Everything we call real
is made of things
that cannot be regarded as real
~Niels Bohr

This is the second part of a multi-part series on quantum mechanics. In part one,  I described and motivated particle-wave duality for light. I demonstrated that light waves are also particles (photons). But does this duality go the other way? Are particles like electrons also waves? As I hinted last time, the answer is yes. These are called matter waves, and their story is very interesting.

The Mystery of the Emission Spectrum

Last time, I discussed the emission spectrum of hydrogen, and the mysterious equation that described it, the Rydberg Formula. As I explained, Max Planck suggested that atoms only emitted light in discrete packets of energy, and Albert Einstein discovered that these packets are particles, photons. A question remains, however, WHY do atoms only emit light in these discrete packets? Furthermore, as I discussed, atoms only emit certain colors of light (and this spectrum of colors is unique to each atom). What determines what colors are emitted by each atom? Niels Bohr offered a solution with his model of the atom.

The Bohr Model of the Atom

Bohr’s model of the atom describes the atom most people are familiar with. Just as the planets orbit around the sun, negatively charged electrons orbit around a positively charged nucleus. The electromagnetic attraction between the electrons and the nucleus takes the place of the gravitational attraction of the planets, and it is balanced by the centrifugal “force” pushing the electrons out away from the atom. However, there’s a secret aspect of the Bohr model that most people don’t know about, a secret that makes it very different from Newton’s model of the solar system: only certain orbits are allowed.

As Sir Isaac Newton predicted, objects like to travel in straight lines–you have to push or pull them to make them deviate. Thus, to make an object travel in a circle, you have to constantly pull it towards the center of the circle, making it turn. The faster an object moves (or the more massive it is), the harder it is to turn, and the stronger you have to pull it towards the center of the circle. This resistance to change is called momentum (or in the case of circular motion, angular momentum). Momentum is usually written as and it is defined as mass (denoted ) times velocity (denoted ):

   

Momentum is closely related to the kinetic energy of a particle (the energy that comes from motion, as opposed to potential to move), which is proportional to the mass times the velocity squared,

   

Bohr proposed that electrons in an atom could only orbit the atom at certain special speeds, which translates to special energies and special momenta. To maintain a perfectly circular orbit, the nucleus of the atom must pull at the electron with certain strength determined by the speed of the rotation of the electron around the nucleus. Because the strength of the electromagnetic force is dependent on the distance between affected objects, Bohr’s proposed restriction means that electrons can only orbit the nucleus at special allowed orbital distances.

The take-home message is that Bohr proposed that electrons can only orbit the nucleus of an atom at special allowed distances, and with special allowed energies, called energy levels, as shown below. The inner orbits have lower energy, and the outer orbits have higher energy. So in the figure below, Level 1, has the lowest energy, and Level 3 has the highest.

As shown below, when an atom absorbs light, the electron takes the energy from the photon and jumps to a different allowed energy level. However, since only certain energies are allowed, the atom can only absorb certain frequencies/colors of light. Similarly, when an atom emits light, an electron loses the energy in the emitted photon and drops down to a lower energy level. And, because only certain energies are allowed, the atom only emits certain frequencies of light.

This behavior completely matches the Rydberg formula described in part 1. This restriction to only certain energy levels is called Bohr’s quantum rule, and Bohr didn’t try to justify it at all. His only justification was that it worked remarkably well–explaining the Rydberg formula was a huge success! It took a brilliant young PhD student in mathematics named Louis de Broglie to justify Bohr’s quantum rule.

The de Broglie Relations

Louis de Broglie justified the Bohr model in fell swoop: his doctoral thesis. De Broglie postulated that, just as light waves are also particles, matter particles are also waves. He argued that an electron has a wavelength inversely proportional to its momentum,

   

where is the wavelength, is the momentum, and is Planck’s constant, a constant of nature. De Broglie argued that the reason an electron in an atom can only orbit the nucleus at special distances from the nucleus is because, for a given radius, only certain wavelengths fit in the orbit. As shown below, if an electron has too short or too long a wavelength, it doesn’t quite fit at a given radius. However, if it has the correct wavelength (or an integer multiple of some minimum wavelength), it fits just fine.