From the square of the displacement we can calculate the strain energy, stored in the material.  From the square of the velocity we can calculate the kinetic energy of the motion.  Adding these two energies gives the total energy at the bottom of the diagram.

Nobody has ever seen any energy.  It is a theoretical idea which greatly simplifies the working out of many problems.  You could argue that all we ever observe is the positions of objects, that everything else is purely theoretical construction.  Zeno, in one of his paradoxes, showed that the idea of motion or velocity is not as simple as it looks.

But these constructions are so useful that it is easier to treat them as real.  If we had to discuss something as simple as velocity only  by reference to positions we would not get far.

In this page we shall meet abstractions that are unfamiliar in everyday life, but no more difficult to grasp than these.

Let's start with the bridge.  The picture shows a spectrum of oscillation of a simple hanging beam, (not to scale).  (Oscillation)

Around the frequencies marked 1 to 4 the beam responds more strongly to a an impressed force than it does at other frequencies.  The fundamental frequency marked 1 is missing because of the way that a suspension bridge is hung, but in most mechanical oscillators it would be strong.  The peak at 3 is wider than the peak at 2, and the peak at 4 is even wider, so that the resonance is ill-defined.

The width of the peaks tells us something about the object.  It tells us how long the object will oscillate after being given an impulse.

A narrow resonance means that the object will vibrate longer than with a wide one.  A bell or a gong has a narrow resonance, for example.  If you hold a metal object such as a bell or a plate, and you strike it repeatedly while holding it in different ways, you can easily hear how this works.  If you hold the object loosely enough at the right place, you can hear a clear tone that lasts well.  But as you hold it more and more tightly, you can hear a less and less well defined tone that lasts a shorter time.

The frequencies in this simplified example are all multiples of the fundamental one, but in real life, they may not be.  Church bells, for example, have overtones that are far enough from the multiples to create a distinctive tone.  To make the overtones become exact harmonics requires a redistribution of the metal.  The necessary calculations to do this correctly are not easy.

The connection between structures in the time domain and the frequency domain is a fundamental property of oscillations and waves.

It means, for example, that if you create a burst of sound, the frequency and the timing cannot both be determined with great accuracy.  A pure sine lasts for ever.  It has a definite frequency, but we don't know "when" it happens.  The product of the two uncertainties cannot be less than a certain value, which can be calculated.  It controls the design of a radio or TV system.  Because the system has to respond to changing signals, it cannot simply broadcast on a single frequency.  It requires a bandwidth.

The next picture shows two examples, made using a digital oscilloscope, showing the inverse relationship between decay time and spectral bandwidth.

All this (not the radio and TV, of course) was well known at the end of the nineteenth century, by which time the theory of mechanical processes was well understood.  The behaviour of electromagnetic radiation, such as light, was also well understood.  But there were huge outstanding problems in physics.  The mechanical and electromagnetic theories were totally incompatible, a problem only resolved by the theory of relativity.  Another great problem was that nobody had the slightest idea why anything existed as it did.  Why were there about ninety chemical elements?  How did they combine to form compounds?  Why did atoms have their particular masses and sizes? 

The clues were there, but nobody find out what they meant.  The spectrum of light from every known element had been analysed, and found to to consist of narrow lines of colour, much like the mechanical spectrum shown above.  Each element and compound had its own distinctive spectrum, acting like the fingerprint test and the DNA test that were invented later.  So people even knew what substances were present in the sun and stars.  But they had no idea why these substances existed.

New knowledge was needed.

Worse still, the spectra of light were incomprehensible.  It was as if we found some animals whose fingerprints were little rectangles or triangles.  In a few simple cases, such as hydrogen, formula for the frequencies were found, but this did not help.  The pattern of the frequencies was weird.  There was no sign of the series of harmonics that were so familiar in mechanics.  Atoms, to scientists, were as alien as the stars.  Here is a photograph of a mercury lamp, taken through a diffraction grating. 

