Everything is made of particles. Pieces of solid matter. All solids, liquids, and gases – you name it. Dirt, pebbles, and red blood cells. Trees, dust mites, planets, and even the air we breath.

That’s obvious and common sense. We even make models of atoms and molecules with wood or plastic manipulatives like this, so that must mean something, right?

Except… we’re going to learn that all solid particles in the universe have a wave-like behavior.

And oh yes, all wave-like behavior has particle-like behavior?! Yup. For real.

This is the inescapable – and verified – result of the quantum mechanical nature of our world.

In the late 19th and early 20th century, when physicists asked hard questions about matter, they came across unexpected, extraordinary results.

We basically went through Alice’s Looking Glass – into the quantum realm. A realm where all particles have wave-like qualities. And further, all waves have particle-like qualities.

To be clear, none of this is a metaphor – we’re being quite literal.

The classical model of matter

The old model of the atom, and of everything in the universe was classical:

Everything is made of solid matter.

Everything has a definite position, mass, and velocity, at any moment in time.

Everything has a definite momentum at any moment in time.

How could it not? That seems to be true by definition.

We envisioned that there was a positively charged nucleus in the center of atoms.

Electrons (hence e^– ) orbited around the nucleus like planets orbit around a start.

And so was everything else in our universe. People, cars, rocks, planets, and stars.

But when we looked more closely at their behavior, we kept seeing evidence that this model couldn’t be correct.

The classical model of atoms was wrong

e^– lose energy by giving off photons (particles of light)

e^– gain energy by absorbing a photon (and its energy)

If e^– behaved like solid objects then they could move to any position (further from nucleus, or closer.)

They should be able to have any amount of energy: From a little to a large amount – and any value in between.

Therefore, when atoms gave off light, it was when e^– dropped from one energy level to another.

If e^– e could exist at any level then they could emit any energy of light, any color.

Thus atoms should be able to produce a continuous spectrum. Continuous means “all possible colors, smoothly going from one to the next, with no gaps.” Like this:

But experiments always showed otherwise! When individual atoms absorb light (energy) they only absorb photons (particles of light) in certain wavelengths. Yet they never absorb energy in others? How is that possible?

And when individual atoms emit (give off) light (energy), they only give off photons in certain wavelengths, never any others. Again, how is this possible?

No one could come with up with any model of the atom which was consistent with classical physics.

By 1913 Niels Bohr realized that nature was telling us something: Our classical intuition about what an atom was, was simply wrong.

We were forced to listen to what nature was telling us. Out of almost desperation, Bohr listened to nature and created a new, semi-classical model of the atom:

Like the old model, Bohr’s model portrays atoms as having a nucleus in the center and e^– orbiting around it.

But in his new model e^– could only exist in orbits of a certain radius. Not in any others.

Sure, they could lose energy, and “fall” from one orbit to a lower orbit – yet they didn’t exist anyplace in between?!

It was like they disappeared from kind of orbit – and reappeared in a different one?!

Quantum jumps

In classical physics any orbit is possible. It doesn’t make sense that only some orbits would be “allowed.”

Think of climbing a ladder. You can climb up from one stair to the next stair… and in doing so you obviously must pass all of the positions in-between.

There are an infinite number of positions between one ladder rung and the next. We don’t just disappear at one rung and then appear up at the next one, right?

(This GIF might be by artist Daniela Sherer)

This guy climbing the ladder, above, shows the classical, normal world we know.

We’re at one place, then at another – but only because we pass through every position in between.

The same thing goes for a car driving down the road. It starts at one place, ends up at another – and by definition the car must pass through every position in between.

But now imagine seeing this: the car literally disappears from the universe at one place, and then reappears in another place further down the road.

Without ever being in any of the positions in between?! That’s not possible, right?

Except – that is precisely what e^– in atoms seem to do.

Worse, all sub-atomic particles have this quantum leap type of behavior.

This violates common sense. But here’s the kicker – when we look closely, this is how the universe works.

Bohr’s model of the atom

An e^– gives off energy in the form of a photon. Photon shown as green squiggly arrow.

Then the e^– disappears from where it was and reappears in a lower orbit – without traveling through any position in-between!

Later, the e^– absorbs energy from a different photon (another green squiggly arrow.)

Once it absorbs the energy the e- jumps up to a higher orbit – again, without traveling through any of the position in between.

These seemingly impossible jumps are called quantum leaps.

(FYI, e- do not actually circular orbits. Bohr’s model was just the first approximation)

(image Bohr atom animation.gif)

This model was the beginning quantum mechanics.

From the Bohr model to the wave model

The following Socratic-style discussion comes from Physics 2000.

Why should an electron’s angular momentum have only certain values?

Why do electrons emit or absorb radiation only when they jump between energy levels?

Bohr’s theory fits experimental results, but it doesn’t explain why atoms behave the way they do.

In 1923, about ten years after Bohr published his results, Louis de Broglie came up with a fascinating idea to explain them: all matter, he suggested, actually consists of waves.

