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