Yet there was one clue.  The frequencies often seemed to occur in additive patterns.  It took the invention of quantum mechanics to solve the problems.  Just as people such as Galileo and Newton made possible a complete understanding of mechanical behaviour, so the quantum theory allowed us to understand the world of atoms and forces.  But the word "understand" is not quite right.  To our mechanically experienced minds the behaviours revealed by quantum theory are utterly strange.  Clever people can do the mathematics and get the right answers, but the processes are not always well understood.

Why should they be?  We experience a world of large objects, such as cows and cars, hockey balls and hockey sticks, which behave in ways that seem reasonable to us.  But there is no reason to suppose that the world would have been created such that every detail would behave in ways that are familiar to us.  This is true even of many of the artefacts of people, such as mobile telephones and colour TV.

Relativity theory dispose neatly of the incompatibility of mechanics and  electromagnetism, but another one took its place.  General relativity described the operation of gravitation in a very neat way, but to this day it remains unique.  It is unlike any other theory that works.  And it is utterly incompatible with quantum theory.  Neither theory has been found wanting in the areas in which they apply, yet they cannot be combined in a way that works.

Analogous to the decay of a classical oscillation is the decay of a quantum state.  Because the mass-energy of a system is related to the frequency of the wave-function, the width of a peak in a mass spectrum is inversely proportional to the mean life of the state.

There is a bit of a problem with this table.  There isn't any room in it for electrons.  Let's ignore that, although it is a pretty fundamental difficulty.

What are protons, anti-protons and mesons?  They are just a few of the many particles that were once thought to be elementary.  They are very small, having a radius of the order of 10^-15 metre.  This length is often called a fermi, in honour of Enrico Fermi, a great physicist of the mid-20th century.  

The particles that are commonly found in ordinary matter are protons, neutrons and electrons.  These were all that were known, for many years, and people naturally thought that they might be elementary, with no constituent parts.  The protons and electrons have equal, but opposite, electric charges.  Again, not knowing any better, people assumed that this quantity of charge was the smallest possible.  Anti-protons are extremely similar to protons, but have opposite charge, and opposite baryon number.  Neutrons and anti-neutrons are corresponding particles without electric charge.

Second table of particles

Here is a little table of particles as they were known at a certain stage in the history of particles.

A certain amount of order has been imposed on the particles.  We see that the neutron and the anti-neutron are not the same, but there is only one neutral pion.  It is its own anti-particle.

What are these anti-particles?  The idea of anti-particles began to emerge in 1928 with Dirac's wave equation for electrons.  He started out with the requirement that the equation would have to be compatible with relativity and with quantum mechanics.  The resulting produced four solutions.  A half of this multiplicity was interpreted as representing the spin of the particles, which hitherto had had to be inserted artificially into the mathematics.  The other half of the multiplicity seemed to predict positive and negative energies.  Eventually it was realised that the solutions referred to particles with opposite properties.  The anti-electron was discovered in cosmic rays in 1931 by Carl Anderson.

This table, with baryon number versus charge, is too simple.  It turns out that there are some particles called kaons, that do not fit into i, because there is another variable.  It was called strangeness because the kaons behaved in what was then thought to be a strange way.

Mesons

Now we need a three dimensional chart.  We will get round this by looking only at the mesons for a while.

Would you believe it - we're still not through - there are eta particles as well.

The two eta particles have no charged partners.  This little periodic table needs an explanation.  It was given independently by Gell-Mann and by Zweig.  Both suggested that all the particles are made of constituents that Gell-Mann called quarks, and Zweig called aces.  The three types of quarks are called up, down and strange.  We can make a little table for these particles.

Using only these particles and their anti-particles, Gell-Mann and Zweig could account for a great many of the known particles and some of their properties.

Mesons were taken as being made of one quarks and one anti-quark.  Baryons were made of three quarks, while anti-baryons were made of three anti-quarks.

