The Wave Nature of Matter

 I'm actually relieved to hear that this Bohr stuff isn't the end of the story. The mathematical part of the Bohr model makes sense to me, but some of its assumptions seem pretty arbitrary. I mean, why should an electron's angular momentum have only certain values? And why do electrons emit or absorb radiation only when they jump between energy levels? I know Bohr's theory fits a lot of experimental results, but it doesn't really explain why atoms behave the way they do.

 That's how most scientists felt about the Bohr model when it was first proposed. In 1923, about ten years after Bohr published his results, Louis de Broglie came up with a fascinating idea to explain them: matter, he suggested, actually consists of waves.

 Uh...will you run that by me again? I know it sounds like a far out notion. At first, de Broglie had no idea what he meant by matter being waves, either; it was just a mathematical construct that unexpectedly turned out to be very helpful.

 Oh, yeah? Explain how thinking of matter as waves can answer my questions about the Bohr model.

 Gladly. First of all, it gives a very nice reason why an electron can only be in certain orbits. What de Broglie did was to assume that any particle--an electron, an atom, a bowling ball, whatever--had a "wavelength" that was equal to Planck's constant divided by its momentum...

 And why would one assume such a thing? I thought we were going to get away from all these out-of-the-blue assumptions.

 Well, this assumption wasn't completely arbitrary; de Broglie knew that the momentum and wavelength of a photon actually were related in just this way. Wait a minute...photons don't have any mass, do they? How can they 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 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, it's both--or it's neither. Sometimes light displays particle-like behavior, and sometimes it acts like a wave; it all depends on what sort of experiment you're doing. This is known as wave/particle duality, and, like it or not, physicists have just been forced to accept it.

 And that's why you've been talking sometimes about "electromagnetic waves" and sometimes about "photons," which seem more like particles.

 That's right. Now, de Broglie's 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 begin to think of electrons as waves, we'll have to change our whole concept of what an "orbit" is. Instead of having a little particle whizzing around the nucleus in a circular path, we'd have a wave sort of strung out around the whole circle. Now, the only way such a wave could exist is if 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 won't cut it.

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

Fitting Waves Around a Circle

Click and drag on the circle to change the circles radius.

Or drag the grey ball around to change the length of the wave.

When an exact number of wavelengths fits around a the circle, the waves will be green. Otherwise they are red.

See how the wave only fits at certain "orbits"?

 Exactly. And if you do the algebra--set the wavelength equal to the circumference of a circle--you'll get precisely the condition that Bohr used: an electron's angular momentum must be an integer multiple of h bar.

 I can show you how to derive Bohr's angular momentum condition from de Broglie's expression for the wavelength.

 I'm impressed; that works out so nicely. But is this just some mathematical trick that happens to work, or do particles actually behave like waves sometimes?

 They actually 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.

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

 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.

 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 "wiggling" 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 wavy electron thing 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 wave-electron up to a higher level.

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

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