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Atomic Structure
The Nature of ...
... Matter-Waves.
Lessons from Angry Bees

Electrons are elusive and light is very mysterious.

Bees are easy to find and not so mysterious - so let's begin by studying the properties of angry bees and see how such a study helps us understand the properties of electrons, light and finally how atoms are constructed.

bees have mass

A bee is a small flying insect that has measurable properties as it interacts with us and our environment. One of these properties is mass, or weight; a bee has a definite, measurable weight that can be determined when it is placed on an appropriate scale and compared to a known mass, such as a gram or an ounce.

bees can fly

Another property of a bee is that is flies. When a bee flies it vibrates its wings up and down to produce lift and thus is able to move through the air at a measurable speed. The velocity, or speed, at which the bee is able to move is another of its measurable properties (you need to measure both the distance a bee travels and the time it takes before you can determine this property).

If the mass property of the bee is combined with the velocity property of the bee, we get a third property; the momentum.

p = m.v
momentum = mass multiplied by the velocity

Clearly, the "momentum" property of the bee will change if it grows bigger (more mass), or flies faster (more velocity). But for the rest of this discussion we are going to assume that all bees fly at exactly the same speed (velocity), so the only way of changing the momentum is to change the mass of the bee. Big bees with more mass will have more momentum than smaller bees with less mass.

The bees in this discussion are different from garden bees in three ways; they all fly constantly, at the same speed (velocity) and in straight lines - until they hit something. What happens when a bee hits something depends on what it hits. If it hits an oncoming car it is flattened, vanishes and all its momentum is transferred to the car. But the momentum of the car is so much greater than the momentum of the bee that the driver of the car hardly notices the difference.

transfer of momentum

If the flying bee hits a thin piece of paper hanging by a thread, however, the paper will react violently, spinning and twisting on its thread and the effect of the transfer of momentum from the bee to the paper will be very noticeable and measurable.

In a similar manner, if the bee crashes into a butterfly, both insects will be effected. Momentum will be transferred and both insects will probably fly away from the encounter in different directions to those they were traveling in originally (they bounced off).

These properties of bees all stem from the fact that the body of the insect is a particle with a mass and a velocity. Other particles could be baseballs, bowling balls, bullets, cars or airplanes; anything with mass that moves.


But when they fly, bees have another property. The beating of their wings up and down (to provide the lift) also produces sound waves. The characteristic "buzzzzzzz" a bee makes is a result of pressure-waves produced by the compression and decompression of the air by the bee's wings as the wings move in a characteristic repetitive motion.

What kind of "buzzzzzzz" sound the bee makes depends on how fast the wings are beating. A fast beat produces a higher pitched sound, a slow beat a lower pitched sound. The rate at which the bee is beating its wings is called the frequency, which can be measured by counting how many times a wing goes up, down and up again in one second. This is usually represented using a unit called a Hertz, where 1Hz = 1cycle/sec.

f = n/t
frequency = number of beats divided by the time interval

A flying bee

Bees beat their wings up and down.

Try this for yourself.

Move your cursor over the squares below the bee.

Try to make the wings of the bee
beat up and down with different frequencies.

A wave of bees

Even if you cannot see a bee you can usually hear it. The wave property of the sound tells you a lot about where the bee is and what it is doing. For example, as a bee flies directly towards you the pitch of the sound it is making (the frequency) goes up, and when it flies away from you the pitch goes down. This is the Doppler effect. As a result you can always tell if the bee is coming or going, a valuable piece of knowledge some times!

There are many different kinds of waves; those in the sea or water, sound waves, electromagnetic waves (radio and television, for example), and another special kind called matter-waves we will see shortly.

The Property of Waves

All waves move outwards from a source with a certain speed (velocity) and with a certain frequency (rate of going up and down). The distance between two crests (or troughs) of such a moving wave is called the wavelength, a property that depends on both the frequency and the velocity.

One wave can effect another wave. When two different waves meet, depending on how they traveling, their troughs and crests can combine to create a new wave which is the sum, or the difference of both the original waves. In the most extreme cases adding two waves together makes a third wave that has double the amplitude (height of crest or depth of trough)...

... or adding the two waves together cancels out both producing a "wave" with no oscillations at all.

Bees in flight

, therefore, show two sets of properties; those of a particle with mass and momentum, and those of a wave with sounds and frequencies. A blind person would only be able to study bees by following their sounds, and thus consider their properties to be that of a wave They would not be able to detect the particle properties of bees because they could not see the insects.

In contrast, a deaf person would not be able to hear the sounds and thus not be able to detect, or appreciate, the wave properties of bees, but would instead consider them to be particles. All the properties of a bee a deaf person could study would be those of a particle with its mass and momentum.

For one observer a bee would be a particle, and for another observer a bee would be a wave; both at the same time. The bee would appear to have particle-wave duality.


In 1924 a scientist called Louis-Victor Pierre Raymond duc de Broglie realized that all objects that could be considered particles could also be considered waves, i.e. have "particle-wave duality". Everything from bowling balls to electrons could be considered to be particles and waves at the same time! This was an amazing discovery, and de Broglie put the relationship in terms of a formula that related the wavelength to the momentum of the particle, thus:

In this formula the Greek letter "lambda" is used to represent the value of the wavelength, and that value in turn is determined by dividing a special constant, known as the Plank constant (more on that later) by the momentum of the particle (mass multiplied by the velocity).

With this de Broglie formula anything that moves can be studied either as a wave - called a matter-wave - or as a particle with a mass and momentum. However, not many of us have ever seen a baseball or a bowling ball suddenly take on the properties of a wave - why?

The answer lies in the numbers. What is called the "Plank constant" is a very small number indeed.

h = 0.0000000000000000000000000066256 erg-seconds
(this is usually written 6.6256 x 10-27 erg-seconds)

Common, everyday objects (such as bowling balls) have large masses and therefore momentums. When the Plank constant is divided by these large numbers the result is a wavelength so small it can not be detected by any known method. So bowling balls can be waves, at least in theory, but their wave properties cannot be detected or studied (yet!). So no one has to worry about a pitcher's fast ball suddenly turning into a wave!

But, as the mass of the particle gets smaller and smaller, so does it's momentum, and when the Plank constant is divided by very tiny numbers, suddenly the wavelengths become long enough to not only be detected, but to have properties we can study. An electron, for example, only has a mass of:

0.00000000000000000000000000091091 grams
(this is usually written 9.1091 x 10-28 grams)

Which, according to the de Broglie formula, should have a matter-wavelength well within the range we can detect, study and use. Indeed it does. Some of the most powerful microscopes, capable of producing the greatest degrees of magnification, use electrons. Instead of light, these microscopes shoot electrons at their specimens and their "rays" can be focused just like light rays (only using magnetic instead of glass lenses).

Tiny objects, like electrons, should therefore have properties of both matter-waves and of particles, both at the same time. Indeed they do.

© 2003, Professor John Blamire