"The fundamental theory that describes the behaviour of nature at and below the scale of atoms."

Table of Contents:

1. Some Educated Predictions.
2. Tearing up Earlier Predictions.
3. Further proof of Particle/Wave Models.
4. Particle Wave Duality.
5. Heisenberg's Uncertainty Principle.
6. Bose-Einstein Condensates.

The Double Slit Experiment.

In the double slit experiment, we have a source of monochromatic light (meaning only 1 colour, frequency and wavelength). This light is directed towards a box.

The box has two slits in the front, allowing light to pass through them. All light that does not pass through the slits is reflected/absorbed by the wall of the box. On the other side of the box we have a piece of photogenic card that registers light by turning a different colour.

This would provide us with the ability to determine conclusively whether light is a particle or a wave because we would expect to see two different and recognisable results:

If Light is a Wave.

If light is a wave, the wave could pass through both slits at the same time. This would lead to a recognisable stripes pattern in the photogenic card as shown below.

The stripes occur because after the wave passes through both slits at the same time, it essentially becomes two new waves. These two waves then have constructive/destructive interference with each other as shown. The concept of constructive/destructive waves is discussed more fully in the electromagnetism chapter in the section on the Aether.

If a peak lines up with a peak, constructive interference occurs and a coloured stripe appears on the card. If peak lines up with trough, destructive interference occurs and no coloured stripe appears. 

This alternating pattern of stripes is called an interference pattern and each wavelength of light has its own unique interference pattern which is quite an elegant way of experimentally determining properties of light

If Light is a Particle.

If light is a particle, it could only pass through one slit. Think of a tennis ball, you would be surprised if you threw it at two holes in a wall and it went through both of them. 

Light also only travels in straight lines, so if light was a particle we would expect to see only two coloured lines in the photogenic card as seen in the diagram below. Furthermore, as particles cannot interfere with each other in the way waves can, we are certain that we will only see two lines

It is possible to see dots of light scattered around these lines because a particle of light could bounce off the wall of the slit but these are very rare so can be largely ignored.

Iteration 1.

1. Use electrons not photons. Rather than fire a beam of light we will excite electrons and fire them towards the double slit. The machine that does this has one of the best names in all of physics: the electron cannon.
2. We will cover up one of the slits such that no electrons can pass through it to get to the second screen. This essentially makes it a single slit experiment.

Before we do the experiment, let's predict its result. Because one slit is covered, and the electron is a particle we would expect to see one thick band of colour in line with the uncovered slit, seen above left. In other words, we would see half the pattern produced from the particle hypothesis.

When we perform the experiment we see that our prediction was correct and the same pattern is produced, seen below left. Off to a good start! 1/1 predictions correct.

Iteration 2.

Let's repeat the experiment but change the parameters slightly again:

1. Keep using electrons firing in a constant stream out of an electron cannon.
2. Have both slits uncovered so that an electron can pass through either one.

Logic tells us that if the electron acted as a particle when one slit was covered, it will continue to exhibit particle-like behaviour when the slit is no longer covered. We therefore predict to see two bold lines as we saw in the first section as seen above right.

However, once we do the experiment we actually see the results shown below right: we see an interference pattern! Down to 1/2 prediction success rate and left with a puzzle. When one slit is covered, the electron acts like a particle. When both slits are available, it acts like a wave.

Iteration 3.

We have a trick up our sleeves though! To definitely say whether the electron is a particle or a wave we will perform another iteration of the experiment, again subtly changing certain aspects:

1. We will continue to use an electron cannon but this time rather than fire a stream of electrons, we will send them out one at a time in such a way that there only one electron passes through the slits at any given time.
2. We will continue to keep both slits open and unblocked- there is no incentive for an electron to choose to go through one slit over the other.

To construct a prediction, we remember that the interference pattern seen when the electron behaves like a wave occurs due to two waves (one from each slit) interfering with each other. If the electron is a particle, which it is, and only one electron passes through the slits at a time, there is no way it can interfere with itself. We therefore predict to see the two (almost) solid lines above left.

Wrong again and down to 1/3 prediction success! When performing this experiment we instead see an interference pattern as presented below left.

Iteration 4.

