wave packets

Wave packets. Use sliders to control group and frequency components and check boxes to toggle visibility. For the group component, the actual value is shaded but the outline is also drawn over opposite side. When the group component value is negative, multiplication by it reverses the sign of the frequency component.

The wave packet is a proposed solution of the wave-particle puzzle that was faced after discovering of both Bohr atom quantization rules and electron diffraction. A speed of the electron, computed from its mass and momentum, is two times higher than the speed of the "electron wave", computed from the estimated values of electric and magnetic fields (mathematics is given at [1]). How can electron stay together with its wave?

The problem

As the electron is also a particle with at least approximate location and momentum, the wave function of the moving electron must be nonzero over some volume, but zero in the places the electron has not possibly reached yet, and zero in the places it has definitely left. The problem is that the plane sine wave extends to infinity in both spatial directions, so does not represent a particle whose wave function is non zero in a limited region of space.

The solution

To represent a localized particle, we must superpose waves having different wavelengths. The principle is best illustrated by superposing two waves with slightly different wavelengths. Describing the wave in space x and time t as

where k and w define spatial and temporal frequencies of the wave (A is the amplitude). w is proportional to the energy of the electron[2].

and using the trigonometric addition formula

This formula describes beats between waves close in frequency. The first term, sin(kx-w t), oscillates at the average of the two frequencies. It is modulated by the slowly varying second term (called "group velocity" in the applet), which oscillates once over a spatial extent. This is the distance over which waves initially in phase at the origin become completely out of phase. Of course, going a further distance of order at some point the waves become synchronized again.

That is, beating two close frequencies together breaks up the continuous wave into a series of packets, the beats, that could represent a chain of travelling electrons.

To describe a single electron moving through space, we need a single packet. This can be achieved by superposing waves having a continuous distribution of wavelengths (rather than oscillating in one single frequency). These distributions can be found from the Fourier transform of the required packet density equation.

References

  1. 1 Michael Fowler lecture, University of Virginia
  2. 2 John.Rodenburg.org. Electron wave interference.