At first, de Broglie had no idea what he meant by matter being “waves.” It was just a mathematical construct that was helpful.

It was only later that physicists realized that this mathematical construction was telling us something about the true nature of reality itself!

de Broglie’s wave model of particles explains why an electron can only be in certain orbits!

de Broglie’s wave model assumes that any particle – an electron, atom, bowling ball, whatever – had a “wavelength”

Yeah, that’s weird – but let’s just roll with it for the moment.

Why assume such a thing? This assumption wasn’t arbitrary; de Broglie knew that the momentum and wavelength of a photon actually were related.

Hmm, wait a minute…photons don’t have any mass, do they? How can photons have momentum?

Photons don’t have mass, but they do have energy – and as Einstein famously proved, mass and energy are really the same thing.

So photons do have momentum – but what exactly is a photon?

For centuries, a heated debate went on as to whether light is made up of particles or waves.

In some experiments, like Young’s double slit experiment, light clearly showed itself to be a wave.

But other phenomena, such as the photoelectric effect, demonstrated equally clearly that light was a particle.

So which is it? Well, sort of both – or better, it is neither.

Light is a thing that sometimes has particle-like behavior, and sometimes has wave-like behavior.

It all depends on what sort of experiment you’re doing.

This is known as wave/particle duality. Like it or not, physicists have been forced to accept it.

That’s why we sometimes talk about “electromagnetic waves” and sometimes about “photons.”

de Broglie’s big idea was that maybe it’s not just light that has this dual personality; maybe it’s everything!

All right…let’s say I accept this idea. How does it explain Bohr’s energy levels?

If we think of electrons as waves, we change our whole concept of what an “orbit” is.

Instead of having a particle whizzing around the nucleus in a circular path, we’d have a wave existing around the whole circle.

Now, the only way that such a wave could exist is if the wave has constructive interference.

It has to have a whole number of its wavelengths fit exactly around the circle.

If the circumference is exactly as long as two wavelengths, say, or three or four or five, that’s great, but two and a half wavelengths won’t cut it.

If we have fractional amounts of wavelengths then there is destructive interference, and the waves cancel out.

So there could only be orbits of certain sizes, depending on the electrons’ wavelengths –which depend on their momentum.

Apps: Modeling electrons with standing waves

Standing waves in Bohr’s atomic model

Standing waves in Bohr’s atomic model Geogebra.org

How to run CDF demonstrations: worlds of math & physics

Seeing constructive & destructive wave interference in 3 dimensions with DESMOS

More from Physics 2000

Student: But is this just some mathematical trick that happens to work, or do particles actually behave like waves sometimes?

Teacher: They actually do behave like waves! Just a few years after de Broglie published his hypothesis, several experiments were done proving that electrons really do display wavelike properties.

Student: So how come when I look at a bowling ball, I don’t notice it acting in a wavelike manner? You said that everything is affected by wave/particle duality.

Teacher: Think about what the wavelength of the bowling ball would be. According to de Broglie, the wavelength is equal to Planck’s constant divided by the object’s momentum.

Planck’s constant is very, very, very tiny, and the momentum of a bowling ball, relatively speaking, is huge.

If you had a bowling ball with a mass of, say, one kilogram, moving at one meter per second, its wavelength would be about a septillionth of a nanometer.

This is so ridiculously small compared to the size of the bowling ball itself that you’d never notice any wavelike stuff going on.

That’s why we can generally ignore the effects of quantum mechanics when we’re talking about everyday objects.

It’s only at the molecular or atomic level that the waves begin to be large enough (compared to the size of an atom) to have a noticeable effect.

Student: If electrons are waves, then it kind of makes sense that they don’t give off or absorb photons unless they change energy levels.

If it stays in the same energy level, the wave isn’t really orbiting or “vibrating” the way an electron does in Rutherford’s model, so there’s no reason for it to emit any radiation.

And if it drops to a lower energy level… let’s see, the wavelength would be longer, which means the frequency would decrease, so the electron would have less energy.

Then it makes sense that the extra energy would have to go someplace, so it would escape as a photon–and the opposite would happen if a photon came in with the right amount of energy to bump the electron up to a higher level.

Teacher: Very good! Now we can look at how Schrödinger extended de Broglie’s idea of waves into a whole new model for the atom…

What happened next, to finally create Quantum Mechanics, was that Schrödinger extended de Broglie’s idea of waves into a whole new model for the atom.

Related apps

Models of the Hydrogen Atom – PhET

Run this PhET app. Click to change from Experiment to Prediction. Press button to start the electron gun.

Under ‘Atomic model,’ the models of the atom most pertinent to this lesson are the Bohr model and the de Broglie model.

External resources

astronomy.nmsu.edu/agso/spectroscopy.pdf

Continuous spectra vs actual spectra

Emission Spectra: How Atoms Emit and Absorb Light

Emission and absorption spectra

Spectral Classification of Stars

Formation of Spectral Lines, Lumen

Physics 2000. University of Colorado by Prof. Martin V. Goldman. This website no longer exists except as an archived copy.

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