|The Quantum World - Atoms
Below are some models of some atoms. These models are based upon precise mathematical theory which is verified by
empirical evidence. First look at the pictures, then Bot will explain more about them...
(Tech Note: These models are not exact mathematical plots, but are constructed from constructive solid geometry in Pov Ray).
What is an atom?
Familiar and 'ordinary' matter, like the wood in a desk, the meat on your bones, the stones in your garden, as well as the
water in your tank and the air in your room, are all made up of atoms. Atoms are particles. Think of smoke, we often talk about
Atoms make up elements
Hydrogen is what we call an element. Iron, carbon, oxygen, sulphur and uranium are all elements. There are currently
something like 112 known elements, though the exact number is under dispute because some of them are so unstable that
their atoms collapse and decay almost as soon as you detect them. Fortunately, however, enough are stable to give the 92
elements found in nature. An iron rod weighing one kilogram (2.2 pounds) contains roughly 10 000 000 000 000 000 000
000 000 atoms of iron (1 x 10^25 or 1E+25 atoms).
Atoms make molecules
Atoms can stick or bond together in certain permissible combinations to form groups of atoms called molecules. In the
element sulphur, atoms group together to form rings of 8 atoms. These rings group together to form a stone of sulphur, such
as what you might find in a rock shop. These rings, and other similar groups of atoms, are called molecules. Atoms of different
elements may bond together to form molecules called compounds, because they are compounded from more than one type
of atom. Water is made up of H2O molecules - molecules made up of two hydrogen (H) atoms and one oxygen (O) atom
Atoms are made up of vibrations or waves
Think of a water wave on the sea. You would be right in thinking that a wave is a repeating or periodic oscillation or vibration,
in this case of the water's surface. Sound waves are vibrations that travel through the air (or water or through a solid wall).
Light waves are electromagnetic vibrations. In fact, just about everything that you can think of is actually made up of
vibrations. Atoms are no exception, so we need to say a little about waves first. Consider a simple wave, either a water wave
or a vibration on a string or slinky spring. In profile it might look something like the diagram below:
As the water moves up and down, we have peaks (the crests of the wave) and troughs (the dips). We define the wavelength
to be the distance between adjacent peaks (or equivalently between adjacent troughs or between any two equivalent points
along the wave). (Note this is not the length of the wave in common terms). For a water wave we may have a wavelength of 10
metres say, though the wave may spread for a thousand kilometres across the Atlantic Ocean. Half the height of the wave we
define as the amplitude. We chose half the height because this is the height of the wave above the undisturbed water
surface, since the peaks move water up and the troughs move water just as far, but downwards. We can see our little orange
buoy bobbing up and down, but note that it doesn't move sideways much, since the water moves up and down but does not
travel along with the wave from say left to right. If we count how often our buoy bobs up in say one minute, then we get the
frequency of the wave - how often a crest passes. For a water wave we might get a frequency of 12 wave crests a minute
(one crest passing the buoy every 5 seconds). If the wavelength is 10 m then the wave speed is (10 x 12 = 120) 120 metres
per minute (or 2 metres per second or 7.2 kph). One can also speak of the wave period as the period of time between two
wave crests, in this case 5 seconds.
(Tech note: Actually the water moves in circular or elliptical orbits, and there is, contrary to what most physics and maths
textbooks say, also some sideways displacement of the water, especially when the waves are deep, since the equations of
motion then become nonlinear).
Our water wave may move along the water though, from say east to west, even if the water does not move sideways with it -
the disturbance spreads across the water, not the water molecules themselves. Such a wave is called a travelling wave.
However, waves can do odd things, like reflect off solid obstacles (light reflects off mirrors, so water waves reflect off harbour
walls). The weird thing about waves is that they interact with one another when they occupy the same space, so the reflected
wave interacts with the incoming wave as the two overlap. If the peak of the reflected wave coincides with the troughs of the
incoming wave (and so the troughs of the reflected wave interact with the peaks of the incoming wave) then what happens?
The waves will cancel out! If, however, the peaks coincide with one another and the troughs coincide then we get a single
wave with twice the amplitude of the incoming wave. This process of waves interacting with each other is called interference.
When they cancel each other out, we have destructive interference, and when they add to give a deeper wave, we have
Inside the harbour walls things could get complicated, with so many waves bouncing around off walls. Not all of these waves
will have the same wavelength either, however, the water does not chop and churn all over the place (except in a storm!).
This is because most of the waves get cancelled out by each other. Only waves of certain wavelengths survive by
constructive interference, the majority are destroyed. We actually end up with a few waves of certain wavelengths that bounce
back and forth and end up going nowhere - these are called stationary waves.
