2 Derivation of the Uncertainty Principle

Consider a sinusoidal wave along the x ‐axis having amplitude a and wavelength λ . Its corresponding wave number k is defined as images . We can express the wave as


If an electron is described by this wave it will have infinite spatial extent. A spatially localised wave packet may be obtained mathematically by adding a series of component sinusoidal waves together, each sinusoidal wave having a unique wavelength. This forms a Fourier series.

Consider the following sum of waves


This may be written as an integral in the case of a continuum of wavelengths:

A2.1 images

Assume, for simplicity, that only a range of wavelengths of the component waves exists such that


and outside of this range a(k) = 0.

Now we obtain


The amplitude of this wave is now in the form of a sinc function. We will approximate the width of the sinc function ...

Get Principles of Solar Cells, LEDs and Related Devices, 2nd Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.