Since there were three different quarks, there could be nine different ways of combining one quark and one anti-quark.  For the baryons, there could be 3 X 3 X 3 = 27 ways of making them.  Using some simple rules, it was possible to break down these totals as follows -   9 = 1 + 8, and 27 = 1 + 8 + 8 +10.  What this means is that the particles fall into multiplets, within which they all have very similar properties.

The rules for adding spins only allow values which are multiples of 1/2 of the unit of spin, which (1/2pi) X Planck's constant.  As a result of this we find that we  get the following multiplets, among others.

Mesons with spin 0        singlet and octet.

Mesons with spin 1        singlet and octet.

Baryons with spin 1/2    octet, including proton and neutron.

Baryons with spin 3/2    decuplet

All the particles to fill these multiplets were known, except that one particle in the decuplet, comprising three strange quarks, had not been seen.   From the properties of quarks, Gell-Mann was able to predict that it would similar to a proton, but with strangeness -3.  The other members of the decuplet had masses of about 1232, 1385 and 1533, which were almost in arithmetic progression.  The new particle, omega minus, was soon found, with a mass of about 1672, as Gell-Mann had predicted.

To accommodate all these particles into a table, with provision for baryon number, charge, strangeness and spin, would require a four-dimensional table.  Worse still, there are three more quark types, charm, bottom and top.  Even if we only include the mesons and forget spin, we need six dimensions.  And the number of mesons would be 3 X 3 X 3 X 3 X 3 X 3 = 729.   Enough these have been found to suggest that the model is likely to work.

What about the eta particles?  They are in the same box as the neutral pion?  How do they differ from it?

If you send a beam of electrons into a target of neutrons, the way they are scattered by the neutrons shows that inside a neutron there are both positive and negative charges.  Neutrons also have a magnetic field, presumably caused by the motion of these charges.  As neutrons are unstable, and extremely difficult to trap in a given place, it is not actually feasible to bombard them.

You have to do every experiment twice, first with hydrogen, and then with deuterium, which is a form of hydrogen with a neutron as well as a proton in its nucleus.  By a suitable subtraction of the two sets of results (which is not very easy) the results for neutrons can be worked out.  It turns out that protons also have a charge structure and a magnetic field.

If you think that something has a structure, it is natural to try to find out what is there.  In earlier times, people could dissect an animal or a plant in order to draw diagrams.  Later, using lenses, they could examine smaller objects.  Later still, the compound microscope enabled people to see the details of objects which were so small that they could not be seen at all with the naked eye.  By this stage it was realised a good knowledge of the theory of light and the optics of the light source and of the image-forming system was needed, in order to understand the images.  It is all too easy to create artefacts with a powerful microscope.

In a  microscope, there is a powerful source of light, which is accurately directed on to an object.  The object transmits or reflects the light, which is then gathered and used to form an image.  We are bombarding the object with light rays, which also behave as photons, or particles.  We collect the scattered particles and examine the results.

The particles of light are too small to have any detectable effect in a microscope, which is usually designed using wave optics or ray tracing.  But at the energies needed to study particles, all forms of energy exhibit particle behaviour very noticeably.

Ernest Rutherford designed experiments to bombard nuclei with alpha particles.  At that time there was a theory which suggested that atoms consisted of a ball of diffuse positive charge with electrons somehow embedded inside.  Electrons were already known to be very light, so the ball would have virtually all the mass.  When the scattering experiments were done, most of the alpha particles were barely deviated.  As they are almost 4000 times as heavy as electrons, this is hardly surprising.  A big deflection would be as amazing as expecting a ping-pong ball to deflect a golf ball.  And the diffuse positive charge could not muster enough electric field to do much either.

But when the angular distribution was drawn, it was not the right shape at all.  It did not decrease fast enough as the angles increased.  Nevertheless, the bigger deviations were still quite rare, and the work required great patience and dedication.  Rutherford made a famous remark about cannonballs and tissue paper.  Somehow there had to be a way of getting a lot of mass and a large electric field.  Rutherford went to the other extreme from the diffuse model, and guessed that all the positive charge was in the middle of the atom in a small ball.