Ok we are nearly ready to understand the shapes of our atoms, but first we need to say a bit more about waves, using
music as our example.
Waves and music
The same is true on a guitar string. A guitar string is a string fixed at both ends. When the string is struck, waves travel along
the string, but soon reflect off both the fixed ends, until only certain wavelengths remain. The string causes the guitar to
vibrate and the guitar causes the air to vibrate in a similar manner and we hear a sound based on the waves on the string.
Notice we hear a definite note not a chaotic clatter - this is because only certain wavelengths, called the harmonics remain.
Which harmonics remain depend upon the length of the string and the reflective properties of the material that fixes the ends
of the string. Of course our guitarist can change the effective length of each string by pinning it to the fret board with their
fingers, altering the note of each string as they play. The waves on the string are standing waves which vibrate up and down
but do not appear to move from side to side.
The diagram below shows the first three harmonics for a vibrating string:
Notice that the number of peaks plus the number of troughs gives the harmonic, so the second harmonic has one peak plus
one trough, the 4th harmonic has two peaks and two troughs, etc. Each note will consist of many harmonics, but the lower
harmonic are the most important. The higher harmonics add quality to the note and depend upon the individual instrument. All
the harmonics actually contain a whole number of half wavelengths. The fundamental is the first harmonic and is just half a
wavelength )because it has one peak but no trough). The exact combination of harmonics is unique to each instrument or
human voice, and is called the timbre (quality) of the sound.
Even if all the strings on a guitar are kept at the same length they all make different notes, why is this? The harmonics are the
same, since the fundamental wavelength is just twice the string length. What differs between strings is not the wavelengths of
the harmonics, but their frequencies. The frequency of a wave depends both upon the wavelength and the speed of the wave
(note standing waves still have speed even though they appear stationary!). The wave speed depends upon the tension and
weight (density) of the string. A lighter or a tenser string produces a higher frequency., A higher pitched string has a higher
wave speed and produces notes of a higher frequency and hence of a higher pitch. Shorter strings also have shorter
wavelengths for their harmonics and so higher frequencies and higher pitches. Thus a violin produces higher pitched notes
than the double bass.
From strings to atoms
Nearly there! Think now of a drum. A drum vibrates when struck just like a string. A drum also vibrates with certain frequencies
or harmonics, but this time the vibrations are those of a sheet (we have a 2D wave). Think of a bell. A bell also vibrates
according to its harmonic frequencies, with larger bells producing deeper sounds, but this time we have a 3D wave.
Atoms are rather like spherical bells that vibrate. As atoms vibrate they change shape. Being tiny objects they adopt shapes
that differ dramatically from one another according to the harmonic or mode of their vibration. Sometimes they vibrate with
several harmonics, but they often have a preference to vibrate with only one harmonic, called the mode of vibration. The
pictures on the previous page showed several hydrogen atoms, in fact they could all be the same hydrogen atom at different
times. Each shape is a standing wave as the atom vibrates, so the first one, labelled 100 is spherical, as if the atom is not
vibrating at all but is at rest. Other modes give very different shapes!
The mode of vibration of an atom depends upon its energy
The more energy we give to an atom (e.g. by hitting it with light or a laser beam) the higher the mode of the vibration (the
higher the frequency of its harmonic). Each vibration mode is assigned three numbers (called quantum numbers), e.g. 100,
321, etc. The first number is the energy of the vibration mode, so 200 is more energetic than 100 and 311 is more energetic
than 200. This is called the principal quantum number and is given the symbol n. The second and third numbers are due to
something called angular momentum. We talk of momentum when an object is moving, the faster it moves or the heavier it is,
the more momentum it has and the harder it is (the more force required) to stop it. This is ordinary linear momentum, when an
object moves in a straight line. However, objects that spin also have angular momentum, as does pirouetting ice skater. An
atom may have angular momentum (though this is somewhat different to the angular momentum of an ice skater). The 2nd and
3rd numbers give us the precise shape of the atom, if they are both zero, as in 100, 200, 300 , ... , then the atom is spherical. If
the 2nd quantum number is equal to 1 or greater, then the atom has a dumb-bell, torus (doughnut) or more complicated shape.
Where do the quantum numbers come from?
The quantum numbers and the shapes of the atom are obtained by solving what is known as Schrodinger's wave equation.
