de Broglie Matter Waves

de Broglie Matter Waves:
Perhaps one of the key questions when Einstein offered his photon description of light
is, does an electron have wave-like properties? The response to this question arrived
from the Ph.D. thesis of Louis de Broglie in 1923. de Broglie argued that since light
can display wave and particle properties, then matter can also be a particle and a wave
too.

One way of thinking of a matter wave (or a photon) is to think of a wave packet.
Normal waves look with this:

having no beginning and no end. A composition of several waves of different
wavelength can produce a wave packet that looks like this:

So a photon, or a free moving electron, can be thought of as a wave packet, having both
wave-like properties and also the single position and size we associate with a particle.
There are some slight problems, such as the wave packet doesn't really stop at a finite
distance from its peak, it also goes on for every and every. Does this mean an electron
exists at all places in its trajectory?
de Broglie also produced a simple formula that the wavelength of a matter particle is
related to the momentum of the particle. So energy is also connected to the wave
property of matter.
While de Broglie waves were difficult to accept after centuries of thinking of particles
are solid things with definite size and positions, electron waves were confirmed in the
laboratory by running electron beams through slits and demonstrating that interference
patterns formed.
How does the de Broglie idea fit into the macroscopic world? The length of the wave
diminishes in proportion to the momentum of the object. So the greater the mass of the
object involved, the shorter the waves. The wavelength of a person, for example, is
only one millionth of a centimeter, much to short to be measured. This is why people
don't `tunnel' through chairs when they sit down.