M402-Chapter1.Fall15.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 (φ,θ,ψ)12.transformation matrix R(φ,θ,ψ)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:εelement ofthere existssuch thatfor everyiffif and only ifx,xi,y,u,v,..(bold face)==>vectorsα,β, ...==>complex numbera,b, ...==>real numbers*complex conjugaterthree dimensional position vector with components (x,y,z)...................................................................................Vector space, S, over a field, F.FIELD:F {α,β,γ,...} whereα,β,γ,... are (in general) complex numbers, and:(1)α+βandα-βare defined and areεF;(2)α+(β+γ) = (α+β) +γ;α·(β·γ) = (α·βγ,α·(β+γ)=αβ+αγ;(3)α+β=β+α,αβ=βα;(4) the element 0 exists whereα+0=α,α·0 =0,(for every)αεF,there exists aβsuch thatα+β=0;(5) an identity, E, exists such that E·α=αfor everyα(E =1);(6) at least one element of F0;(7) for everyαεF,βεFαβ= E(βα-1)....................................................................................SPACE:S{x,y,z,v,...}wherex,y,z,v,... are mathematical objects ("vectors") over field, F and:(1)x+yεS;αxεS(αεF,xεS,yεS) ;(2)x+y=y+x;(3)x+ (y+z) = (x+y) +z;(4)α(x+y) =αx+αy;(5) the "zero" or null vector,0, exists (and isεS)x+0=xandα0=0;[note:xεSyεSx+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= (α1,α2,...αn);(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,ψ(x), -< x < ,ψ*ψdx 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 functionsψ(r), |r| ,where the integral over all space,|ψ(r)|²d3x is finite.
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 whichxεSa quantity defined to be the norm ofx, ||x||exists.The norm must satisfy:(1) ||x|| 0(2) ||αx|| = |α|·||x||(3)Minkowski Inequality:||x+y||||x||+||y||.

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Linear Algebra, Vector Space, Euler angles, Hilbert space

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