Perturbation Theory allows us to find approximate solutions to quantum systems that deviate slightly from exactly solvable ones. The anharmonic oscillator has Hamiltonian:
H = p²/2m + ½mω²x² + λx⁴
The first-order energy correction is ⟨n|λx⁴|n⟩ = λ(ℏ/2mω)²(6n²+6n+3). For a cubic perturbation λx³, the first-order correction vanishes by symmetry; second-order gives negative corrections proportional to λ². The wavefunctions shown are harmonic oscillator eigenfunctions offset to their perturbed energy eigenvalues, with the potential curves overlaid.