GATE 2024 PH – Question 61
A mass m in an infinite well of width a has perturbation $V^{\prime}=h^2/(40ma^2)$ over $3a/4<x<a$, as shown. The first-order shift of its fourth energy eigenstate is $h^2/(Nma^2)$. The integer N is _____
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Correct answer: 160
Explanation
The unperturbed energy eigenfunctions of an infinite well of width $a$ are $\psi_n=\sqrt{\tfrac2a}\sin\dfrac{n\pi x}{a}$.
**First-order shift:**
$$\Delta E_n^{(1)}=\langle\psi_n|V^\prime|\psi_n\rangle=V_0\int_{3a/4}^{a}\frac2a\sin^2\frac{n\pi x}{a}\,dx,\qquad V_0=\frac{h^2}{40ma^2}.$$
**For $n=4$** (the fourth state):
$$\int_{3a/4}^a\frac2a\sin^2\frac{4\pi x}{a}dx=\left[\frac xa-\frac{\sin(8\pi x/a)}{8\pi}\right]_{3a/4}^{a}=\left(1-\frac34\right)-\frac{\sin8\pi-\sin6\pi}{8\pi}=\frac14 .$$
The probability of finding the particle in the last quarter of the well is exactly one quarter.
$$\Delta E_4^{(1)}=\frac14\cdot\frac{h^2}{40ma^2}=\frac{h^2}{160\,ma^2}\;\Rightarrow\;N=\mathbf{160}.$$