Lecture 5: Vector Spaces I - Definitions

Transcription

1 Lecture 5: Vector Spaces I - Definitions 1 Key points Many mathematical objects used in physics are elements of a Hilbert space Definition of vector spaces Definition of inner products Basis sets and expansion in a basis set Maple commands vector<x,y,z> DotProduct(a,b) CrossProduct(a,b) LinearAlgebra package Vector Norm(V,2) Physics[Vectors] package 2 Vector space Dedfinition: All elements of a vector space, Addition 1 Closure: If and are elements of V, then is also an element of V 2 Commutativity: 3 Associativity: 4 Zero vector: There exists 0 such that 5 Additive inverse: There exists such that ( Scalar multiplication, 6 Closure: if is an element of, is also an element of 7 Vector distributivity: 8 Scalar distributivity: 9 Associativity: 10 and Examples can be many different mathematical objects

2 Euclidean vectors ( used everywhere in physics) force: Matrices (used in quantum mechanics, classical mechanics) Scalar functions (used in quantum mechanics and E&M),, Rotation is not a vector directionaxis of rotation, magnitudeangle of rotation Linear Dependence can be satisfied if and only if are independent Otherwise they are dependent for all j, then the vectors Suppose that If then Then and are NOT linearly independent but they are in the "same line" Examples and are linearly dependent since 3 Basis Any element in a vector space can be expanded in a superposition of basis vectors, N basis set The vector space is said to be spanned by the basis set Examples (3 dimension)

3 (2 dimension) Third order polynomicals: (4 dimension) Fourier series: e where Fourier transform: 4 Hilbert Space Inner product space Dual space (complex number) iff ( is a norm of ) and Examples Euclidan vectors: Its dual is itself Matrix: Dual of a column vector: Lattice vectors: ( ) Its dual is a reciprocal space where,, Complex functions Its dual is complex conjugate Innerproduct is defined as The norm of the function is defined as

4 Orthonormal basis set Normality: Orthogonality: Using the orthonormality, one can easily compute the expansion coefficients 5 Examples Position vector In bracket notation: Norm: Orthogonality: Coefficients are determined by innderproduct:,, Exercise Find the norm of Answer Fourer series In bracket notation: Orthonormalty is defined by innerproducts Expansion coefficients are also determined by innerproducts, and

5 Legendre expansion In bracket notation: Coefficients are determined by innerproduct: 6 Exercises 1 Show that for the above lattice vectors, and Answere Define base vectors in real space (1) (2) (3) Define base vectors in the reciprocal space (4) (5) (6)

6 (7) (8) (9) Check if Find where 0, 0 0, 0 0, 0 Now we evaluate using the above properties (10) (11)

7 simplify (12)

8 You can show this easily by hand calculation Maple (Physics [Vectors]) (13) Dotproduct: 5 Norm: 26 or 26 (14) 2 Find the norm of, where and are positive constants Answer Hence, the norm of is 3 Find the norm of

9 Answer (15) To compute innerproduct, we need to find the dual which is transpose of the original vector 26 Maple (Linear Algebra) Column vectors Transpose 0 Dotproduct (Innerproduct) 5 5 (Note that D and P must be in upper case) Norm

10 Show the norm of is where and is an orthonormal basis set Answer 5 Show that the basis and is orthonormal Then, expand using the basis Answer Define the base vectors Check the orthonormality

11 Define the traget vector Hence, 6 For, show that basis functions,, are orthonormal Then, expand Answer using the basis set (16) (17) 0

12 Hence, and form an orthonormal basis set Hence, 7 Construct an orthonormal basis set based on Legendre polynomials Answer The Legendre functions is orthogonal but not normalized Noting that, we can define normalized functions which satisfy orthonormal relation 8 Consider two rotations in a three-dimentional Euclidean space, one is rotation about x axis

13 and the other rotation about y axis We denote the coressponding operators and, respectively Find the commutation relation Answer Hence, and do not commute indicating that the final vector depends on the order of the two rotations Homework: Due 9/18 11am 51 A vecoter in a two dimensional Hilbert space is expressed as where and are orthonormal base vectors When expressed in matrix forms, the base vectors are given as and 1 Confirm that the base vectors are orthonormal 2 Find the matrix expression of

14 52 A vector in three dimensional Hilbert space is given in a matrix form as The following basis vectors span the space,, 1 Check if the basis vectors are orthonormal 2 Expand in this basis set 53 A function is defined on We want to expand it in the following basis functions,,, 1 Check if the basis functions are orthonormal 2 Expand in the the basis functions 3 What is the dimension of the Hilbert space