$$\require{cancel}$$

# 11.10: 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

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

$$\newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}$$ $$\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}$$ $$\newcommand {\btau}{\mbox{\boldmath\tau}}$$ $$\newcommand {\bmu}{\mbox{\boldmath\mu}}$$ $$\newcommand {\bsigma}{\mbox{\boldmath\sigma}}$$ $$\newcommand {\bOmega}{\mbox{\boldmath\Omega}}$$ $$\newcommand {\bomega}{\mbox{\boldmath\omega}}$$ $$\newcommand {\bepsilon}{\mbox{\boldmath\epsilon}}$$