Q87 · CSIR-NET Chemistry, June 2013

Paper: CSIR-NET June 2013 · Subject: Physical Chemistry · Chapter: Quantum Chemistry · Topic: Quantum Chemistry – General · Marks: 2 · Difficulty: Medium

For the particle-in-a-box problem in $(0, \mathrm{L})$ an approximate wave function is given as $\mathrm{x}(\mathrm{L} / 2-\mathrm{x})(\mathrm{L}-\mathrm{x})$. The average energy E for such a state will obey
(a)$\frac{\mathrm{h}^{2}}{8 \mathrm{mL}^{2}}<\overline{\mathrm{E}}<\frac{\mathrm{h}^{2}}{2 \mathrm{mL}^{2}}$
(b)$\overline{\mathrm{E}}>\frac{\mathrm{h}^{2}}{2 \mathrm{mL}^{2}}$
(c)$\frac{\mathrm{h}^{2}}{4 \mathrm{mL}^{2}}<\overline{\mathrm{E}}<\frac{\mathrm{h}^{2}}{2 \mathrm{mL}^{2}}$
(d)$0<\overline{\mathrm{E}}<\frac{\mathrm{h}^{2}}{8 \mathrm{mL}^{2}}$
Answer
Answer: B ✓ checked by 4AB · confidence medium

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

Explanation
x(L/2 − x)(L − x) has a node at L/2 and is odd about the centre, so it is orthogonal to ψ₁. By the variation principle, Ē ≥ E₂ = h²/2mL² (b).

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