A certain 2-level system has stationary state energies E₁ and E₂ (E₁<E₂) with normalized wave functions φ₁… · ABC26PH1274

Subject: Physical Chemistry · Chapter: Quantum Chemistry · Topic: Perturbation Theory · Exam: CSIR NET JUNE 2014 · Marks: 2 · Difficulty: Medium

A certain 2-level system has stationary state energies $\mathrm{E}_{1}$ and $\mathrm{E}_{2}\left(\mathrm{E}_{1}<\mathrm{E}_{2}\right)$ with normalized wave functions $\phi_{1}$ and $\phi_{2}$ respectively. In the presence of a perturbation V , the second-order correction to the energy for the first state $\left(\phi_{1}\right)$ will be
(a)$\frac{\left\langle\phi_{1}\right| \mathrm{V}\left|\phi_{2}\right\rangle}{\mathrm{E}_{1}-\mathrm{E}_{2}}$
(b)$\frac{\left\langle\phi_{1}\right| \mathrm{V}\left|\phi_{2}\right\rangle}{\mathrm{E}_{2}-\mathrm{E}_{1}}$
(c)$\frac{\left.\left|\left\langle\phi_{1}\right| \mathrm{V}\right| \phi_{2}\right\rangle\left.\right|^{2}}{\mathrm{E}_{1}-\mathrm{E}_{2}}$
(d)$\frac{\left.\left|\left\langle\phi_{1}\right| \mathrm{V}\right| \phi_{2}\right\rangle\left.\right|^{2}}{\left(\mathrm{E}_{1}-\mathrm{E}_{2}\right)^{2}}$
Answer
Answer: C ✓ checked by 4AB · confidence high

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

Explanation
Second-order energy for state 1: E₁⁽²⁾ = |⟨φ₁|V|φ₂⟩|²/(E₁ − E₂) (negative, since E₁ < E₂).

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