1. Fourier transforms and
Lecture 2
Fourier transforms and
conjugate variables

2. Recap…. Fourier analysis…
… involves decomposing a waveform or function into its
component sinusoids.
… spans practically every area in physics.
… converts from the time to frequency domain (or vice versa),
… or from real space to reciprocal space.
NB There is a typo. on p. 9 of the lecture notes handout. In the last paragraph on that page, wt=2p should be wt=p.

3. Fourier Analysis: Gibbs phenomenon?
At what points in the waveform is the Fourier series representation of the function poorest?
At discontinuties.
Note that the Fourier series representation overshoots by a substantial amount.
..but we know that we get closer to the correct function if we include more harmonics. Can’t the approximation be improved by adding in more terms?
NO!

4. Fourier Analysis: Gibbs phenomenon
The inclusion of more terms does nothing to remove the overshoot – it simply moves it closer to the point of discontinuity.
Therefore, we need to be careful when applying Fourier analysis to consider the behaviour of a function near a discontinuity
N. D. Lang and W. Kohn, Phys. Rev. B (1970)
Gibbs phenomenon, however, is not just of mathematical interest. The behaviour of electrons near a sharp step in potential (e.g. at a surface) is fundamentally governed by Gibbs phenomenon.

5. Outline of Lecture 2 From Fourier series to Fourier transforms
Pulses and top hats
Magnitude and phase
Delta functions, conjugate variables, and uncertainty
Resonance

6. Luckily, the vast majority of functions of interest
Dirichlet conditions
Are there functions for which we can’t find a Fourier series?
There are a number of conditions (the Dirichlet conditions) which a function must fulfil in order to be expanded as a Fourier series:
must be periodic
must be single-valued
must be continuous or have a finite number of finite discontinuities
integral over one period must be finite
must have a finite number of extrema
Luckily, the vast majority of functions of interest
in physics fulfil these conditions!

7. Fourier Analysis: Spectra?
Sketch the frequency spectrum (i.e. Fourier coefficients) for the pure sine wave shown below.

8. Fourier Analysis: Spectra
NB Note presence of 0 Hz (i.e. DC term) – the value of A0 is given by: Bn 4
Frequency (Hz) 1 2 3

9. Complex Fourier series
As discussed (and derived) in the Elements of Mathematical Physics module:
Practice/revision problems related to both the trigonometric and the complex forms of the Fourier series feature in this week’s Problems Class.

10. From Fourier series to Fourier transforms?
For what type of function is it appropriate to use a Fourier transform as opposed to a Fourier series analysis?
Consider an aperiodic function as a limiting case of a periodic signal where the period, T, → ∞.
If T → ∞, how does the spacing of the harmonics in the Fourier series change? ?

11. From Fourier series to Fourier transforms
f(t)
Continuous frequency spectrum
for aperiodic waveform t
f(t) t
Discrete frequency components
for periodic waveform
Fourier transform of isolated pulse (top-hat function) is a sinc function:
where 2t is the width of the top-hat function.

12. Beyond time and frequency
Fourier analysis is not restricted to time  frequency transformations.
Consider a periodic (1D) lattice – e.g. a ‘wire’ of atoms – with lattice constant a. a
If a is the lattice period, then the spatial frequency associated
with this lattice is 2p/a.

13. Beyond time and frequency
Just as we can transform from time to frequency, we can transform from space to spatial frequency (inverse space):
s → s-1, m → m-1
Remarkably, a diffraction pattern is the Fourier transform of the real space lattice. (Holds true for any scattering experiment – photons, X-rays, electrons....)
Diffraction pattern for a rectangular aperture (i.e. a 2D top hat function) is a 2D sinc function.

14. Crystallography and Fourier transforms
STM image of Si surface showing real space lattice
Electron diffraction pattern – reciprocal space lattice (Fourier transform of real space lattice)

15. Time-limited functions and bandwidth
HWHM ~ 25 Hz

16. Time-limited functions and bandwidth
HWHM ~ 60 Hz

17. The ultimate time-limited function: Dirac d-function?
So, in the limit of the pulse width → 0, what happens to the pulse’s Fourier transform?
Fourier transform becomes broader and broader as pulse width narrows.
In the limit of an infinitesimally narrow pulse, the Fourier transform is a straight line: an infinitely wide band of frequencies.
Dirac delta-function, d(t): ?
Calculate the Fourier transform of d(t).

18. Magnitude, phase, and power spectra
Fourier transform is generally a complex quantity.
- Plot real, imaginary parts
- Plot magnitude
- Plot phase
- Plot power spectrum:|F(w)| 2
Take two quantities, z = e-ikx and y = sin w ?
The complex conjugate of y is:
(a) -sin w, (b) sin2 w, (c) cos w, (d) none of these ?
Write down the magnitude (or modulus) of z.

19. Magnitude, phase, and power spectra
A complex number, z, can be written in the form z = reif
where r is the magnitude (or modulus) of z and f is the phase angle. ?
What is the phase angle of z = 4 cos x?
For a delta function, d(t), F(w) is:
What can you say about the phase spectrum for the delta function, d(t)? ?

20. Magnitude, phase, and power spectra?
Calculate the Fourier transform of d(t-t0). ?
How is the magnitude of the Fourier transform affected by the shift in the function? ?
How are the phases affected?

21. Why ‘power’ spectrum?
The power content of a periodic function f(t) (period T) is:
If f(t) is a voltage or current waveform, then the equation above represents the average power delivered to a 1 W resistor.
Parseval’s theorem states:

22. Why ‘power’ spectrum?
For aperiodic signals, Parseval’s theorem is written in terms of total energy of waveform:
Total power or energy in waveform depends on square of magnitudes of Fourier coefficients or on square of magnitude of F(w). (Phases not important).

23. Fourier Transforms and the Uncertainty Principle
Note the ‘reciprocal’ nature of the characteristics of the function and those of its Fourier transform.
Narrow in time  wide in frequency: Dt   Df 
Compare this with
From the Fourier transform of a Gaussian function we can derive a form of the uncertainty principle.

24. Fourier Transforms and Resonance?
Write down a mathematical expression for the response of the damped harmonic oscillator shown in the graph above.

25. Localisation and wavepackets?
There is something fundamentally wrong with using a plane wave description (eg ) to describe a quantum mechanical particle. What is it?
What do you think the Fourier transform of the function shown below will look like? ?

26. Wavepackets To localise the particle in space we need a spread of
momentum (or wavevector) values.