37.7 - Gotchas
More massive particles always have shorter de Broglie wavelengths than less massive particles. Not necessarily. A particle’s wavelength
depends on its momentum, which equals its mass multiplied by its velocity. A more massive particle may have a longer wavelength than a
lighter particle, if its velocity is small enough.
Only particles like protons and electrons have a de Broglie wavelength associated with them. “Matter waves” do not apply to larger things like
baseballs. No. The wavelengths of objects with relatively large momenta, such as moving baseballs, are so small that experiments do not
reveal their wave-like properties. But these objects are still subject to the laws of quantum physics. For large objects moving at ordinary
speeds, the predictions of quantum physics and those of Newtonian mechanics are identical for all intents and purposes, much as special
relativity essentially agrees with Newtonian mechanics at slow enough speeds.
37.8 - Summary
Further evidence that light has properties of particles í in addition to properties of
wavesí is provided by the Compton effect. When a photon and an electron collide,
they behave much like two colliding billiard balls. The photon’s frequency is
reduced, because it has transferred some energy to the electron. The Compton
effect provides evidence that though photons do not have mass, they do have
momentum.
One of the basic tenets of quantum physics is that matter, like light, also exhibits
wave-particle duality. The wave nature of matter explains why the energies of
electrons in an atom are quantized.
The equation Ȝ = h/p relates a particle’s wavelength to its momentum, and applies
to particles of matter or of light.
Matter wave interference can be observed using an experiment similar to the
double-slit experiment for light. However, in the case of matter waves, the
wavelengths of even the smallest particles (generally those with the least
momentum), such as electrons, are very small and so require very small slits to
observe. Such small, closely-spaced slits are difficult to manufacture, but certain
crystals naturally have atomic spacing similar to the required slit spacing. Matter
waves were first observed using such crystals.
Matter waves are probabilistic. The wave- and particle-like properties of matter are reconciled in the form of the particle’s wavefunction. The
value of the wavefunction is related to the probability that the particle will be found at a given location at a given time. Mathematically speaking,
a particle is described by a probability density function that tells the probability of observing the particle in any region.
One consequence of the wave-particle duality is that it is impossible to know both the position and momentum of a particle to infinite precision
at the same time. In fact the product of the uncertainties in position and momentum has a precise lower limit: half of ʄ. This is the Heisenberg
uncertainty principle.
Momentum of a photonWavelength of a matter particleHeisenberg uncertainty principle