notes10 - 5.73 Lecture #10 10 - 1 Matrix Mechanics should...

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10 - 1 5.73 Lecture #10 updated 9/18/02 8:55 AM Matrix Mechanics should have read CDTL pages 94-121 read CTDL pages 121-144 ASAP Last time: Numerov-Cooley Integration of 1-D Schr. Eqn. Defined on a Grid. 2-sided boundary conditions nonlinear system - iterate to eigenenergies (Newton-Raphson) So far focussed on ψ (x) and Schr. Eq. as differential equation. Variety of methods {E i , ψ i (x)} V(x) Often we want to evaluate integrals of the form ψφ * () () x x dx a ii = overlap of special ψ on standard functions { φ } a is “mixing coefficient” { φ } is complete set of “basis functions” OR φ i * ˆx n φ j dx x n () ij expectation values transition moments There are going to be elegant tricks for evaluating these integrals and relating one integral to others that are already known. Also “selection” rules for knowing automatically which integrals are zero: symmetry, commutation rules Today: begin matrix mechanics - deal with matrices composed of these integrals - focus on manipulating these matrices rather than solving a differential equation - find eigenvalues and eigenvectors of matrices instead (COMPUTER “DIAGONALIZATION”) * Perturbation Theory: tricks to find approximate eigenvalues of infinite matrices * Wigner-Eckart Theorem and 3-j coefficients: use symmetry to identify and inter- relate values of nonzero integrals * Density Matrices: information about state of system as separate from measurement operators H I G H L I G H T S
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10 - 2 5.73 Lecture #10 updated 9/18/02 8:55 AM First Goal: Dirac notation as convenient NOTATIONAL simplification It is actually a new abstract picture (vector spaces) — but we will stress the utility ( ψ | relationships) rather than the philosophy! Find equivalent matrix form of standard ψ (x) concepts and methods. 1. Orthonormality 2. completeness ψ (x) is an arbitrary function A. Always possible to expand ψ (x) uniquely in a COMPLETE BASIS SET { φ } ψ i * ψ j dx ij ψ (x) = a i φ i i a i i * ψ dx mixing coefficient — how to get it? Always possible to expand in { } since we can write in terms of { }. So simplify the question we are asking to What are the b j Multiply by j * ˆ ˆ ?
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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notes10 - 5.73 Lecture #10 10 - 1 Matrix Mechanics should...

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