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Lecture 2

Lecture 2. Fourier transforms and conjugate variables. 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),

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Lecture 2

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  1. 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=2pshould be wt=p.

  3. ..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? Fourier Analysis: Gibbs phenomenon ? At discontinuties. At what points in the waveform is the Fourier series representation of the function poorest? Note that the Fourier series representation overshoots by a substantial amount. NO!

  4. N. D. Lang and W. Kohn, Phys. Rev. B1 4555 (1970) 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 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. http://www.almaden.ibm.com/vis/stm/images/stm6.jpg

  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. Are there functions for which we can’t find a Fourier series? Dirichlet conditions • 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) 0 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. 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? 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?

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

  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. Diffraction pattern for a rectangular aperture (i.e. a 2D top hat function) is a 2D sinc function. www.ece.utexas.edu/~becker/diffract.pdf 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....)

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

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

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

  17. Dirac delta-function, d(t): ? Calculate the Fourier transform of d(t). 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.

  18. ? 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. 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

  19. ? 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)? 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 fis the phase angle.

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

  21. Parseval’s theoremstates: 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.

  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. Compare this with From the Fourier transform of a Gaussian function we can derive a form of the uncertainty principle. 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 

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

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

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

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