multiply both sides of eq. by a and projection overlap

Transcription

1 Fourier Series n x n x f xa ancos bncos n n periodic with period x consider n, sin x x x March. 3, 7 Any function with period can be represented with a Fourier series Examples (sawtooth) (square wave) 4 mx nx cos cos dx mn,, m, n mx nx sin sin dx mn, nx nx sin dx cos dx nx mx cos sin dx for all n, m x showing that the leading term, which is the constant a, is orthogonal to the other terms. f, x, x

2 How does one determine the coefficients? n x n x f xa ancos bnsin f x dxa dxa n x f xcos dx an a n x f xsin dx bn a a f xdx n x an f xcos dx a n x bn f xsin dx a multiply both sides of eq. by and cos n x sin n x multiply both sides of eq. by a and multiply both sides of eq. by a and The coefficients in the expansion are given by the projection of f(x) onto the basis functions projection overlap

3 Even if the function is not periodic, we can represent it on (, ) with a Fourier series Even function: all b n = Odd function: all a n = If we only want to represent a function on (, ), we can use either the sine or cosine series f(x) = x for < x < (Sawtooth function) n x bn xsin dx n x xsin dx n f x n n x sin n n n x x x 4x sin sin sin sin 3 4 Note, that in the text an integration symbol (Ex.) is missing) Note this expression, at x = and

4 An alternative representation Euler relations cos sin e i e i So we can also use i e e i i / f x cne n in x for a real function, a n, b n real, c n complex * imx inx exp exp dx im x cm exp f xdx nm

5 Many other classes of orthogonal functions The orthogonal functions form a basis set in which other functions can be expanded f cn n n a set of orthogonal functions that obey the same boundary conditions as f b again cn f d, where a, b are the relevant integration a limits, and d represents the variable(s) integrated over scalar product of two functions b a * f xg x dx assuming D readily generalized to D or 3D If the functions are also normalized b a dx * n m nm

6 V V a V V = potential energy QM particle in box eigenfunctions n x n sin a a f x x ax Expand for <x <a in terms of the eigen functions a x a x a nx c x axsin dx x sin dx axsin dx a a a a a a 5/ 4 a 5/.844a 3 8a x f x ax sin leading term of 3 a a Fourier series

7 Fourier transform As, the sum in the Fourier series goes over to an integral (all wavelengths become possible) n et k, k continuous as ikx f x Fke dk this replaces c n in the sum ikx Fk f xe dx we can work in x or k space

8 The Fourier transform converges if f x dx In that case, f(x) is said to be "square integrable" This requires f x as x Consider: f x e x / Gaussian function with = the mean of x, and is the standard deviation Evaluate the Fourier transform of f x e ax ax ikx Fk e e dx ax i ax e coskxdx e sin kxdx k /4a e even odd a so integral is zero

9 The FT of a Gaussian is a Gaussian Now consider a e x k e / : e x / very spread out; k / e very narrow opposite behavior as very large Note: You will see in P. Chem. that k p This illustrates the "uncertainty principle" between x and p. We can also use Fourier transforms to go between the frequency and time domains

10 FTs are also very using in going from frequency to time it f () t Fe d circular frequency = ν it F f t e dt FTIR and NMR measure signals as a function of time and use FT to get spectrum in the frequency domain Actually, one measures the signal at discrete values of t and uses an algorithm called "fast Fourier Transform (FFT)" at Example: generate the sine transform of e sin bt typical of a signal from an NMR measurement at F e sin btsin tdt ab a b a b at at if f t e sin bt e sin ct f(t) for a =, b=5 Sin FT get two peaks in the spectrum

11 aplace transform st F s f te dt F s f t f t F s in general, s is real denotes the aplace transform F s s s s a s s a a s a t cos sin f t at e s a at at

12 Shifting theorem e at f t Fs a Derivative theorem Integral theorem df f t s f t f dt ' f s f sf f '' ', etc. t f udu f t s

13 Integral Theorem Example: t f u du f t s s s s a t au at e du e cos( ku) s s k k k k k t t Example: sin( ku) du cos( kt) k s s k Using information in the table, we see that the FT of cos kt cos( kt) s k s s k s s s k k s s s k s s k s s k s

14 Note, some texts/web pages use non symmetrical definition of FTs. E.g., it F e f t dt it f t e f d Some interesting cases sin at f t F i a a sin(at) goes on forever in time gives a delta function in the frequency domain f t F t delta function in time all frequencies equally probable

15 x = F i Heaviside step function Note that the derivative of the Heaviside function is the delta function This is a sum of two Heaviside functions