Time-independent perturbation theory

Time-independent perturbation theory expands the eigenvalues and eigenstates of $H_0+V$ in powers of a small perturbation $V$.

For a differentiable normalized eigenstate of a differentiable self-adjoint Hamiltonian, differentiate $H\psi =E\psi$ and pair with $\psi$. The terms involving ${\psi }^{\prime }$ cancel because $H$ is self-adjoint, leaving the stated formula. It turns energy derivatives into observable expectations. A smoothly chosen eigenbranch and suitable operator-domain regularity are required, particularly at degeneracies or crossings.

The first-order energy correction is the coefficient of the perturbation parameter in the expansion of an energy eigenvalue.

An energy eigenvalue is nondegenerate when its eigenspace is one-dimensional. Its normalized eigenstate is then unique up to a complex phase.

For a normalized eigenstate $|n⟩$ belonging to a nondegenerate energy eigenvalue,

For a quantum harmonic oscillator perturbed by $\lambda X^4$, write $X=\sqrt{\hslash /(2m\omega )}(A+A^{†})$. Sincefirst-order nondegenerate perturbation theory gives

For a normalized nondegenerate eigenstate $|n⟩$,For a ground state, every denominator is negative, so the correction is nonpositive.

Forthe exact energies areThe perturbation series around $\lambda =0$ has radius $\Delta /2$, set by the branch points $\lambda =\pm i\Delta /2$.

Suppose one state $|0⟩$ couples to degenerate states $|1⟩,\dots ,|m⟩$ with amplitudes $v_1,\dots ,v_m$, while the degenerate states do not couple to each other. The normalized bright state proportional to ${\sum }_j{v}_j|j⟩$ is the only combination coupled to $|0⟩$. Every orthogonal combination is a dark state and remains an exact eigenstate at the degenerate energy. The remaining spectrum comes from a two-dimensional Hamiltonian whose off-diagonal entry is $({\sum }_j|v_j{|}^2{)}^{1/2}$.