12_variation_perturbation_matrixhq

8 27 in atomic units 8 8 1072013 variational method

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Value of Parameter 2 27 8 27 16 0 effective nuclear charge; formal charge is 2 plug this back into energy expression , 27 8 27 16 27 27 8 16 27 16 Most Accurate Value (ignoring repulsion 2.8477 2.9037 4.000 ) Effect of Electron-Electron Interaction H-like 1s with Z=2 •whenever you add electrons to energy diagrams, the energy levels change •we generally don’t redraw these diagrams because it is understood that the energies change 9 10/7/2013 Wavefunction as a Column Matrix Given a basis set of functions, , that are orthonormal ( ∗ , ), wavefunctions ( can be expressed as coefficients multiplying the basis functions. as analytical function or as a column matrix ∑ ∗ Ket as vector | ∗ ⋮ ⋮ Recipe for getting the coefficients 10 10/7/2013 Hamiltonian Operator as a Square Matrix Given an orthonormal basis set, matrix elements are 1,2, … , the Hamiltonian for ∗ , Recipe for getting the matrix elements The Hamiltonian matrix will be ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Hamiltonian Operator as a Square Matrix Particle-in-Box Example: and evaluate matrix elements ∗ , ∗ 2 1 0 2 0 0 2 0 2 ⋮ 0 ⋯ 0 2 0 2 3 2 0 0 0 Eigenvalues, 0 ∗ 2 , When basis set is eigenfunction of , then H is diagonal (with the eigenvalues on the diagonal). 0 ⋱ ⋯ ⋮ 2 A diagonal H is a solved problem. Solving a problem is equivalent to diagonalizing H. , on diagonal 11 10/7/2013 Ramp Potential Hamiltonian Matrix whe...
View Full Document

Ask a homework question - tutors are online