Matter Waves and the de Broglie Hypothesis

The Bohr postulates were more or less without justification. The concept of a stationary state
seemed useful, but the reasons for the existence of such a thing were not readily apparent. The view
of particles as classical objects did not seem to imply stationary states. However, such is not really
the case for wave-like quantities such as light. It was known, for example, that light could make
'standing wave cavities' which looked something like what a stationary state should look like. In
one dimension, such a standing wave cavity is easy to comprehend. We have light of wavelength
lambda that must 'fit' inside a box of length L. This only happens when an integral number of
waves fits within the box. Otherwise, the wave would 'interfere' with itself. This leads to a natural
quantization - only certain wavelengths are allowed in the box. Lasers are predicated upon this
fact. All of this begs the issue of the wave-particle duality of light, but remember that light does in
fact exhibit both wave and particle properties.

The de Broglie Hypothesis

Recall the Planck/Einstein energy/momentum/frequency relationships for photons/light waves:

These relationships embody the essence of the wave-particle duality - they relate wave-like
quantities like frequency and wavelength to particle-like quantities like photon energy and
momentum. Given that light has particle-like qualities, it may not be so surprising that particles
might have wave-like properties. After all, we can think of a photon as a particle with zero rest
mass. In his doctoral thesis, de Broglie had the insight that if one could associate wave-like
properties with particles, then the quantization postulated by Bohr in his explanation of atomic
spectra might be justified. de Broglie hypothesized relationships for particles which are formally
very similar to those above for light:

A significant difference, of course, is that the photon relationship between energy and momentum,
E = cp, is more complicated for particles. Probably the most revolutionary aspect of the de Broglie
hypothesis is the first of these equations - that every particle has a wavelength which is inversely
related to its momentum.

This hypothesis, which is now generally accepted, obviously does not match our everyday
experience - massive particles do not oscillate like a wave. Let us see why.

Find the de Broglie wavelength of an object with a mass of 10^-6 g and a speed of 10^-6 m/s. Note
that this is a very small particle that is moving very slowly and thus has very small momentum. We
would expect that the de Broglie wavelength might be substantial. In fact, given that h =
6.6x10^-34 J s, we find that the de Broglie wavelength is 6.6x10^-19 m! This is four orders of
magnitude smaller than diameter of a typical nucleus (not an atom, which is 6 orders of magnitude
larger yet). The value of h is just so small that anything larger than an atom will always have an
unimaginably small de Broglie wavelength. It would be difficult indeed to detect such a small
wavelength.

Quite the opposite is true for a low energy electron. Consider, for example, an electron with an
energy of 13.6 eV. We have found that this is the binding energy of the n = 1 electron in hydrogen,
and it is thus an energy which is typical of electrons in atoms. This energy is small compared to the
rest mass of the electron, so we can calculate the momentum classically:

Substituting K = 13.6 eV, we find a de Broglie wavelength of 0.33 nm = 3.3 Angstroms. This is
small, but it is comparable to atomic dimensions so that it can in fact be detected and measured,
as we will discuss and demonstrate shortly.

De Broglie Waves in Atoms

Let us presume that the de Broglie is correct, and that the electron orbiting the nucleus in
hydrogenic atoms follows this relationship. In order to have a 'stationary state', we need to satisfy
the same sort of quantization condition as light in the 1D box discussed above. The difference is
that our box is not linear, it is in a circle. Specifically, we require that we have an integral number
of de Broglie wavelengths in one orbit:

The left side of this equation is, for a circular orbit, simply the angular momentum. We thus
recover the Bohr quantization hypothesis from the de Broglie relationship.

You should not take this calculation too seriously. Despite its ad hoc nature, however, it must have
some bearing on reality. The wave nature of the electron must be related to the quantization
apparent in atomic spectra. In later years, you will learn much more about this relationship - this
is the basis of quantum mechanics.