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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(l). Expectation value and eigenvalue

  • 5 marks
  • Topic: Quantum chemistry
  • Answered
Question

Define the expectation value of an operator. How does it differ from an eigenvalue? Explain with reference to energy measurements.

Answer in brief

A=(∫ψ^* Aψ dτ)/(∫ψ^*ψ dτ) is the average of many measurements. An eigenvalue a satisfies Aψ=aψ and is a single possible result. If ψ is an eigenfunction, A=a.

Key points

  • eigenvalues Eₙ are the only results of a measurement. For Ψ=c₁ψ₁+c₂ψ₂ each measurement gives E₁ (probability |c₁|²) or E₂ (|c₂|²), never an intermediate value.
  • E=|c₁|²E₁+|c₂|²E₂ lies between E₁ and E₂ and is generally not an eigenvalue.
  • In an eigenstate the spread Δ E=0 and E=Eₙ.

The full model answer

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

  • Energy: eigenvalues Eₙ are the only results of a measurement. For Ψ=c₁ψ₁+c₂ψ₂ each measurement gives E₁ (probability |c₁|²) or E₂ (|c₂|²), never an intermediate value.
  • E=|c₁|²E₁+|c₂|²E₂ lies between E₁ and E₂ and is generally not an eigenvalue.
  • In an eigenstate the spread Δ E=0 and E=Eₙ.
Model answer to Q1(l), 2026 Mains Paper II: solution page 8
Solution page 8 of 22
Read the whole 2026 solution2026 question paper

CoverspostulatesExpectation value and eigenvalue

All parts of Question 1