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
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
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
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
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
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
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
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
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
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
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
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
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
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
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