11.1: Exercises

1. Consider the two-state system investigated in Section 1.3. Show that the most general expressions for the perturbed energy eigenvalues and eigenstates are \[\begin{aligned} E_1'&= E_1 + e_{11}+\frac{|e_{12}|^{\,2}}{E_1-E_2}+{\cal O}(\epsilon^{\,3}),\nonumber\\[0.5ex] E_2' &= E_2+e_{22}- \frac{|e_{12}|^{\,2}}{E_1-E_2}+{\cal O}(\epsilon^{\,3}),\nonumber\end{aligned}\] and \[\begin{aligned} \psi_1' &= \psi_1+ \frac{e_{12}^{\,\ast}}{E_1-E_2}\,\psi_2 + {\cal O}(\epsilon^{\,2}),\nonumber\\[0.5ex] \psi_2' &= \psi_2 -\frac{e_{12}}{E_1-E_2}\,\psi_1+{\cal O}(\epsilon^{\,2}),\nonumber\end{aligned}\] respectively. Here, \(\epsilon = |e_{12}|/(E_1-E_2)\ll 1\). You may assume that \(|e_{11}|/(E_1-E_2)\), \(|e_{22}|/(E_1-E_2)\sim {\cal O}(\epsilon)\).
2. Consider the two-state system investigated in Section 1.3. Show that if the unperturbed energy eigenstates are also eigenstates of the perturbing Hamiltonian then \[\begin{aligned} E_1'&= E_1 + e_{11},\nonumber\\[0.5ex] E_2' &= E_2+e_{22},\nonumber\end{aligned}\] and \[\begin{aligned} \psi_1' &= \psi_1 \nonumber\\[0.5ex] \psi_2' &= \psi_2\nonumber\end{aligned}\] to all orders in the perturbation expansion.
3. Consider the two-state system investigated in Section 1.3. Show that if the unperturbed energy eigenstates are degenerate, so that \(E_1=E_2=E_{12}\), then the most general expressions for the perturbed energy eigenvalues and eigenstates are \[E^\pm = E_{12}+e^\pm,\] and \[\psi^\pm= \langle 1|\psi^\pm\rangle\, \psi_1+\langle 2|\psi^\pm\rangle\,\psi_2,\] respectively, where \[e^\pm = \frac{1}{2}\,(e_{11}+e_{22})\pm \frac{1}{2}\left[(e_{11}-e_{22})^2+4\,|e_{12}|^{\,2}\right]^{1/2},\] and \[\frac{\langle 1|\psi^\pm\rangle}{\langle 2|\psi^\pm\rangle}=-\left(\frac{e_{12}}{e_{11}-e^\pm}\right)=-\left(\frac{e_{22}-e^{\pm}}{e_{12}^{\,\ast}}\right).\] Demonstrate that the \(\psi^\pm\) are the simultaneous eigenstates of the unperturbed Hamiltonian, \(H_0\), and the perturbed Hamiltonian, \(H_1\), and that the \(e^\pm\) are the corresponding eigenvalues of \(H_1\).
4. Calculate the lowest-order energy-shift in the ground state of the one-dimensional harmonic oscillator when the perturbation \[V = \lambda\,x^{\,4}\] is added to \[H = \frac{p_x^{\,2}}{2\,m} + \frac{1}{2}\,m\,\omega^{\,2}\,x^{\,2}.\]

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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