A particle is in a state φ=ψ₁+3 ψ₂ where ψ₁ and ψ₂ are eigenfunctions of the Hamiltonian of the particle with eigenvalues E₁ and E₂, respectively. The average energy of the particle in the state φ is

(a) (E₁+9 E₂) / 10. Explanation: ψ = ψ₁ + 3ψ₂ normalises to (ψ₁ + 3ψ₂)/√10, so ⟨E⟩ = (E₁ + 9E₂)/10.

Q92 · CSIR-NET Chemistry, June 2016

Paper: CSIR-NET June 2016 · Subject: Physical Chemistry · Chapter: Quantum Chemistry · Topic: Postulates Operators · Marks: 2 · Difficulty: Medium

A particle is in a state $\phi=\psi_{1}+3 \psi_{2}$ where $\psi_{1}$ and $\psi_{2}$ are eigenfunctions of the Hamiltonian of the particle with eigenvalues $\mathrm{E}_{1}$ and $\mathrm{E}_{2}$, respectively. The average energy of the particle in the state $\phi$ is

(a)$\left(\mathrm{E}_{1}+9 \mathrm{E}_{2}\right) / 10$

(b)$\left(\mathrm{E}_{1}+3 \mathrm{E}_{2}\right)$

(c)$\left(\mathrm{E}_{1}+9 \mathrm{E}_{2}\right) / 14$

(d)$\left(\mathrm{E}_{1}+3 \mathrm{E}_{2}\right) / 10$

Answer

Answer: A ✓ checked by 4AB · confidence high

The source book printed no answer; this one was worked out and checked — see the explanation.

Explanation

ψ = ψ₁ + 3ψ₂ normalises to (ψ₁ + 3ψ₂)/√10, so ⟨E⟩ = (E₁ + 9E₂)/10.

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