From this assumption he could calculate the distribution of angles.  He was brilliant, and he was lucky.  He was moving into an era in which relativity theory would be needed to do proper calculations.  But because the alpha particles were moving very slowly compared with the speed of light, Rutherford could get away with Newtonian calculations.  Another nice feature of the work was that all the alphas from a given source had the same energy, which simplified the sums.  The calculations agreed with the measurements up to a certain angle.  After that, the measured scattering was too feeble.  Wrong theory?  No - these particles were penetrating the nucleus.  Now a part of the nucleus was outside a part of their trajectories, and so was not fully contributing to the deflection.  

Rutherford was able to calculate a value for the radius of the nuclei.  The result was astonishing.  The radius of a nucleus was not even 1 / 10000 of the radius of an atom.  By volume it therefore occupied less than 1/1000000000000 of the volume of the atom.  As virtually all the mass of the atom is in the nucleus, this object must be extremely dense.  Electrons are also very small, so atoms, like the solar system, are mainly empty space.

Here is a kaon-pion mass-plot for the reaction  K^- + p  >  K⁰ + pi⁺ + p.  What is going on here?  A beam of negative kaons struck a target of liquid hydrogen, containing mainly protons.  The projectiles had a high kinetic energy, enough to create extra particles. 

In this case, one pion was made.  By taking the outgoing particles in pairs, it is possible from their energies and momenta to calculate the mass of the combinations.

In this example the kaon and pion were combined, and their masses are plotted horizontally on the chart.

The curve shows the result that would be expected from statistical probability.  The small deviations are entirely attributable to random fluctuations, but the two larger peaks are not.  If we compare this with the first diagram on this page, showing the resonances in a bridge beam, the resemblance is clear.  It is repeated below the mass spectrum.

Resemblance, of course, proves nothing, but in fact these peaks in the K-pi mass spectrum are indeed sometimes called resonances.  At certain masses of the K-pi system, the combination is favoured.  If we could have a target of pions, and we could bombard it with kaons, with variable energy, then we would find that as we swept the energy upwards, there would be an increase in the probability of collisions at these particular energies.  As both pions and kaons are unstable, they cannot be used as targets.  But protons are stable.  Using proton targets (hydrogen) and pion beams, a whole series of resonances has been discovered.

With energy E and frequency F,   E  =  h X F, where h is Planck's constant.  Because h is tiny, the frequency F is unimaginably high, and cannot be directly observed.

From the case of the bridge oscillation we saw that the width of the peak was related to the decay time of the oscillations.  So it is here.  The two peaks are at 890 MeV (width 50 MeV) and 1450 MeV (width 140 MeV).  The lifetime of these K-pi combinations is very short, of the order 10^-23 seconds, a typical time for strong interactions.  These structures can in fact be regarded as particles with very short life-times.  They are no more and no less elementary than those which live longer, and can equally be regarded as composed of quarks.

In 9124, Louis de Broglie, influenced by Einstein's idea that light waves consisted of photons, boldly suggested in his doctoral thesis that all particles should behave as waves.  This idea was not inconsistent with any known observations, because the wavelengths for any normal object would be unobservably small, and the frequencies would be unobservably high.  But for elementary particles the values would be reasonable.  In 1927, Davisson and Germer used a piece of nickel as a diffraction grating, and observed the expected effects of waves.  Since then, the technique has been turned around, and the known properties of waves have been used to discover the structure of materials and molecules.   

(Please click here for an account of diffraction.)  

(Please click here for an account of X-ray diffraction.)

Particles can not only be diffracted - they can apparently undergo a transformation at the same time.  If incoming particles have enough energy they can create more particles in collisions.  If the final group of particles have the same quantum numbers as the incoming projectile, they can behave as if diffraction has occurred.  In the example below, the projectiles were K^- particles (mass 498 MeV).  The final state particles were pions (mass 140 MeV) and K* particles (mass 89).  As the final state is more massive than the initial state, the K^- cannot normally "decay" in this way.  But the target proton recoils in just the right way to balance energy and momentum.  The diagram below shows the mass spectrum of K* plus pi in such collisions.

