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GATE 2022 PH – Question 39

Quantum Mechanics · variational method, time independent perturbation theory · 2 marks · Multiple choice

Consider a particle in three different boxes of width L. The potential inside the boxes vary as shown in figures (i), (ii) and (iii) with V଴≪ ħమగమ ଶ௠௅మ . The corresponding ground-state energies of the particle are Eଵ, Eଶ and Eଷ, respectively. Then

Three weak perturbations inside infinite wells.
  1. $E_2>E_1>E_3$
  2. $E_3>E_1>E_2$
  3. $E_2>E_3>E_1$
  4. $E_3>E_2>E_1$

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Correct answer: (C) $E_2>E_3>E_1$

Explanation

First-order shifts weight the perturbation by the unperturbed sine-squared ground density. The central third (ii) has the largest weight; (iii) includes two outer sixths and exceeds the left third (i).