Chapter3_16 - ) ( ) ~ ( | ~ * * 2 * 1 * n T v v v v v v L =...

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PHY4604 R. D. Field Department of Physics Chapter3_16.doc University of Florida Matrices as Operators (1) An important class of operators are matrices. = nn n n n n T T T T T T T T T T L M M M M L L 2 1 2 22 21 1 12 11 ~ where T ij are in general complex numbers. Column and Row Matrices: Matrices operate on column matrices (or column “vectors”). >= = n v v v v v M 2 1 | ~ The “Transpose” Matrix: The “transpose” of a column matrix is a “row” matrix, ) ( ~ 2 1 n T v v v v L = and = nn n n n n T T T T T T T T T T T L M M M M L L 2 1 2 22 12 1 21 11 ~ ( i.e. T ij T ji ) The “Hermitian Conjugate” or “Adjoint” Matrix: The hermitian conjugate is the transpose-complex-conjugate matrix. = = * * 2 * 1 * 2 * 22 * 12 * 1 * 21 * 11 * ) ~ ( ~ nn n n n n T T T T T T T T T T T T L M M M M L L and
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Unformatted text preview: ) ( ) ~ ( | ~ * * 2 * 1 * n T v v v v v v L = = =< The Inner Product: The norm of a state is <v|v> = |v 1 | 2 +|v 2 | 2 +...+|v n | 2 . The Unit Matrix: The unit matrix takes every vector into itself. = 1 1 1 L M M M M L L I ( i.e. I ij = ij ) Orthogonal and Unitary: A matrix O is orthogonal if O-1 = O T and a matrix U is unitary if U-1 = U : OO T = O T O = 1 UU = U U = 1. Not all operators can be expressed in terms of functions and differentials! Ket vector! Bra vector! Orthogonal Unitary...
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