Lecture 2

Fourier transforms and

conjugate variables

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.

Recap….

NB There is a typo. on p. 9 of the lecture notes handout. In the last paragraph on that page, t=2 should be t=.

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!

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.

N. D. Lang and W. Kohn, Phys. Rev. B1 4555 (1970)

http://www.almaden.ibm.com/vis/stm/images/stm6.jpg

• From Fourier series to Fourier transforms• Pulses and top hats• Magnitude and phase• Delta functions, conjugate variables, and uncertainty• Resonance

Outline of Lecture 2

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

Are there functions for which we can’t find a Fourier series?

Luckily, the vast majority of functions of interestin physics fulfil these conditions!

Fourier Analysis: Spectra

Sketch the frequency spectrum (i.e. Fourier coefficients) for the pure sine wave shown below.

??

Fourier Analysis: Spectra

0 1 2 3Frequency (Hz)

Bn4

NB Note presence of 0 Hz (i.e. DC term) – the value of A0 is given by:

Tt

tdttf

TA

0

0

)(2

0

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.

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???

Discrete frequency components

for periodic waveform

From Fourier series to Fourier transforms

t

f(t)

Continuous frequency spectrum

for aperiodic waveform

t

f(t)

Fourier transform of isolated pulse (top-hat function) is a sinc function: where 2 is the

width of the top-hat function.

Beyond time and frequency-20 -10 0 10 20

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

B

Y A

xis

Titl

e

X axis title

Fourier analysis is not restricted to time frequency transformations.

Consider a periodic (1D) lattice – e.g. a ‘wire’ of atoms – with lattice constant a.

If a is the lattice period, then the spatial frequency associated with this lattice is 2/a.

a0.000 0.002 0.004 0.006 0.008 0.010

-4

-2

0

2

4

Vo

ltag

e (

V)

Time (s)

0.000 0.002 0.004 0.006 0.008 0.010

-4

-2

0

2

4

Vo

ltag

e (

V)

Time (s)

0.000 0.002 0.004 0.006 0.008 0.010

-4

-2

0

2

4

Vo

ltag

e (

V)

Time (s)

-20 -10 0 10 20-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

B

Y A

xis

Titl

e

X axis title

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

Beyond time and frequency

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.

www.ece.utexas.edu/~becker/diffract.pdf

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)

Time-limited functions and bandwidth

HWHM ~ 25 Hz

Time-limited functions and bandwidth

HWHM ~ 60 Hz

The ultimate time-limited function:

Dirac -function

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.

So, in the limit of the pulse width → 0, what happens to the pulse’s Fourier transform???

?? Calculate the Fourier transform of (t).

Dirac delta-function, (t):

Magnitude, phase, and power spectra

Fourier transform is generally a complex quantity.

- Plot real, imaginary parts - Plot magnitude - Plot phase - Plot power spectrum:|F()|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.

Magnitude, phase, and power spectra

?? What is the phase angle of z = 4 cos x?

A complex number, z, can be written in the form z = rei

where r is the magnitude (or modulus) of z and is the phase angle.

What can you say about the phase spectrum for the delta function, (t)?

??For a delta function, (t), F(w) is:

Magnitude, phase, and power spectra

?? Calculate the Fourier transform of (t-t0).

?? How is the magnitude of the Fourier transform affected by the shift in the function?

?? How are the phases affected?

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:

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).

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: t f

From the Fourier transform of a Gaussian function we can derive a form of the uncertainty principle.

Compare this with

Fourier Transforms and Resonance

?? Write down a mathematical expression for the response of the damped harmonic oscillator shown in the graph above.

Localisation and wavepackets

What do you think the Fourier transform of the function shown below will look like? ??

?? There is something fundamentally wrong with using a plane wave description (eg. ) to describe a quantum mechanical particle. What is it?

Wavepackets

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