The statistically expected spectrum is roughly half an ellipse, but the observed results are clearly biassed to low mass.  This is hardly surprising, because the mass of the incoming particle is off to the left of the graph.  Most of the shape can be accounted for simply by multiplying the statistical curve by a mass factor of 1/(m - m_K)².  What this factor says is that the further we are from the real mass of the K*, the harder it is to create the products.

The curves below represent the statistical phase space, the mass factor, and the product of the two.  The curves above represent various attempts to fit the data using more advanced calculations, including details of the interactions.  Since other processes, such as a resonance around 1350 MeV, may be contributing to the result, a perfect fit would be unlikely.

There is nothing sacrosanct about the mass of an object.  Although all electrons appear to have the same mass within the current experimental uncertainty, this is only the mass in empty space.  In situations where virtual particles can be created, or within a solid substance, the effective mass, as calculated from what the particle does, may be very different from the value in the text books. 

One peculiarity of diffraction scattering arises when people try to describe it by the mechanism of virtual particle exchange which works so well for other processes.  The problem is that apart from mass and momentum, no other properties, such as spin, are exchanged.  No particle is known with the right properties - the name pomeron is ready if it is ever discovered, named after the physicist Pomeranchuk.

In atomic nuclei, beta decay sometimes produces a negative electron, and sometimes a positive anti-electron (positron).  How can this be.  An isolated neutron decays into a proton, and electron and an anti-neutrino, because it is more massive than these three put together.  But in a nucleus the energy relationships are such that the decay looks like proton to neutron, positron and neutrino, as if the proton were heavier than the neutron.

There are not many particle properties that can be simulated in your home, but one particular ability of K⁰ mesons is one of those.  

The diagram represents a simple system that you set up at home.  Between two rigid supports, which could be chairs, we place a thread.  From this thread, two equal weights, small but reasonably heavy, are hung by threads of equal length.  The two weights are to swing at right-angles to the plane of the diagram, in and out of the picture.

If the two weights swing in phase, the effective length of the pendulums will be L2.  If they swing exactly out of phase, the length will be a little more than L1.  These two modes are stable, and are called normal modes, if the weights swing with equal amplitude.  Both modes decay exponentially, just as a single isolated pendulum would do.

But if you hold one weight and start the other one swinging, and then let go of the stationary one, a peculiar thing happens.  The amplitude of the swinging weight dies away, and the other one starts to swing.  Eventually the motions are almost completely interchanged.  Then the system slowly reverts to the original situation.  Click here to download a simulation of the double pendulum..

What happens is that the oscillation of one of the pendulums represents a superposition of the two normal modes.  These, as we have seen, have slightly different frequencies, and as the two modes progress, they go in and out of phase, making the phenomenon that in sound  we would call beats.  The next diagram attempts to show how this happens.

The amplitude of the yellow waves depends on the relative phases of the red and the green waves.  The frequency response of a coupled system like this is wider than that of either oscillator alone.  Most people would not guess that that would happen.  By choosing a suitable amount of coupling between the oscillators, the two frequency peaks can be made to overlap.  This effect has been used in radio receivers to widen the bandwidth in order to detect the signal properly.

If the signal gets bigger and smaller, we must ask where the energy goes when the amplitude is small.  For two sound waves, the phase difference between the signals depends on the position of the detector, so the timing of the beats is different in different places.  The total energy remains constant.

How can we relate this to kaons?  Earlier we saw that the mesons occur in nonets, in which there are four strange particles, two of which are neutral.  

The lightest strange particles are the kaon and anti-kaon.  The neutral kaons have strangeness + 1 and -1 and are therefore not their own anti-particles.  They can both decay into pi⁺ pi^-.  These real processes are irreversible, but they are also possible as virtual ones.  This means that the transition kaon to pions to anti-kaons is possible, albeit very weakly.  What are the consequences?

