GATE 2020 PH – Question 60

Quantum Mechanics · Variational method, time independent perturbation theory · 2 marks · Numerical answer

Consider $\hat H=\hat H_0+\hat H\prime$, with $$\hat H_0=\begin{pmatrix}E&0&0\\0&E&0\\0&0&E\end{pmatrix},\qquad\hat H\prime=\begin{pmatrix}0&k&0\\k&0&k\\0&k&0\end{pmatrix},\quad k>0.$$ If the maximum eigenvalue is 3 eV when E = 2 eV, find k in eV, to three decimal places.

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Correct answer: 0.706 to 0.708

Explanation

The perturbation characteristic polynomial is $-\lambda(\lambda^2-2k^2)$, giving eigenvalues $0,\pm\sqrt2k$. Adding $E I$ shifts them to $E,E\pm\sqrt2k$. The largest is $2+\sqrt2k=3$ eV, so $k=1/\sqrt2=0.707106$ eV, rounding to 0.707 eV.

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