Perturbation Theory Only a few QM systems that can be solved exactly: Hydrogen Atom(simplified), harmonic oscillator, infinite+finite well solve using perturbation theory which starts from a known solution and makes successive approx- imations start with time independent. V’(x)=V(x)+v(x) V(x) has solutions to the S.E. and so known eigenvalues and eigenfunctions let perturbation v(x) be small compared to V(x) As yl form complete set of states (linear algebra) Sometimes Einstein convention used. Implied sum if 2 of same index P460 - perturbation

Plug into Schrod. Eq. know solutions for V use orthogonality multiply each side by wave function* and integrate matrix element of potential v is defined: P460 - perturbation

One solution: assume perturbed wave function very close to unperturbed (matrix is unitary as “size” of wavefunction doesn’t change) assume last term small. Take m=n. Energy difference is expectation value of perturbing potential **** P460 - perturbation

Redo compact notation eigenvalues/functions for a “base” Hamiltonian want to solve (for l small)(l keeps track or order) define matrix element for Hamiltonian H finite or infinite dimensional matrix. If finite (say 3x3) can use diagonalization techniques. If infinite can use perturbation theory write wavefunction in terms of eigenfunctions but assume just small change P460 - perturbation

compact notation- energy look at first few terms (book does more) which simplifies to (first order in l) rearranging take the scalar product of both sides with first approximation of the energy shift is the expectation value of the perturbing potential P460 - perturbation

compact notation- wavefunction look at wavefunction and repeat equation for energy take the scalar product of both sides with gives for first order in l note depends on overlap of wavefunctions and energy difference P460 - perturbation

Time independent example know eigenfunctions/values of infinite well. Assume mostly in ground state n=1 P460 - perturbation

Time independent example Get first order correction to wavefunction only even Parity terms remain (rest identically 0) as gives Even Parity P460 - perturbation