M402-Chapter1.Fall09

M402-Chapter1.Fall09 - MATHEMATICAL PHYSICS: SEMESTER I...

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MATHEMATICAL PHYSICS: SEMESTER I University of Missouri-Rolla Barbara Hale University of Missouri-Rolla
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CHAPTER I - VECTOR SPACES Page I-n 1. Vector spaces, S, over field, F field, space definitions examples of vector spaces symbols defined 2. Normed vector space Unitary vector space inner product 3. Orthogonality of vectors linear dependence metric space completeness 4. basis vectors Schmidt orthogonality procedure 5. Linear transformations properties of linear operations identity 6. powers of operators: A n definition of bound of a linear transformation Hermitian conjugate Hermitian operator unitary operator orthogonal operator 7. properties of unitary operators properties of Hermitian operators Infinitesimal linear transformation 9. Eigenvectors and eigenvalues of Hermitian operators 10. Rotations in 3-dim Euclidian Space, ± 11. Euler angles ( φ , θ , ψ ) 12. transformation matrix R( φ , θ , ψ ) 14. Use of Euler Angles 15. Orthogonality of Rotation Matrix 18. Transformation of unit vectors 19. det ± =+1 20. Rotation about a general axis
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I-1 CHAPTER I: VECTOR SPACES First Some Notation : ε ± element of ² ± there exists ³ ± such that ´ ± for every iff ± if and only if x , x i , y , u , v ,..( bold face ) ==> vectors α , β , . .. ==> complex number a,b, . .. ==> real numbers * ± complex conjugate r ± three dimensional position vector with components (x,y,z) ................................................................................... Vector space, S, over a field, F. FIELD: F ± { α , β , γ ,...} where α , β , γ ,... are (in general) complex numbers, and: (1) α + β and α - β are defined and are ε F; (2) α +( β + γ ) = ( α + β ) + γ ; α ·( β · γ ) = ( α · β γ , α ·( β + γ )= αβ + αγ ; (3) α + β = β + α , αβ = βα ; (4) the element 0 exists where α +0= α , α ·0 =0, ´ (for every) αε F, there exists a β such that α + β =0; (5) an identity, E, exists such that E· α = α for every α (E =1); (6) at least one element of F µ 0; (7) for every αε F, ² β ε F ³ αβ = E ( β ± α -1 ). ................................................................................... SPACE: S ± { x , y , z , v ,...} where x , y , z , v ,... are mathematical objects ("vectors") over field, F and: (1) x + y ε S ; α x ε S ( ´ α ε F, x ε S , y ε S ) ; (2) x + y = y + x ; (3) x + ( y + z ) = ( x + y ) + z ; (4) α ( x + y ) = α x + α y ; (5) the "zero" or null vector, 0 , exists (and is ε S ) ³ x + 0 = x and α 0 = 0 ; [note: ´ x ε S ² y ε S ³ x + y = 0 and y ± additive inverse of x ] ................................................................................... Some examples of vector spaces: (1) 3-dim. Euclidian space [ This is the r coordinate space and F = real numbers] ; (2) n-dimensional vector space over field of complex numbers, x = ( α 1 , α 2 ,... α n ); (3) set of all real, continuous functions, f(x), on [0,1] ..... (note f(x) = y , a vector element of S ); (4) set of all complex functions, ψ (x), - < x < , ³ ± ψ * ψ dx is finite ; (sometimes called "L²", a (Hilbert) space of all square integrable functions); (5) set of solutions to · ²f( r ) = 0 (or · ²f( r )= k², real k); (6) set of functions ψ ( r ), | r | ¸ , where the integral over all space, ± | ψ ( r )|²d 3 x is finite.
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I-2 n-dimensional vector space over the field of real numbers: x = x 1 e^ 1 + x 2 2 +x 3 3 + ..... + x n n = (x 1 ,x 2 ,.... ,x n ) 1 = (1,0,0. ....0) n = (0,0,0. ....1) the i are called basis vectors.
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This note was uploaded on 12/20/2010 for the course PHYS 402 taught by Professor J. during the Fall '09 term at Missouri S&T.

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M402-Chapter1.Fall09 - MATHEMATICAL PHYSICS: SEMESTER I...

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