Quantum Physics of Atoms

Year 3

Perturbation Theory

• appropriate if dealing with system similar to one that can be solved exactly

• eigenvalues and eigenfunctions form a series that can converge/diverge

Equation for first order shift in energy

E_{n}^1=\langle\phi_{n}^0\vert\hat{V}\vert\phi_{n}^0\rangle

Equation for first order correction to eigenstates

\vert\psi_{n}^1\rangle=-\sum_{p\neq n}\frac{\langle\phi_{p}^0\vert\hat{V}\vert\phi_{n}^0\rangle}{E_{p}^0-E_{n}^0}\;\vert\phi_{p}^0\rangle

Equation for second order shift in the energy

E_{n}^2=-\sum_{p\neq n}\frac{|\langle\phi_{p}^0\vert\hat{V}\vert\phi_{n}^0\rangle|^2}{E_{p}^0-E_{n}^0}