The Charming Doubleness: Particle-Wave Duality

But the beauty here lay in the duality,
in the charming doubleness…
~ Thomas Mann (Felix Krull)

Thomas Mann
Thomas Mann saw the beauty in duality, though it was probably not particle-wave duality (source).

I apologize to those of you who have requested a topic. The current requests are all pretty in-depth and I want some time to think about how to explain them properly. So, in a bid to buy time, I’m going to do a multi-part series on quantum mechanics. In this part, I’ll describe some of the experimental results motivating the fundamental principle of quantum mechanics: particle wave duality. As amazing as it may seem, quantum mechanics tells us that every particle is also a wave.

Physics at the Turn of the Century

Around 1900, physicists believed their job to be almost done. James Clerk Maxwell had unified electromagnetism. Boltzmann statistics had revolutionized thermodynamics, and Newton’s equations were ever-faithful. There were just a few mysteries left to solve: the problem of atomic spectra and the mysterious photoelectric effect. Little did the physicists of the past know that these unsolved mysteries were only the tip of the quantum iceberg.

Atomic Spectra

By running electric current through a gas, physicists could make the gas emit light. The physicists would then send that light through a prism and look at the different colors produced.

Emission Spectroscopy: Physicists run current through some gas (red), which causes it to emit light (white bar). The physicists then pass that light through a prism (blue triangle) and look at what colors of light emerge. This tells the physicists something about the atomic structure of the gas.

Weirdly, most atoms only produced a few specific colors of light. For example, the emission spectrum of hydrogen looks like this:

The Emission Spectrum of Hydrogen
The emission spectrum of hydrogen (source).

Johannes Rydberg even discovered a formula describing the wavelength of each visible line in hydrogen. The famous Rydberg formula is as follows:

    \[\frac{1}{\lambda} = R\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right),\]

where R is the Rydberg constant–a constant Rydberg guessed at–and n_1 and n_2 are arbitrary numbers. All n_1 and n_2 seemed to work. Although this phenomenon wouldn’t be fully explained until Neils Bohr came up with his model of the atom, Max Planck had a stopgap suggestion: that atoms only emitted light in discrete packets of energy. (Planck suggested this to solve the Ultraviolet Catastrophe, but that’s a bit more than I want to get into right now.) Though this explained why the Rydberg formula gives only discrete colors of light, it didn’t explain the formula itself. But Planck’s hypothesis was the first hint of quantum mechanics, and it inspired Einstein’s explanation of the photoelectric effect.

The Photoelectric Effect

The modern photoelectric effect was discovered by Philipp Lenard, although the conclusive experiment was performed by the genius experimentalist Robert Millikan. As shown below, Lenard shone light on one of two metal plates connected by copper wire. Maxwell’s theory of electromagnetism tells us that light carries energy, so the light energized some photons and kicked them out of the metal (red). Eventually the electrons flew through the air to the other plate (black), then traveled through the wire back to the red plate. Lenard used an ammeter to measure the current (number of electrons per second) running through the wire. He also placed a generator on the wire, which he used to create a voltage that resisted the electric flow. Lenard figured that the voltage required to stop a current would tell him the energy of the electrons in that current.

The Photoelectric Effect
The photoelectric effect. Two metal plates (red and black) connected by wire are contained inside a vacuum tube. Light rays shine on the red plate, which kick some electrons (yellow) out of the metal. Eventually, the elecrons find their way to the black plate and, through the wire, the red plate. An ammeter (green A) measures the current, which is the flow of electrons traveling through the wire. Lenard hooked up a generator to the wire and used it to apply a voltage to resist the current. This way he could measure the amount of energy required to stop the electrons from flowing through the wire.

Lenard varied the color and brightness of the light. He measured the number of electrons ejected from the metal by measuring the current (with the voltage generator off), then found the energy of each individual electron by measuring the voltage required to resist that current. Since the energy of light increases with its brightness and its frequency (i.e., how purple it is), Lenard expected the energy of the electrons to react similarly if he increased the brightness or frequency of the light. However, the results were extremely surprising:

  • As expected, making the light more purple increased both the number of ejected electrons and the energy of those electrons.
  • But as the brightness of the light increased, the number of electrons ejected from the metal increased. But their energy remained completely unchanged.
  • And if the light became red enough, current simply stopped. No electrons were ejected from the metal, no matter how intense the light was.

In his Nobel Prize-winning paper, Albert Einstein proposed that there was a simple explanation for this. The light only carried energy in small packets, or quanta. (Incidentally, this is where the word “quantum mechanics” comes from.) As the color of the light came closer to purple, the energy of each quantum increased. As the brightness of the light increased, so did the number of quanta. But each quantum can only give its energy to one electron. The higher the energy of a quantum, the higher the energy of its recipient electron; the more quanta there are, the more electrons that are ejected.

Einstein also suggested that the metal holds on to its electrons tightly, so that each electron must receive a certain amount of energy to break free. This is why low-energy light will never kick any electrons out of the metal. Each quantum simply doesn’t have enough energy to kick out an electron.

Of course, the only way that light can carry quanta is if light is made of particles. This is a staggering development! It was already well-known that light is a wave. Refraction and diffraction readily demonstrated this. Lenses only work because light is a wave. The weird truth, then, is that the quantum of light, the photon, is both a particle and a wave!

This really really weird phenomenon is a founding element of all quantum mechanics. If light, which we traditionally believed to be a wave, is also a particle, then are objects we traditionally believed to be particles–like electrons–also waves? This question inspired Geoffrey Taylor to perform the famous double slit experiment, which showed that electrons are in fact waves. I’ll discuss the double-slit experiment and its interpretation next time.

Further Reading

Questions? Comments? Hatemail?

As always, if you have questions, corrections, comments, or just want to share something cool, please post in the comments!

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