Schrodinger proposed this equation based upon our knowledge of maths and physics. This equation has more than one
solution, one for each possible value of the three quantum numbers. Each such solution is called a wave function and the wave
function tells us most of what we can know about the shape and behaviour of the atom (I say 'most' because there are various
extra corrections for things we call spin, relativity, internal and external magnetic and electric fields and quantum
electrodynamics). Actual observations and measurements verify the Schrodinger wave equation, so we can be reasonably
confident that our physical model of the atom is correct. In fact, quantum mechanics, the branch of science that uses the
Schrodinger equation amongst many others, can describe the hydrogen atom very precisely indeed! This is a major triumph for
science. I am not going to give the equation or its solutions here - to explain these would take a whole book, and plenty of
people have already written such books! The key point is, that these mathematical equations allow us not only to accurately
describe the shapes of atoms, but also to accurately and reliably predict their behaviour - so we can make predictions and then
carry out experiments to test those predictions, and refine the theory if need be and learn new physics.
What does 'quantum' mean?
Notice that the atom can not vibrate in any old manner, just as our guitar string cannot, rather it can only vibrate in certain ways
(the harmonics). Since each harmonic has a certain well defined frequency and so energy, this means that the atom cannot
possess any old energy, but only certain values of energy. The energy is divided into a series of discrete 'energy levels' and we
say that the energy is quantised (made up of packets of a certain size). It turns out that many properties of many things in
physics are quantised according to the same principles (almost everything is made up of vibrations). The study of these
quantised processes is called quantum mechanics or quantum physics.
What exactly is an atom made of - what is actually vibrating?
So far we have talked about atoms as tiny bells that vibrate. This is an over-simplistic model. The real world of the atom is very
very different from the large-scale world that we are familiar with from everyday experience. Atoms can do odd things, like
behave as if they are in two or more places at once, or teleport from one place to another in an instance. The world
of the atom is simply different!
Atoms are made up of smaller particles, called electrons, protons and neutrons. The protons and neutrons cluster together
in the centre of the atom, forming the nucleus. The electrons form the vibrating shells that we have seen pictures of, around
the nucleus. Hydrogen atoms only have one electron and one proton (and usually no neutrons) but other atoms contain many
of these particles. The nucleus contains almost all of the atom's mass, but is very tiny indeed - about one hundred thousandth
the diameter of the atom (1 x 10^-15 metres diameter). Being so small, but containing most of the atom's mass, the nucleus is
extremely dense! People used to think of the atom like the Solar System, with the electrons representing the planets orbiting
the nucleus which represented the Sun. This is not correct, however. The model that we have discussed is far more correct, but
requires a deeper understanding. The problem is a phenomenon called wave-particle duality.
We have talked about atoms as particles, like tiny solid balls (or bells). However, an atom can also behave like a wave. We saw
that an atom is made up of waves, but it can also be a wave! This confuses most people (if not everybody!) but it shouldn't
really, because almost everything is made up of vibrations anyway. What one takes for granted to be a solid table, is made up
of atoms which vibrate and the electrons inside the atoms vibrate - the whole table consists of energy vibrations. So atoms, and
indeed electrons, protons, neutrons, in fact any particle, can also be a wave, this is the phenomenon of wave-particle duality.
Really, particles are waves that sometimes resemble particles.
When you lean against the table it feels 'solid' because the electrons in the atoms in your body are repelled by the electrons in
the atoms of the table. This is because electrons carry negative electric charge and like charges repel. The protons in
the nucleus carry positive electric charge and opposite charges attract, so the electrons are attracted to the nucleus and
tend to stay close to it. Thus we can imagine electrons forming negatively charged shells that surround the positively charged
central nucleus. The reason that electric charges repel or attract one another is that they exchange photons. Photons are
particles of light, so the electrons and protons exchange a kind of light, however, it is not the kind of light we can see, because
the photons exist for a period of time that makes it impossible in principle to ever detect them. It is as if these photons never
exist, and so they are called virtual photons. The light that you see consists of photons of the non-virtual variety! So, in a
sense an atom is made up of waves (particles called electrons, protons and neutrons) bound together by (virtual) light, which is
itself a wave. Thus, waves hold the waves together!
So what vibrates? Well, everything just about, but the electrons that form shells (or orbitals) around the nucleus are vibrating to
give the atoms their shapes. However, this is not quite correct! Atoms are far more mysterious. If you measure the electron in a
hydrogen atom, to see how it is vibrating, then you will instead find that it exists as a single discrete particle at some point near
to the nucleus (or sometimes inside it and sometimes far away), so what happened - is our model wrong? No, not exactly, what
is vibrating is not the electron itself, as such, but the wave function that describes its properties, such as its position and speed.
However, that is not all, the electron actually does not have any position or speed until we measure it! (Or it may have any,
every or no position!).
Discover more of the mysteries of the atomic electron.