R. I. Badran Complete set of orthogonal basis Mathematical Physics Complete set of orthogonal functions Discrete set of vectors: ˆ ˆ ˆ ˆ ˆ ˆ The two vectors A AX i A y j Az k and B BX i B y j Bz k are 3 orthogonal if A B 0 or Ai Bi 0 . i1 Continuous set of functions on an interval (a, b): a) The two continuous functions A(x) and B (x) are orthogonal b on the interval (a, b) if A(x)B(x)dx 0. a b) The two complex functions A (x) and B (x) are orthogonal on b the interval (a, b) if A (x)B(x)dx 0 , where A*(x) is the a complex conjugate of A (x). c) For a whole set of functions An (x) (where n= 1, 2, 3,) and on the interval (a, b) b 0 if m n An (x)Am (x)dx a , const.t 0 if m n An (x) is called a set of orthogonal functions. Examples: 0 m n sin nx sin mxdx i) , m n 0 where sin nx is a set of orthogonal functions on the interval (-, ). Similarly 0 m n cos nx cos mxdx if if m n 0 R. I. Badran Complete set of orthogonal basis Mathematical Physics ii) sin nx cos mxdx 0 for any n and m 0 (einx ) eimxdx iii) 2 1 vi) P (x)Pm (x)dx 0 unless. m 1 [Try to prove this; also solve problems (2, 5) of section 6]. 1 v) P (x) (any polynomial of degree < ) dx = 0. 1 [Solve problems (6 of section 6) & (4, 5 and 6 of section 7)]. ˆ ˆ ˆ Conclusion: The vector A AX i A y j Az k in 3-D is expanded in terms of unit vectors iˆ, ˆj and . kˆ Then and form complete set of orthogonal basis vectors. Thus any vector (or function) can be expanded in terms of a set of orthogonal basis vectors (or functions). Normalization of functions: The scalar product A A A2 gives the square of the length of a vector), where the length of a vector is also called the norm of a vector. What is the norm of a unit vector? How can we find the norm of a function A (x) on the interval (a, b)? b b 2 Answer: A (x)A(x)dx A(x) dx N 2 N is called the norm of the a a function over the interval (a, b). Can we get a normalized function? A(x) Answer: Yes, we just take if asm a normalizedn function where N if m n 0 N-1 is called the normalization factor. R. I. Badran Complete set of orthogonal basis Mathematical Physics Exercise: a) What is the norm of the function sin nx on (0, )? 2 Answer: Firstly, we find sin nx dx and then the norm is . 0 2 2 b) Find the normalized function. 2 Answer: The normalized function is sin nx . [How can you be so sure of that?! Orthonormal set of vectors: The unit vectors and are a set of orthonormal vectors because they are orthogonal to each other and each has a norm ˆ ˆ ˆ equal 1. i, j k Note: Any set of normalized and orthogonal functions is called 2 orthonormal (e.g. sin nx is an orthonormal set of basis functions on (0 ,). Any function (or a vector in vector space) like f (x) can be expanded on (0, ) in a Fourier sine series as 2 f (x) Bn sin nx n Here f(x) is considered a vector (or a function) with components Bn while are the basis vectors. [Note: In Q. M. a physical system is expressed as either a state function or a state vector]. Normalization of the Legendre polynomials: What is the norm of P (x) on the interval (-1, 1)? R. I. Badran Complete set of orthogonal basis Mathematical Physics 1 2 [P (x)]2 dx Answer: The norm of is. 1 2 1 Proof: Use the identity xP(x) P1(x) P (x) and multiply both sides by and integrate to get 1 1 1 2 [P (x)] dx xP (x)P(x)dx P (x)P1(x)dx 1 1 1 The last integral is zero (See problem 4 of section 7). Use the method of integration by part to find the 1st integral on R. H. S of the equation as: 1 1 1 2 xP (x)P(x)dx 1 [P (x)] dx 1 2 1 [Hint: to reach this answer you may need the identity ( P (1) (1) ](See problem 2, section 2). Substitute this result back into the previous equation to get: 1 2 [P (x)]2 dx Q. E. D. 1 2 1 Orthonormal set of Legendre functions: 2 1 a. The functions P (x) form an orthonormal set of functions 2 n (-1, 1). 1 2 1 2m 1 0 if m b. P (x) Pm (x)dx m , where is 1 m 2 2 1 if m called Kronecker delta. Reminder for suggested problems: Solve problems (6.2, 6.5, 6.6), 7.4, (8.1, 8.2, 8.5) P (x).