Advanced Solid State Physics

Outline

Electrons

Magnetic effects and
Fermi surfaces

Magnetism

Linear response

Transport

Crystal Physics

Electron-electron
interactions

Quasiparticles

Structural phase
transitions

Landau theory
of second order
phase transitions

Superconductivity

Quantization

Photons

Exam questions

Appendices

Lectures

Books

Course notes

TUG students

      

Numerical Calculations of Fourier Transforms

Typically a Discrete Fourier Transform (DFT) is used to numerically calculate the Fourier transform of a function. A DFT algorithm takes a discrete sequence of $N$ equally spaced points $(g_0,g_1,\cdots,g_{N-1})$ and returns the Fourier components of a continuous periodic that passes through all of those points. There are infinitely many periodic functions that will pass a discrete sequence of points. Here we restrict ourselves to the periodic function that can be constructed using only those complex exponentials in the first Brillouin zone.

The Fourier transform of a function $g(t)$ is $G(f)$. The values of $g(t)$ at equally spaced points can be input into the textbox in the lower left as three columns. If the data you have is not equally spaced, use linear interpolation, or a cubic spline to generate equally spaced points. Alternatively, the functional form of $g(t)$ can be given and equally spaced points will be calculated. If is also possible to specify $G(f)$ by providing equally spaced points or by giving its functional form in the first Brillouin zone.

Real space

Reciprocal space

$g(t)$


$G(f)$

$t$ [s]

$f$ [Hz]

$t$ [s] $\text{Re}[g(t)]$ $\text{Im}[g(t)]$ 

$f$ [Hz] $\text{Re}[G(f)]$ $\text{Im}[G(f)]$ 

$\text{Re}[g(t)]=$ 
$\text{Im}[g(t)]=$ 
from $t_1=$  to $t_2=$  for $N=$  points

  

$\text{Re}[G(f)]=$ 
$\text{Im}[G(f)]=$ 
for $N=$  points using a spacing $\Delta f=$  Hz