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Advanced Physics Homework Help
Second-order correction to energy in degenerate purturbation
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[QUOTE="Haorong Wu, post: 6275604, member: 658919"] [B]Homework Statement:[/B] If ##H^{'}## is diagonalised, could I treat the problem with non-degenerate purturbation theory? Here is the problem. ##H_0## has two eigenvalues of ##E_1^0## and ##E_2^0##. The eigenstate of ##E_1^0## is ##\psi _1##, and the two eigenstates of ##E_2^0## are ##\psi_2## and ##\psi_3##. In the space of ##H_0##, ##H^{'}=\begin{pmatrix} 0 & b & c \\ b & -a & 0 \\ c & 0 &a \end{pmatrix}##. What is the second-order corrections to the energies? [B]Relevant Equations:[/B] none Let's focus on the degenerate case. Since in the subspace of the denegerate eigenvectors, ##H^{'}=\begin{pmatrix} -a & 0 \\ 0 &a \end{pmatrix}## is diagonal, then I treat ##\psi _2## and ##\psi _3## with the undegenerate perturbation theory. And I got ##E_2=E_2^0+\left < 2 \right | H^{'} \left |2 \right > +\frac {{\left | H^{'}_{12} \right | }^2} {E_2^0 - E_1^0} = E_2^0 -a +\frac {b^2} {E_2^0 - E_1^0}## ##E_3=E_3^0+\left < 3 \right | H^{'} \left |3 \right > +\frac {{\left | H^{'}_{13} \right | }^2} {E_3^0 - E_1^0} = E_3^0 +a +\frac {c^2} {E_2^0 - E_1^0}## Or should I still appy the second-order correction equations for degenerate perturbation? Such that, I got ##\begin{pmatrix} G_{22} - E_2^2 & G_{23} \\ G_{32} & G_{33} - E_2^2 \end{pmatrix} \begin{pmatrix} c_1 \\ c_2 \end{pmatrix} =0## where ##G_{22}=\frac {{\left | H^{'}_{21} \right | }^2} {E_2^0 - E_1^0}=\frac {b^2} {E_2^0-E_1^0}##, ##G_{33}=\frac {{\left | H^{'}_{31} \right | }^2} {E_2^0 - E_1^0}=\frac {c^2} {E_2^0-E_1^0}##, ##G_{23}=\frac { H^{'}_{21} H^{'}_{13} } {E_2^0 - E_1^0}=\frac {bc} {E_2^0-E_1^0}##, ##G_{32}=\frac { H^{'}_{31} H^{'}_{12} } {E_2^0 - E_1^0}=\frac {bc} {E_2^0-E_1^0}## Solving the equation then ##E_2^2=\frac {b^2+c^2} {E_2^0-E_1^0}## or ##0##. So ##E_2 = E_2^0 -a +\frac {b^2+c^2} {E_2^0-E_1^0}## ##E_3 = E_2^0 +a +\frac {b^2+c^2} {E_2^0-E_1^0}## I am no sure which one is correct. A textbook reads, if the degenerate eigenvectors are splited with the first-order correction, then I should treat the corrected good states with undegenerate perturbation. Could this apply to this problem? Thanks! [/QUOTE]
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Second-order correction to energy in degenerate purturbation
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