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Unformatted text preview: Lecture 5 Mathematical Tools: Complex numbers Vectors of complex numbers Hermitian Operators Eigenvalues and Eigenvectors of Hermitian Operators Commutators Math Chapters A,B,C McQuarrie Simon September 18 2009 Complex Numbers Wave functions and operators can take on complex values , even though expectation values , i.e., the results of physical measurements, are always real numbers . Every complex number c can be represented as (51) where a and b are real numbers and i is the square root of 1 (the unit for the socalled imaginary part of complex numbers). We refer to the complex conjugate of c , which is written c *, as (52) c = a + bi c * = a " bi Square Modulus of Complex Numbers The square modulus of the complex number c is (53) Note that one way to view complex numbers is as vectors in a 2dimensional cartesian coordinate system . The x axis represents the real component of c (written Re[ c ]) and the y axis the imaginary component of c (written Im[ c ]). In that case, it becomes clear that the square modulus of c is simply the square of the length of the corresponding vector in the complex plane . c * c = c 2 = a + bi ( ) a " bi ( ) = a 2 + b 2 y = Im[ c ] x = Re[ c ] ( a , b ) ! c = a + bi b a ( a 2 + b 2 ) 1/2 Vectors of complex numbers Imagine that we have a vector whose elements are complex numbers . Dirac notation in quantum mechanics provides a shorthand connection between vectors and matrix algebra. We refer to a column vector of complex numbers as a ket> , which is written as  > and defined as (54) " # c 1 , c 2 , c 3 , K ( ) # c 1 c 2 c 3 M $ % & & & & ( ) ) ) ) The term in parentheses is written like a vector specification in an n dimensional coordinate system ( n may be infinite) and the term in brackets is a matrix of n rows and 1 column (also called a column vector). The Transpose of a Vector We may define the analog of the square modulus for such a column vector. In general, the square of the length of a vector may be computed as the product of the transpose of the vector with itself. The transpose of a column vector is defined as the row vector having the same elements (in order) as the column vector. We might write (55) The transpose definition is true in reverse as well (i.e., the column vector is the transpose of the row vector; in general, the transpose of a matrix with an arbitrary number of rows and columns is defined to be the matrix formed by converting every i , j th element of the original matrix to a j , i th element of a new matrix). " # c 1 c 2 c 3 L [ ] The Transpose of a Vector/Matrix: Examples Matrix Multiplication The rules of matrix multiplication define that when an m x n matrix (where m defines the number of rows and n the number of columns) multiplies an n x l matrix ( n must be the same for both matrices!), then the result is an m x l matrix. Matrix Multiplication So, for a row vector (1 row, n columns) multiplying its transpose column vector (...
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This note was uploaded on 12/14/2010 for the course CHEM 3502 taught by Professor Staff during the Fall '08 term at Minnesota.
 Fall '08
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