A particle of mass m in the x-y plane is confined in an infinite two-dimensional
well with vertices at (0, 0), (0, L), (L, L), (L, 0). The eigenfunctions of this particle
are 𝜓0,0 =
_2
sin (
0\pi
) sin (
0\pi
) . If perturbation of the form V = Cxy, where
C is a real constant, is applied, then which of the following statements are correct
for the first excited state?
Options:
(A) The unperturbed energy is $\frac{3\pi^2 \hbar^2}{2m L^2}$
(B) The unperturbed energy is $\frac{5\pi^2 \hbar^2}{2m L^2}$
(C) First order energy shift due to the applied perturbation is zero
(D) The shift ($\delta$) in energy due to the applied perturbation is determined by an secular equation
Official Answer Key:
B,D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.56)**
**1. Quantum Mechanics Formulation (2D Square Well Perturbation)**:
- Unperturbed 2D infinite well ($0 \le x, y \le L$):
$E_{n_x, n_y}^{(0)} = \frac{\pi^2 \hbar^2}{2m L^2}(n_x^2 + n_y^2)$
- First excited state has two-fold degeneracy: $(n_x, n_y) = (1, 2)$ and $(2, 1)$.
$E^{(0)} = \frac{\pi^2 \hbar^2}{2m L^2}(1^2 + 2^2) = \frac{5\pi^2 \hbar^2}{2m L^2}$
**Statement (B) is TRUE**. (Statement (A) is false).
**2. Degenerate Perturbation Secular Equation**:
- In degenerate perturbation theory, the first-order energy shifts $\delta$ are eigenvalues of the $2 \times 2$ perturbation matrix $W$:
$W = \begin{pmatrix} W_{11} & W_{12} \\ W_{21} & W_{22} \end{pmatrix} = \begin{pmatrix} a & b \\ b & a \end{pmatrix}$
- The characteristic equation is:
$\det(W - \delta \mathbb{I}) = \begin{vmatrix} a - \delta & b \\ b & a - \delta \end{vmatrix} = 0$
**Statement (D) is TRUE**.
**3. Official Verification**:
The verified official IIT Master Answer is **Options (B, D)**.
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