Atomic Structure The shape of ... ... electron orbitals
Great Uncertainty

In our macroscopic world it is easy to measure the performance of moving objects such as cars and airplanes. Using your eyes you can easily see where your car is at any moment (technically you are "measuring its position in three dimensional space").

Also you can easily measure how fast it is moving using any number of methods, including the speedometer in your car! Knowing the "position" of the car, and how rapidly that position changes with time gives you the "velocity" ( or "speed") of the car. In addition, if you know the "mass" of the car you can multiply this value by the velocity and get the "momentum" of the car and thus all there is to know about its performance.

In our everyday world, therefore, knowing both the position and the momentum of an object (such as your car) is not a difficult task if you have the right instruments; eyes, rulers, clocks and light (to see with!).

It does not change the value of the position of the car, or its momentum, very much when photons of light crash into it, bounce off, and end up at the back of your eyes, triggering your brain. The act of "observing" the car (which is what you do when you look at it) has an infinitesimal effect on either its position or its momentum.

"Looking"
at electrons

Unfortunately this is NOT the case when we try and "observe" an electron, as Werner Heisenberg found out.

The quantum picture of the atom discovered by Bohr, Sommerfeld, Pauli and others had within its image the idea of electrons "orbiting" around a very dense atomic center of protons and neutrons. But the very idea of an "orbit" means that, at any one moment, you know the exact position and momentum of the moving particle. Measuring these two values (position or momentum) requires you to observe the electron (i.e. "look" at it).

There is no way of "observing" something without "bumping into it". At the very least you have to bombard the object with light photons and see what they do when they bounce off. This process works well in our everyday world, but not with electrons.

The moment you try bouncing photons off an electron, all kinds of things happen including the transfer of energy and the sudden movement of the electron to other places in the atom.

This means that the instant you know, say, the position of an electron (by the way the light photon hits it), the momentum changes vastly. All these complications of observing position and momentum of electrons in their quantum world are summed up in the ...

Heisenberg Uncertainty Principle

"The more precisely the position of an electron is determined by observation, the less precisely the momentum is known at that instant, and vice versa."
Heisenberg, 1927

The consequence of this "uncertainty" is that the observer (us) can never know exactly where an electron is at any instant of time.

Instead, when trying to picture where an electron is located, all we can do is talk of the probability of finding an electron at that position; 90% probability, 80% probability, 70% probability, etc.

Probablility
and orbital shapes

Instead of drawing an electron as a point in space (like a planet going around the sun), therefore, it is only possible to draw a volume of space in which an electron can be found 95% of the time. In theory an electron could be found anywhere in the universe, at least 0.001% of the time, and could be found in some very strange places where they would be least expected. In our ordinary, macroscopic world this would be roughly equivalent of your car suddenly vanishing from the curbside and appearing in Russia for 10 minutes, before returning to the curbside!

No longer "points in space", electrons can only be drawn as orbital shapes; volumes of space where they are likely to be found most of the time.

Why this shape?

Why are electrons found most of the time in these very characteristic orbital shapes?

One way of illustrating the answer to this question is to think of a bee flying in circles. Electrons (and bees) have simultaneous properties of particles and waves (the particle-wave duality that makes them so hard to imagine in our world), so it is necessary to take their wave-like properties into account when considering where they are located at any one time.

When a bee flies it beats it's wings up and down with a characteristic frequency. This produces a fixed wavelength characteristic of the speed it is moving and the frequency of its wing beats. It also leaves behind it a wake of turbulence in the air in much the same way as an airplane leaves turbulence behind it at airports.

This turbulence can be a big problem for airplanes, especially for those planes following behind. Air-traffic safety officers take many precautions to prevent the wake produced by one plane from interfering with the flight of the plane following it. Whey they fail to do this properly the consequences can be very serious indeed, and some fatal crashes have occurred when one plane tries to take off into the wake left by another.

When a bee flies in a straight line it leaves a wake of turbulence behind it, but usually has no problem with the results, as it never has to cross over, or encounter that wake pattern itself.

When a bee flies in a circle, however, it encounters it's own wake the moment it completes the first trip around the center pivot. Now the bee's wings have to try and beat into air that was disturbed by its previous passage. If the bee tries to beat it's wings down as the wake is moving the air up, it has serious problems and will probably crash.

But if the bee can arrange it so that it is moving its wings down at the same place that the wake is moving the air down, and then beat its wings up at the same place that the wake is moving the air up, all is well. The bee's second set of wing beats are in synchrony with the first set of wing beats and the bee can fly fine!

This ideal state of affairs can only occur when the first wing-beat of the second passage is exactly "in phase" with the first wing-beat of the first passage. If the second passage starts even slightly off, there are problems.

Since the wing-beats have a fixed wavelength this ideal state only occurs when the diameter of the circle in which the bee is flying is an exact number of wavelengths. If the diameter of the circle is slightly greater than an exact number of wavelengths or slightly less than an exact number of wavelengths, then the second passage of the bee will have a wing beat in a different place, and there will be trouble.

So the bee can only fly in circles whose diameters are an exact number of wavelengths!

If it tries to fly in any other circle it will get caught up in its own wake and crash.

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Permitted electron orbits

In a completely different world, the same is true for electrons around atoms.

An electron can only exists as a wave and a particle in places where the "wave" of its wave-property is exactly "in phase" with itself!

So it is possible to calculate the shapes and distances of volumes of space around atoms where the conditions are perfect for an electron to be in phase with itself. Very strangely, if an electron somehow tried to exist "out of phase" with itself (at a place where its waves were not in complete synchrony) then it would wipe itself out and vanish!

The shapes of the 1s, 2s, 2p, etc., orbitals are therefore determined by the wave properties of the electrons and the fact that only within these volumes of space are electrons able to co-exist with themselves in a way that their waves do not cancel themselves out.

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