{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

M402-Chapter1.Fall09

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

This preview shows pages 1–5. Sign up to view the full content.

MATHEMATICAL PHYSICS: SEMESTER I University of Missouri-Rolla Barbara Hale University of Missouri-Rolla

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 24

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online