Eigenvalues of the energy, eigenvectors (50 min)

Calculate the two eigenvectors $\vert{i}\rangle$ and eigenvalues $\varepsilon_{i}$ of $\hat{H}$, eq. (3.29), that is the solutions of

$$\hat{H}\vert{i}\rangle = \varepsilon_{i} \vert{i}\rangle,\quad i=1,2.$$ (30)

Show that

$$\vert 1\rangle = \frac{1}{N_1}\left[-2T \vert L\rangle + (\Delta +\varepsilon) \ve... ...quad \varepsilon_1=\frac{1}{2}\left(\varepsilon_L+\varepsilon_R - \Delta\right)$$

$$\vert 2\rangle = \frac{1}{N_2}\left[\phantom{-}2T \vert L\rangle + (\Delta -\varep... ...quad \varepsilon_2=\frac{1}{2}\left(\varepsilon_L+\varepsilon_R + \Delta\right)$$

$$\varepsilon := \varepsilon_L-\varepsilon_{R},\quad \Delta:=\varepsilon_2-\varepsilon_1=\sqrt{\varepsilon^2+4\vert T\vert^2}$$

$$N_{1,2} := \sqrt{4\vert T\vert^2+(\Delta\pm \varepsilon)^2}.$$ (31)