The GATE Grind

GATE 2025 PH – Question 44

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

A two-level quantum system has eigenvalues $E_1,E_2$. A perturbation $H^{\prime}=\lambda\Delta\sigma_x$ is introduced, where $\Delta$ has energy dimensions, $\lambda$ is small and dimensionless, and $$\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix}.$$ The magnitudes of first and second order corrections to $E_1$, respectively, are

  1. $0,\lambda^2\Delta^2/|E_1-E_2|$
  2. $|\lambda\Delta|/2,\lambda^2\Delta^2/|E_1-E_2|$
  3. $|\lambda\Delta|,\lambda^2\Delta^2/|E_1-E_2|$
  4. $0,\lambda^2\Delta^2/(2|E_1-E_2|)$

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Correct answer: (A) $0,\lambda^2\Delta^2/|E_1-E_2|$

Explanation

**Unperturbed energies:** $E_1$ and $E_2$. **Perturbation matrix:**
$$H^\prime=\lambda\Delta\begin{pmatrix}0&1\\1&0\end{pmatrix}.$$

**First-order correction to $E_1$:**
$$E_1^{(1)}=\langle1|H^\prime|1\rangle=0,$$
since the diagonal elements of $\sigma_x$ are zero.

**Second-order correction:**
$$E_1^{(2)}=\frac{|\langle2|H^\prime|1\rangle|^2}{E_1-E_2}=\frac{\lambda^2\Delta^2}{E_1-E_2},$$
whose magnitude is $\dfrac{\lambda^2\Delta^2}{|E_1-E_2|}$.

So the magnitudes are $0$ and $\lambda^2\Delta^2/|E_1-E_2|$ (option A).