Earlier we saw that coupling two identical mechanical oscillators had the effect of creating two normal modes with different frequencies, which decayed exponentially.  And coupling the two kaons has a similar effect.  Two states are produced which decay exponentially.  They have slightly different frequencies, which means slightly different masses, and they are both superpositions of the original objects, just like the pendulums.  

These objects are represented by K1 = (K⁰ + anti-K⁰)/sqrt(2), 

and K2 = (K⁰ - anti-K⁰)/sqrt(2).  All the symbols represent the wave-functions. not the particles.  Because they are superpositions, they do not have strangeness.  It turns out that only K1 can decay into two pions.  K2 has to find another way, which is to decay into three pions.  This has a much lower probability, and so K2 has a much longer life.  Therefore if you start with a beam of pure K0 or pure anti-K0, both of which are mixtures of K1 and K2, the K1 part will decay rapidly, leaving a pure K2 beam.

Now K⁰ ani-K⁰ behave differently on hitting a nucleus.  This is because anti-K⁰ can create more final states.  The strange relations of the proton and the neutron have negative strangeness, and so does the anti-K.  Hence a process which turns a proton into a hyperon can start with an anti-K, which loses its own strangeness and becomes a pion.  Therefore, when a K2 beam hits a target, anti-K will be preferentially removed from it, leaving a preponderance of K.  This is called regeneration.  If we had started in the first place with anti-K, we would have finished up with the opposite.

The tau-theta puzzle dates from the early days of strange particle physics.  It involved the realisation that two apparently different things might actually be the same.  To read about other examples click here.

There are two kinds of upper case letters in the alphabet - ones which look the same in a vertical mirror, and ones which do not.  Objects can be divided up in this way as well.

A I H M O T U V W X Y           N S Z

With particles the situation is a little different.  What counts here is not the shape of the particle itself, but the mathematical function that describes it.  This may be composed of several parts.  There must be a part that gives the probability of finding the particle at a given place at a given time.  If the particle has spin there will be a part for that also.  This function can be referred as a wave-function.  It may have mirror symmetry or mirror anti-symmetry, as in the examples below.  These symmetries are referred to as even or positive parity, and odd or negative parity.

These wave-functions do not give the probability directly.  They have to be squared, so that the result is always positive.  They also have to be normalised, that is, the total probability must be one, when all the parts of the function are added together, corresponding to the particle be 100 % certain to exist somewhere.  Squaring is an over-simplification, though the effect is similar.  In the very first example on this page - an oscillating bridge, the total energy was found by adding the potential energy and the kinetic energy.  These in turn were formed by squaring the displacement and the velocity.

The wave-function is treated somewhat similarly, except that no actual object is oscillating, and the function isn't about energy.  The two parts are conventionally called real and imaginary.

Why is parity important?  In strong interactions, those that happen between baryons and mesons, parity is conserved.  So it is with electromagnetic interactions.  Conservation here means the the parity of the total wave-function does not change during a process.  In the absence  of any information to the contrary, people assumed that parity was always conserved.

But the kaons made them think.  The problem was that after the strange mesons were discovered, it was found that some decayed into two pions, and some into three.  Now these two states have opposite parity.  This wasn't much of a problem in the early days when few events were known, and the measurements were not very accurate.  But as time went on, and techniques improved, the mass and the mean lives of the two types of strange meson seemed to converge, and it began to look as if the values were equal.  This seemed unlikely, until some people started to think that they were examples of the same particle, and that some decays did not conserve parity.

Eventually C S Wu set up an experiment in which nuclei with spin were aligned in a magnetic field.  This had to be done at a very low temperature, to prevent thermal randomness overcoming the alignment.  if parity were conserved, the electrons from the decay would come out with a distribution that would be symmetrical with the magnetic field.  But it didn't.  It was asymmetrical, and as the apparatus was allowed to warm up slowly, randomizing the spins, the asymmetry slowly disappeared.  Nature had a handedness for weak interactions.

Note that symmetry implies a conservation law.