P460 - perturbation1 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 (derived slightly differently in Griffiths) 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  l form complete set of states (linear algebra) Sometimes Einstein convention used. Implied sum if 2 of same index

P460 - perturbation2 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 - perturbation3 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 - perturbation4 Time independent example know eigenfunctions/values of infinite well. Assume mostly in ground state n=1

P460 - perturbation5 Time independent example Get first order correction to wavefunction only even Parity terms remain (rest identically 0) as gives Even Parity

P460 - perturbation6 Time Dependent Perturbation Theory Many possible potentials. Consider one where V’(x,t)=V(x)+v(x,t) V(x) has solutions to the S.E. and so known eigenvalues and eigenfunctions let perturbation v(x,t) be small compared to V(x) examples:finite square well plus a pulse or atoms in an oscillating electric field Solve Schrodinger.Eqn

P460 - perturbation7 Time Dependent First line (in brackets[]) is equal to 0 take “dot product” of other two terms with wave function Wave functions are orthogonal

P460 - perturbation8 Time Dependent :Example This is exact. How to solve depend on the initial conditions and perturbation take v(x,t)=0 t 0. Also assume all in k-state at t=0 and perturbation is small Probability to be in state m depends on the square of amplitude a. Only large when energies are ~same

P460 - perturbation9 Time Dependent :Example The probability for state k to make a transition into any other state is: evaluate P by assuming closely spaced states and replacing sum with integral Seen in Plank distribution. We see again later

P460 - perturbation10 Fermi Golden Rule Assume first 2 terms vary slowly.Pull out of integral and evaluate the integral at the pole this doesn’t always hold--ionization has large dE offset by larger density n are states near k. Conserve energy. Rate depends on both the matrix element (which includes “physics”) and the density of states. Examples later on, especially in 461