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[QUE/QM-23015]

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Use the first order non-degenerate perturbation theory to compute the correction to the $\ell=1$ level of ( use units so that $\hbar=1$ ) $$ H = L^2 + \alpha L_z  + \beta L_x $$ Use the splitting of $H$ as $$ H_0=L^2 + \alpha L_z, \qquad \mbox{and} \qquad H^\prime =  \beta L_x $$ in terms of unperturbed Hamiltonian and perturbation Hamiltonian $H^\prime.$

  1. Find the corrections to the $\ell=1$ energy level using  perturbation theory to  the lowest non-vanishing order in $H^\prime$. Compare your answer with the exact answers.
  2. Obtain the eigenvectors of $H$ upto lowest order in $\beta$.

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