Degenerate Perturbation Theory

Assume a $d$-fold degenerate energy level $E$ with $d$ degenerate eigenstates of $\hat{H}_0$

$$\vert 1\rangle, \vert 2\rangle, \dots, \vert d\rangle,\quad \hat{H}_0\vert i\rangle = E\vert i\rangle.$$ (3.16)

The perturbation $\hat{H}_1$ leads to new eigenfunctions

$$x_1 \vert 1\rangle + x_2 \vert 2\rangle + ...+ x_d \vert d\rangle... ...quad {\bf d}^T \equiv = (\vert 1\rangle, \vert 2\rangle, \dots, \vert d\rangle)$$ (3.17)

where the notation ${\bf x} \cdot {\bf d}^T$ is just an abbreviation using the coefficient vector ${\bf x}$ and the vector of the degenerate states ${\bf d}^T$. The coefficient vectors ${\bf x}$ are then determined from the matrix eigenvalue equation

$$\underline{\underline{H}} {\bf x} = E' {\bf x},\quad \underline{\underline{H}}_{ij} \equiv \langle i \vert \hat{H}_1 \vert j \rangle$$ (3.18)

with the Hermitian $d$ times $d$ matrix $\underline{\underline{H}}$ of the matrix elements of the perturbation $\hat{H}_1$ in the sub-space of the degenerate eigenstates $\vert i\rangle$ of $\hat{H}_0$.

The solutions for $E'$ are determined from $\det \left( \underline{\underline{H}} - E' \underline{\underline{1}}\right)=0$ or

$$\left\vert \begin{array}{cccc} \langle 1 \vert \hat{H}_1 \vert 1\... .... & \langle d \vert \hat{H}_1 \vert d \rangle - E'\\ \end{array}\right\vert=0,$$ (3.19)

which is an algebraic equation with $d$ real solutions $E'_i$, $i=1,...,d$. Correspondingly, one obtains $d$ coefficient vectors ${\bf x}_i$ leading to $d$ new linear combinations ${\bf x}_i \cdot {\bf d}^T$, $i=1,...,d$, of states within the $d$-dimensional subspace spanned by $\vert 1\rangle, \vert 2\rangle, \dots, \vert d\rangle$.

Exercise 1: Revise if necessary Gasiorowicz [3] cp. 11.2, plus the corresponding math background: eigenvalues, eigenvalue equations, vector spaces, matrices etc. !

Exercise 2: Revise degenerate perturbation theory by applying it to the 2-level system ${\mathcal H}_{\rm TLS}$ from section II.2 for the case $\varepsilon=\varepsilon_L-\varepsilon_R=0$. How good is first order perturbation theory in this case?