GATE 2021 PH – Question 44

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

A particle is in a one-dimensional infinite potential well with walls at $x=0$ and $x=L$. A perturbing potential rises linearly from $V(0)=0$ to $V(L)=V_0$, as shown. The first-order correction to its energy eigenvalue is

$V_0/4$
$V_0/3$
$V_0/2$
$V_0/5$

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Correct answer: (C) $V_0/2$

Explanation

Inside the well, $V\prime(x)=V_0x/L$. First-order perturbation theory gives $\Delta E_n^{(1)}=\langle n|V\prime|n\rangle=(V_0/L)\langle x\rangle_n$. The probability density of every infinite-well eigenstate is symmetric about $L/2$, so $\langle x\rangle_n=L/2$. Hence $\Delta E_n^{(1)}=V_0/2$ for every level, option C.

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