M402-Chapter1.Fall13b.pdf - MATHEMATICAL PHYSICS SEMESTER I...

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MATHEMATICAL PHYSICSSEMESTER IMissouri Universityof Science & TechnologyBarbara HaleUniversity of Missouri-Rolla
CHAPTER I - VECTOR SPACESPage I-n1.Vector spaces, S, over field, Ffield, space definitionsexamples of vector spacessymbols defined2.Normed vector spaceUnitary vector spaceinner product3.Orthogonality of vectorslinear dependencemetric spacecompleteness4.basis vectorsSchmidt orthogonality procedure5.Linear transformationsproperties of linear operationsidentity6.powers of operators:Andefinition of bound of a linear transformationHermitian conjugateHermitian operatorunitary operatororthogonal operator7.properties of unitary operatorsproperties of Hermitian operatorsInfinitesimal linear transformation9.Eigenvectors and eigenvalues of Hermitian operators10.Rotations in 3-dim Euclidian Space,11. Euler angles (, , )12.transformation matrix R(, , )14.Use of Euler Angles15.Orthogonality of Rotation Matrix18.Transformation of unit vectors19. det=+120. Rotation about a general axis
I-1CHAPTER I:VECTOR SPACESFirst Some Notation:element ofthere existssuch thatfor everyiffif and only ifx,xi,y,u,v,..(bold face)==>vectors,, ...==>complex numbera,b, ...==>real numbers*complex conjugaterthree dimensional position vector with components (x,y,z)...................................................................................Vector space, S, over a field, F.FIELD:F{,,,...} where,,,... are (in general) complex numbers, and:(1) + and - are defined and areF;(2) +( + ) = ( + ) + ;·( · ) = ( · )· , ·( + )=+;(3) + = + ,=;(4) the element 0 exists where +0= , ·0 =0,(for every)F,there exists asuch that + =0;(5) an identity, E, exists such that E·=for every(E =1);(6) at least one element of F0;(7) for everyF,F= E(-1)....................................................................................SPACE:S{x,y,z,v,...}wherex,y,z,v,... are mathematical objects ("vectors") over field, F and:(1)x+yS;xS(F,xS,yS) ;(2)x+y=y+x;(3)x+ (y+z) = (x+y) +z;(4) (x+y) =x+y;(5) the "zero" or null vector,0, exists (and isS)x+0=xand0=0;[note:xSySx+y=0andyadditive inverse ofx]...................................................................................Some examples of vector spaces:(1) 3-dim. Euclidian space [ This is thercoordinate space and F = real numbers] ;(2) n-dimensional vector space over field of complex numbers,x= (1,2,...n);(3) set of all real, continuous functions, f(x), on [0,1].....(note f(x) =y, a vector element ofS);(4) set of all complex functions,(x), -< x < ,*dx is finite ;(sometimes called "L²", a (Hilbert) space of all square integrable functions);(5) set of solutions to²f(r) = 0 (or²f(r)= k², real k);(6) set of functions(r), |r| ,where the integral over all space,| (r)|²d3x is finite."L²", a (Hilbert) spaceof all square integrable functions)
I-2n-dimensional vector space over the field of real numbers:x= x1e^1+ x2e^2+x3e^3+.....+xne^n= (x1,x2,....,xn)e^1= (1,0,0.....0)e^n= (0,0,0.....1)thee^iare called basis vectors.................................................................................................................................NORMED VECTOR SPACEA normed vector space is a vector space in whichxSa quantity defined to be the norm ofx, ||x||exists.The norm must satisfy:(1) ||x|| 0(2) ||x|| = ||·||x||(3)Minkowski Inequality:||x+y||||x||+||y||.note also that ||x|| = 0 iffx=0;example:Inan n-dimensional Euclidian space,x= (a1,a2,...an) and||x|| = [|a1|²+ |a2|² + ... + |an|²]½................................................................................................................................

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Linear Algebra, Vector Space, Euler angles, Hilbert space

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