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 integration and differentiation

This page contains some programs for integrating and differentiating numerical data. A function $f(x)$ is described by the two columns of data in the textbox at the top-left. You can paste your own data into this textbox. When the "calculate" button is pressed, the data is plotted on the right. Below the data and plot of $f(x)$, the derivative $\frac{df}{dx}$ and the second derivative $\frac{d^2f}{dx^2}$ are tabulated and plotted. Below the derivatives, the integral of $f(x)$ is shown as well as the integral of the integral. The integration routines assume that the measurements are equally spaced with an interval $\Delta x$.

 $x$   $f(x)$

  

$f(x)$

$x$

The dervative
The dervative of $f(x)$ is calculated as,

$\Large \frac{df}{dx}\approx \frac{f(x+\Delta x)-f(x)}{\Delta x}.$

 $x$   $\large \frac{df}{dx}$

  

$\large \frac{df}{dx}$

$x$

The second dervative
The dervative of $f(x)$ is calculated as,

$\Large \frac{d^2f}{dx^2}\approx \frac{\frac{df}{dx}(x+\Delta x)-\frac{df}{dx}(x)}{\Delta x}.$

 $x$   $\large \frac{d^2f}{dx^2}$

  

$\large \frac{d^2f}{dx^2}$

$x$

The integral of $f(x)$
There is always an integration constant when an integral is performed. The values in the table below are,

$\large \int fdx +C_1,$

where $C_1$ is the integration constant.

$C_1=$

The integral of $f(x)$ was calculated numerically using a method called Simpson's rule.

$\large \int \limits_a^b f(x) dx \approx \frac{b-a}{6}\left(f(a) + 4f\left(\frac{a+b}{2}\right)+f(b)\right).$

 $x$   $\large \int fdx$

  

$\large \int fdx$

$x$

The integral of the integral of $f(x)$
Simpson's rule was used a second time to calculate the integral of the integral of $f(x)$.

$\large \int \limits_{x_0}^{x_2} \left(\int \limits_{x_0}^{x_1} f(x)dx +C_1\right) dx_1+ C_2.$

$C_2=$

 $x$   $\large \int\left(\int fdx\right)dx'$

  

$\large \int\left(\int fdx\right)dx'$

$x$