Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 35B (Principles of Quantum Mechanics)

Let $\Delta =E_{+}-E_{-}>0$. Hermiticity givesso in the ordered basis $(|-⟩,|+⟩)$,For a nondegenerate unperturbed level, the first-order nondegenerate perturbation theory formulas areThe diagonal matrix elements vanish, so both linear energy corrections are zero. The state corrections areThus, through linear order,Normalization changes only at quadratic order.

The characteristic equation isHence the imaginary-coupled two-level Hamiltonian has exact energiesChoose a real angle $\theta$ satisfyingNormalized exact eigenstates areSincethese states and energies reproduce exactly the linear-order perturbative results from part (i).

The exact energies and mixing coefficients containAs a function of complex $\lambda$, its nearest branch points to the origin occur when its argument vanishes:Their distance from the expansion point $\lambda =0$ determines the radius of convergence. ThereforeThe real-axis perturbation series converges for $|\lambda |<(E_{+}-E_{-})/2$.