Particle/Wave Duality of Matter

The Electron is a Particle

Certainly, the electron is a particle. It travels through the wire because of an applied electric field. The electron has mass, momentum, and energy. In most experiments, the location of the electron is measurable.

The Electron is a Wave

The Bohr Model for Hydrogen

The emission line spectrum for hydrogen was observed by Balmer. He observed the pattern of bright lines shown below.

The pattern of emission lines can be described by the equation:

Equation 1

where R is the Ryberg constant, determined observationally is R = 1.097 X 10⁷ m^-1. When n_i = 3, the red line in the above photograph is produced. Similar, the next line occurs when n_i = 4 etc. Later, Lyman discovered a similar series in the ultraviolet (where the 2 is replaced by 1) and Paschen discovered another series in the infrared (where the 2 is replaced by 3). The general equation for all hydrogen spectra is given by:

Equation 2

But why should the hydrogen spectrum behave this way? If the classical picture of the electron orbiting the proton (similar to the moon orbiting the earth), then the electron accelerates around the proton according to

and the electron will remain in orbit around the proton in a circular orbit given by:

Equation 3

But an accelerating charge always produces light. The energy removed by the photon comes from the kinetic energy of the electron. The electron will spiral into the nucleus, emitting a flash of continuous light (white light). If the electron actually did this crash into the nucleus, the earth would reduce to a ball made of neutrons with a radius of 0.1 km! Balls of neutrons do exist in nature. When a star with a mass of more than about 3 times the mass of the sun dies, the electrons are driven into the nucleus creating a neutron star the size of the city of Santa Clara with a mass >2 times the mass of the sun! In normal matter, the electron stays away from the nucleus for some reason. The understanding of this peculiar property of the electron lets us to the modern view of the world of the small.

Nelis Bohr created a model that keeps the classical model, but adds one arbitrary assumption to fit Equation 2. Assuming Coulombs law and the electron orbits the proton in a circular orbit, Equation 3 yields,

The potential energy of the “orbit” is given by U = -ke²/r so that the total Energy is,

Light is emitted by the atom when the electron falls from one energy to another energy. The energy of the photon is hu = E_i-E_f or

Equation 4

The radical step that Bohr took to find the radius of the orbit was to assume that the angular momentum cannot be any arbitrary value, but has to be discrete or quantized value. The angular momentum of a particle moving around a fix point is L = mvr. Bohr said let the angular momentum be quantized as,

Equation 5

so that

And finally we arrive at equation 2,

Usually we define the radius of the inner orbit as the Bohr radius, a₀,

Equation 6

Figure 1

This result agrees exactly with Equation 2. The numerical calculation for R agrees with the observational value. This amazing result led people to believe that they understood the nature of the hydrogen atom – classical physics with one modification in the angular momentum. Not only did this model work for Hydrogen, in worked for any ion with only one electron (He⁺, C⁺⁵ …).

Figure 1 shows the relation between the quantum number n and the radius and energy. Notice that as n increases, the radius of the “orbit” increases as n² while the energy approaches 0. Classical physics of Newton is recovered when n becomes very large.

Figure 2 shows the relation between the radius and the energy for the electron in the hydrogen atom. The potential energy goes as 1/r. The energies allowed by the Bohr model are shown by marks of the graph. Again, notice that as the radius increases, the energy increases only by a very small amount. For large n, the atom is very large and the difference in the energy levels is very small.

Figure 2

The Bohr model is wrong. The modern view of the interaction between the electron and the proton did not mature until the Schrödinger Equation.

De Broglie Waves

Inspired by the wave/particle duality for light, de Broglie advanced a bold idea in his Ph.D. thesis. Why can’t the electron behave as a wave as well as a particle? The momentum of a photon of light is given by (see Relativity section, equation 8): p =h/l. So why can’t the electron behave in the same manner? De Broglie stated that the wavelength of a particle could be given by,

Equation 7

As the momentum of an object increases, its wavelength decreases. Only for the very low mass electron, is the wavelength large enough to actually diffract and interfere as a wave with realistic size silts.

Standing Waves

De Brogile’s idea of the wave nature of the electron was then applied to Bohr model for the Hydrogen atom. If the electron formed a standing wave pattern around the nucleus, then he could derive the Bohr equation for the angular momentum. A standing wave occurs when the end points of a wave are tuned to the wavelength and the speed of the wave.

Figure 3

Figure 4

Electron Diffraction

For 100 keV electrons, the wavelength is of the order 10^-12 m. If these electrons are passed through a lattice of atoms with a separation of this order, then the electrons should diffract and interfere. Observations of the diffraction pattern confirm de Broglie wavelength for the electron.

As with the photons, one electron travels through the lattice and hits the screen at one location. Where exactly the electron will hit the screen is not predictable, but the probability of where the electron will go is predictable. Where the image on the screen is bright in the above photograph, the probability is the highest. Where the image is dark, the probability is zero.

The typical wavelength of the electron is very small. On the larger classical physics scale, the electron acts like a particle. However, when you conduct an experiment on the electron where the location at any time is required to the accuracy of the electron’s wavelength, the wave nature of the electron becomes apparent. As the electron falls toward the proton, the electron reaches a distance comparable to the wavelength of the electron and the wave nature of the electron dominates. In the de Broglie picture of the electron in the hydrogen atom, the electron wraps itself around the proton in a standing wave pattern. As amazing as the successes of these models are in predicting the spectra of the hydrogen-like atoms, they still are not the final modern view of the electron in the atoms.