We will do the experiment once more with a new set of rules:

1. Keep both slits open.
2. Fire electrons out one at a time.
3. Place a detector above the top slit that flashes every time it detects an electron passing through the slit. The detector does not change the motion of the electron in anyway and is solely there to observe which slit each electron passes through.

The purpose of the detector is to pin down exactly which slit the electron passes through to further examine how it can pass through both simultaneously. Because it doesn't change the path of the electron, our prediction in this stage is easy: we must get the same result as the previous iteration of the experiment and produce an interference pattern.

This arises from one of the most fundamental laws of physics: experiments are repeatable. We haven't changed any of the inputs of the experiment (apart from watching exactly which slit the electron chooses) so we wouldn't expect to see a change in outputs.

Wrong again!! Down to the dangerously low 1/4 success rate. We actually see two solid bars which suggest the electron is a particle, as seen below left.

Particle - The Photoelectric Effect.

Before the photoelectric effect can be understood, we have to discuss a new equation known as the Planck-Einstein equation: Energy (E) = Planck's constant (h) x frequency (f). Planck's constant is 6.63 x 10^-34 Joule-seconds. It is the smallest constant in physics and is a good logic check for quantum mechanics: if you get an answer smaller than h, its wrong!

The photoelectric effect is a quantum phenomena observed when monochromatic light (light of a single wavelength) is shone onto a metal surface and electrons are excited out of the surface - these are known as photoelectrons. A strange effect on the number of photoelectrons produced is seen as the frequency of the incident light is changed:

When the frequency of the incident light is low (and therefore it's energy is low according to E = hf), there are absolutely no photoelectrons removed from the surface of the metal. 

As frequency (and therefore energy) increases linearly, there are still no photoelectrons produced until suddenly at a certain frequency - known as the threshold frequency - the number of photoelectrons detected jumps up to a positive constant.

As frequency then increases beyond the threshold frequency, the number of photoelectrons produced never goes beyond this constant even if the frequency of the incident light is incredibly high.

This is a strange outcome and it baffled scientists at the time - the experiment was first done in 1887 by German physicist Heinrich Rudolf Hertz.

The understanding of light at the time was that light was a continuous wave. Newton himself was the champion of this line of argument and his world-wide fame lent it credibility but this model could not explain the strange effects of frequency on photoelectrons produced.

It was predicted that a low frequency and low energy wave would still be able to produce photoelectrons as long as the light was incident on the metal for a longer time. The idea was the low energy light would have less power but could still transfer the energy need to excite an electron as long as it could act over a long time.

Very confused by what they saw, scientists at the time also looked at the kinetic energy of the photoelectrons that were emitted:

As frequency of the incident light changes, the number of photoelectrons does not change but the kinetic energy of them does. As seen in the above graph, the kinetic energy of the photoelectrons increases linearly as frequency does above the threshold frequency.

This is modelled by the second important equation of the photoelectric effect:

E (hf)  = Φ + KEmax

Φ - Work Function. It is a property inherent to the metal used in the experiment and is officially defined as the minimum energy required to produce photoelectrons. This therefore defines the threshold frequency as Φ = hf1, where f1 is threshold frequency.

KEmax - The maximum kinetic energy of the photoelectrons. The photoelectrons released have a small range of possible KE but assuming perfect energy transfers, we know that any energy left over once the work function has been overcome must go into the KE of the photoelectrons. This is modelled by the above equation.

Wave - Electron Diffraction.

Diffraction is officially defined as 'The deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture'. Electron diffraction is the process of firing light (photons) at crystals and observing the interference produced by ionised electrons. It works by:

A crystal is composed of a lattice of atoms arranged into neat ordered rows. This essentially makes the crystal lattice a diffraction grating just like the slits in the double slit experiment. The crystal lattice acting as a diffraction grating can be seen below:

If we fire a focused stream of electrons at the crystal, we would expect to see a predictable scattering of the electrons as they are pass through the lattice.

The truly interesting part of this experiment is what happens if we place a fluorescent screen behind to the crystal. The experimental set up and results are shown below:

This is another interference pattern! Just like in the double slit experiment where two waves overlap each other to produce the characteristic crests and troughs of a single wave, the electrons scattered through the crystal have interfered with each other to produce this pattern. 

