Lecture11 - Waves Physical Chemistry Lecture 11 Waves...

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1 Physical Chemistry Lecture 11 Waves, Matrices, Operators, and Eigenvalue Equations Waves λ 0 - wavelength in vacuum n( λ 0 ) - refractive index of the medium, which varies with wavelength (dispersion) Wave moving through space Dispersion relation: Not simple, because v is a function of ݇ ൌ ߱ ݒ Waves in complex notation Wave equation Matrices Matrix: an array of numbers or functions. Position in the array is important Labeling of matrix elements requires two indices Rank – the number of rows or columns 9 8 7 6 5 4 3 2 1 A 33 32 31 23 22 21 13 12 11 a a a a a a a a a A Special types of matrices Square matrix Row rank equal to column rank Symmetric matrix S Off-diagonal elements across principal diagonal are equal Diagonal matrix D All off-diagonal elements are zero Real matrix R All elements are real numbers or real functions Complex matrix C Some elements are complex numbers or functions Transpose of a matrix A matrix formed from another by exchanging elements across the principal diagonal xy i iy x iy x y x a a a a C B C a A B A a 3 6 44 2 3 25 1 ) , ( 9 8 7 6 5 4 3 2 1 0 0 0 0 0 0 33 22 11 33 22 11 C R D S
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2 Matrix mathematics Equality Additivity and subtraction Multiplication by a scalar ij ij b a B A ij ij ij b a c B A C ij ij kb a k B A Matrix multiplication Multiplication by a matrix is defined as Must be commensurate for multiplication Row rank of first must equal column rank of second Example Multiplication is not necessarily commutative Order matters k kj ik ij b a c B A C
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This note was uploaded on 02/02/2012 for the course CHEM 444 taught by Professor Dybowski,c during the Fall '08 term at University of Delaware.

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Lecture11 - Waves Physical Chemistry Lecture 11 Waves...

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