined Geo-Scientist

Chemist

  • 3 descriptive papers
  • 200 marks each
  • 3 h per paper

Mains Paper II covers physical chemistry.

UPSC Combined Geo-Scientist, Mains · 2026 · Mains Paper II

Q1(k). Uncertainty in an energy eigenstate

  • 5 marks
  • Topic: Quantum chemistry
  • Answered
Question

A student says “If a particle is in an energy eigenstate, its energy is fixed, so there is no uncertainty in any quantity”. Is this statement correct? Explain your answer with a suitable example.

Answer in brief

Incorrect. Only the energy has no uncertainty. Position and momentum do not commute with H, so ψₙ is not an eigenfunction of x or p and they have a spread. Example: particle in a box, n=1: Δ x=0.18 L, Δ p=πħ/L.

Key points

  • A property is sharp only if ψ is an eigenfunction of its operator, i.e., Aψ=aψ. ψₙ=√(2/L)sin(nπ x/L) is an eigenfunction of H only.
  • For n=1: x=L/2, Δ x=L112-12π²=0.181 L. p=0, Δ p=√( p²)=πħ/L.
  • Δ x Δ p=0.181πħ=0.568 ħ>ħ/2: the uncertainty principle is satisfied.

The full model answer

Worked out step by step, as it should go in the answer booklet. The solution sheet follows it.

  • A property is sharp only if ψ is an eigenfunction of its operator, i.e., Aψ=aψ. ψₙ=√(2/L)sin(nπ x/L) is an eigenfunction of H only.
  • For n=1: x=L/2, Δ x=L112-12π²=0.181 L. p=0, Δ p=√( p²)=πħ/L.
  • Δ x Δ p=0.181πħ=0.568 ħ>ħ/2: the uncertainty principle is satisfied.
Model answer to Q1(k), 2026 Mains Paper II: solution page 8
Solution page 8 of 22
Read the whole 2026 solution2026 question paper

Coversuncertainty principleUncertainty in an energy eigenstate

All parts of Question 1