Variational Method Variational Principle Depends on variational parameters The more we use: The more complicated the function The closer we get to the true value the lowest eigenvalue of an arbitrary wavefunction is greater than or equal to the minimum Variational Method Proof:

Hydrogen Example

Hydrogen Example S-orbital & guess function,  s-orbital &  with more variables

Helium Example

Secular Determinants Getting all the players in order VM & Linear Combos of Functions We work our way into a matrix representation & solve the system with determinants Secular Determinants The Generalized Case

Simplified N=2 Example

Matrices & Determinants where we solve Matrices & Determinants

Applied to 1D PIB Overall nice fit (better than average) The 51.065 root is an approximation for 1st e.s.

Used to Solve for Excited States

Perturbation Theory The Setup Applied Example Use perturbation theory to calculate the first - order correction to the ground-state energy of a quartic oscillator whose Use Harmonic Oscillator for the unperturbed system.

SKIP