GATE 2022 PH – Question 56
A 2D square well has $\psi_{n_xn_y}=2\sin(n_x\pi x/L)\sin(n_y\pi y/L)/L$. For perturbation $V=Cxy$, which statements apply to its first excited state?
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Correct answer: (B) $E=5\pi^2\hbar^2/(2mL^2)$; (D) The secular equation is $\det\begin{pmatrix}a-\delta&b\\b&a-\delta\end{pmatrix}=0$, with nonzero real a,b
Explanation
**Unperturbed energies:** $E_{n_xn_y}=\dfrac{\pi^2\hbar^2}{2mL^2}\left(n_x^2+n_y^2\right)$.
- Ground state: $(1,1)$, $E=2\epsilon_0$ with $\epsilon_0=\dfrac{\pi^2\hbar^2}{2mL^2}$.
- **First excited level:** $(1,2)$ and $(2,1)$, both with $E=5\epsilon_0=\dfrac{5\pi^2\hbar^2}{2mL^2}$ (doubly degenerate). So B is true and A is false.
**Degenerate perturbation theory** for $V=Cxy$ in the doublet $\{|12\rangle,|21\rangle\}$:
- Diagonal elements: $\langle12|xy|12\rangle=\langle x\rangle_1\langle y\rangle_2=\dfrac L2\cdot\dfrac L2=\dfrac{L^2}{4}$, so $a=\dfrac{CL^2}{4}$ (the same for $|21\rangle$).
- Off-diagonal element: $\langle12|xy|21\rangle=\langle1|x|2\rangle\langle2|y|1\rangle\neq0$, a non-zero real number $b$.
The first-order shifts are the roots of
$$\det\begin{pmatrix}a-\delta&b\\b&a-\delta\end{pmatrix}=0,\qquad\delta=a\pm b ,$$
which are non-zero, so the first-order shift is not zero (C is false) and D is true.
Answer **B and D**.