One crucial difference is the number of individual interferences occurring: in the double slit experiment only two waves are interfering but electron diffraction can have many more. This leads to a much thinner interference pattern as the chance of constructive interference is significantly lower.

This is therefore a piece of evidence for electrons (and it follows matter) being a wave because only waves can form interference patterns - to constructively or destructively interfere the electrons must have a wavelength which is in an inherent property of waves.

Wavefunctions.

In classical physics, particles are described as tiny points of space with definitive dimensions (albeit small ones!) and easily measurable properties like mass and velocity.

However, quantum physics takes a very different view on particles, describing them through mathematical models called wavefunctions. A wavefunction for a given particle has a well defined value for every point of space in the universe - this value could be negative or imagery but never exactly 0.

The symbol for a wavefunction is the greek symbol: Ψ. If you write 

Classical Uncertainty.

In the macroscopic world we live in, any measurement we take is guaranteed to have some level of uncertainty. This type of uncertainty is called classical uncertainty and it arises from a simple conjecture: all measurements are imperfect because a perfect measuring system is not achievable.

For example if you measure the width of a book with a ruler that has a resolution (the smallest reading it can take) of 1mm, you will always have an uncertainty of +-2mm because you measure either end of the book to within 1 mm. If you wanted a more accurate measure, you could use a micrometer (resolution 0.01mm) but the problem still remains - you are left with a +-0.02mm uncertainty.

All measuring systems in the macroscopic world suffer from this type of uncertainty - nothing you measure from time to distance to mass to charge is ever truly 100% accurate.

The first misconception in the uncertainty principle comes from the thought that the uncertainty arising from certain quantum measurements according to the principle is due to the same effect that causes classical uncertainty.

Quantum Uncertainty.

The uncertainty referred to by Heisenberg is subtly different to classical uncertainty.

It is independent of human or machine error in the measurement. Even if a hypothetical perfect measuring system was in place, quantum uncertainty would still exist because it is a limit on what can be determined and measured about a system, regardless of how it is measured.

One long well-defined peak that shows 100% probability of finding the particle in that position and 0% in any other positions. This is no good for our requirements though because it has no wavelength. The works of de Broglie show us that LAMDA = h/mv. In other words, the de Broglie wavelength (remember this is the wavelength of the wave function) is inversely proportional to the objects momentum.

Because this wave function obviously has no wavelength (it is just one peak as oppose to an oscillating wave), we can infer absolutely nothing about the momentum of the particle. To conclude, we could very accurately model the position of the particle but if we did, we would not be able to say anything about its momentum.

To try the opposite, we could imagine a wave function as shown below with a clear oscillating nature and an obvious wavelength. We can measure the wavelength and so know the momentum. Not all is well though! Because there are multiple peaks, there are now multiple points in space where the particle is equally likely to be found - we have lost track of its position.

The solution is to take the middle ground between these two and form what is known as a wave packet, as seen to the right. A wave packet has one maximum peak and then smaller peaks oscillating either side. Each subsequent maxima is less extreme than the one before it.

IMPORTANT NOTE: even though the maxima either side of the centre get smaller and smaller, they never vanish entirely. The normal distribution has similar properties: it has a central bell curve and trails off to either side, although it never quite touches 0.

The wave packet satisfies our original criteria:

1. It (almost) describes a particle because it has one central maxima that contains most of the probability the particle will exist there.
2. It describes a wave because although the maxima reduce in height they maintain constant spacing and so it still expresses a constant wavelength.

As it describes a wave with constant wavelength, we can use de Broglie's equation to determine the momentum of the particle.

States of Matter.

In the macroscopic world of everyday life, there exist four states of matter: solid, liquid, gas and plasma. All particles can exist in any of these states depending on external factors such as temperature and pressure.

However, in the microscopic scale of quantum phenomena, a fifth state can be achieved. Objects in this state are given the name Bose-Einstein condensates. 

When a group of particles are bound together into a Bose-Einstein condensate (henceforward called a BEC), they exhibit the quantum behaviour of one particle. This means they show strange properties such as a macroscopic wave function, superfluidity and an anomalous magnetic response (much more on this later).