**By **Robert M. Hazen, Ph.D.**, **George Mason University

## Wave-particle duality brings us full circle to the Bohr atom and its strange non-accelerating electron orbits. Those electrons that seem to be at discrete energy levels, that just disappear from one and appear at the other but never are anywhere in between. Electrons are not particles or waves but a combination of both of these things.

### Wave-Particle Duality: How Electrons Behave

One way to differentiate between particles and waves is to see how they behave when they are thrown or directed at a parallel pair of narrow slits. There are two vertical slits, so when solid particles like ping-pong balls are directed at this pair of slits, they go through the first slit, or they go through the second slit, or they bounce back right.

But when waves of sound or water are directed at two slits, they create what’s called an interference pattern, with many maxima and minima at the opposite side of the slits. Well, this experiment has been tried many, many times with electrons. When a single electron is fired at the slits, a single spot on a photographic film is seen.

That suggests that electrons are like isolated particles. But when thousands of electrons are fired at pairs of slits, there are thousands of separate spots, and they gradually merge into what looks like an interference pattern. The conclusion is that electrons must behave in some way, both as particles and as waves. And at the atomic scale, this is called the wave-particle duality.

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### When Electrons Behave As Particles

If the electron is thought of as a particle, then Newton’s equations for motion can be used to analyze possible orbits. That is, for a given energy, the given mv^{2}, which is the kinetic energy of the electron, how far out does it need to be to achieve a stable orbit because there is a positive charge and a negative charge, and therefore an attractive force? So this can be analyzed just like a little Newtonian universe.

If a planet is going to achieve a stable circular orbit, then the gravitational force pulling down the planet has to be exactly balanced, of course, by the gravity and the outward thrust. So this is a real analogy to a planetary system, even though the Bohr atom is not like a planetary system.

So, by the same token, if an electron’s going to achieve a stable orbit, then this attractive electrostatic force has to be balanced by the momentum of the electron. These calculations, by the way, lead to an infinite number of possible, stable electron orbits, and these can be cataloged down from the first orbit on up.

Learn more about an why atoms bond to one another.

### When Electrons Behave As Waves

Now, the electron can be thought about in a completely different way. When a wave is restricted to lie in a circle, it needs to adopt a pattern, which is called a standing wave. There has to be an integral number of wave crests fitting into that circular orbit. This feature can be illustrated by a vibrating string.

There is the ground state, which is just a simple wave of the lowest energy. This could then go to the first stable orbit, the first excited state. This is a little bit faster, and it’s divided into two units, and that fits perfectly in the same length cord. Each of these higher excited states requires more energy.

Shorter wavelengths, higher energy, higher excited states. These nodes, these perfect harmonics of a vibrating string, correspond in a very simple way to the different orbits, the stable orbits of electrons in the Bohr atom. The ground state has zero nodes. There are one node, two nodes, three nodes, depending on how many vibrations fit into that circular pathway.

### How Electrons Choose Their Orbit

Now, as it turns out, the electron orbits of hydrogen atoms correspond to solutions of two completely different equations: one for particle behavior and one for wave behavior. In fact, the Bohr orbits correspond exactly to the only possible solutions that are present both for an electron if it’s considered as a particle and for an electron if it’s considered as a wave.

Somehow the electron knows it has to have both of these attributes, and it chooses those orbits which fit both possible models. Think about these results in terms of energy. Every electron orbit can be represented by energy. In the particle formulation, this energy has the kinetic energy of a particle.

Each orbit has a different energy, with orbits closer to the nucleus having lower energies because the charged particles are a lot closer together. In the wave formulation, the energy corresponds to the wave energy of the electron. The orbital energies, it turns out, are exactly the same.

Learn more about electromagnetic radiation.

### Wonder of the Quantum World

Both formulations fit together beautifully, but it requires someone to think about this strange idea: wave-particle duality. The subatomic world is rather different than a person’s intuition might have suggested. It all fits and makes a consistent picture, albeit a strange, quantum mechanical picture.

The quantum world is strange. The role of science, though, is to formulate ever more exact descriptions of the physical world and events in that world. Some situations, like the orbit of a planet, are amenable to exact mathematical descriptions. Other situations, like the flipping of a coin, are better described in terms of probabilities.

### Common Questions about Wave-Particle Duality

**Q: What is the interference pattern?**According to the wave-particle duality, if a single electron is fired at the slits, it makes a single spot on the photographic film. But when many electrons are fired at the slits, they make many spots. The spots merge together to make an interference pattern.

**Q: What is the wave-particle duality?**Wave-particle duality is a term in the world of electrons and subatomic particles. Electrons have neither particle nor wave behavior, but something between these two behaviors. Hence, they always choose an orbit that has both wave and particle models.

**Q: How are possible orbits for an electron chosen if one thinks about the electron as a particle?**It’s clear that electrons use the wave-particle duality behavior to choose the ideal orbit. But by considering the electron’s behavior as a particle, Newton’s equation of motion (mv^{2}) can be utilized to guess possible orbits.