Tuesday, 31 January 2017

The Exclusion Principle

Schrodinger showed that electrons in atoms could have certain energies, and they could jump between energy levels by emitting or absorbing light energy. His equation was a brilliant success, apart from one obvious question- why didn't all the electrons in an atom just sink to the lowest energy level and stay there?
We don't really know the answer to the question, but, simplifying things somewhat, one can explain it by introducing a rule that says you can't have more than two electrons with the same shape wave in an atom. This means that only two electrons can sink to the lowest energy level in an atom. After that, the electrons have to gradually fill the higher energy levels.
The rule is known as The Pauli Exclusion Principle, after Wolfgang Pauli, the Austrian physicist who first proposed it, in 1925.
The theory also says that two electrons that share the same shape wave in an atom must have opposite spins.

How quantum theory explains colours

When Schrodinger developed his famous equation he used it to model the waves of electrons in atoms. He and others found that for any given element only certain waves fitted with his equation, and those waves had particular energy values associated with them.
What was amazing was that the differences between the energies of the waves in an atom exactly match the energies of the photons of light that appeared in the element’s spectrum. Here was an explanation of colour!
It seemed that electrons could jump between energy levels in an atom. An electron in a low energy level could jump to a higher energy by absorbing a photon that had exactly the right energy to make up the difference between the two levels.  Likewise an electron in a higher energy level could drop to a lower one, giving off the energy difference in an emitted photon.

Going back to the spectrum of hydrogen, the energies of the photons that make up different coloured bands of light correspond exactly to the jumps between the energy levels of electrons in a hydrogen atom that are predicted by Schrodinger’s equation.
Imagine how exciting it must have been for Schrodinger to have made calculations using his new equation  to find that they predicted something so fundamental as the colours of elements, something that physics had never been able to explain before.

Colours and spectra

Before we can finish our exploration of spin, we need to understand a bit more about light.
We learned in an earlier post that light is made of photons, that photons are tiny ripples of electromagnetism, that each photon vibrates at a particular rate or frequency, and that the frequency of a photon determines the colour it appears to be when it interacts with our eye.

Well before the development of quantum theory scientists had found that each element had a distinctive set of colours which it would absorb when light shone upon it. The element would give off exactly the same set of colours if it was heated up sufficiently.  The set of colours is called the element’s spectrum. When we talk about the colours being given off by an element we use the phrase ‘emission spectrum’. When we talk about the colours being absorbed by the element we say ‘absorption spectrum.’ The actual colours are the same in both cases.
The pictures below show the emission spectra of hydrogen and iron, as well as the spectrum of all the frequencies of light.


In the 1920s very sensitive experiments showed that individual electrons act like little magnets. At the time, it was known that the effect of a magnet was created whenever electricity went in a circular path (through a coil of wire, for example), so physicists developed the idea that the electron created its magnetic effect by spinning like a top. That idea  was quickly found to have problems. For example, the strength of the magnetic effect seems too high to be accounted for by spinning unless the electron was spinning so rapidly that its surface (assuming it has one!) was moving faster than the speed of light (which would break another set of rules called relativity). In truth quantum theory doesn’t provide a physical picture of what is going on with an electron to account for the magnetic effect, but we still use the word ‘spin’ as a shorthand for whatever it is.

We’ve since discovered that nearly all types of fundamental particle have the ‘spin’ property. Each type of particle has a certain spin, which can never be increased or reduced.

Spin seems to come in multiples of a half. Electrons, for example  have half a unit of spin, while photons have one unit. The other non-zero values that have been observed are one and a half, two, and two and a half (other multiple of a half are possible, according to the theory, but we haven’t seen them yet).

The ‘Higgs Boson’ (of which more later) is the only fundamental particle that has been found experimentally with zero spin.

Spins don’t add-up straightforwardly. For example a helium atom has no spin, even though it is made of protons and electrons with spin.

The most thought-provoking aspect of spin (to me) is that particles with spins that are not whole numbers (which are called Fermions) behave in a very different way from particles whose spins are whole numbers (which are called Bosons), and the difference is very striking. In fact, the difference is fundamental to the behaviour of matter, as we will see in the next couple of posts.

Monday, 30 January 2017

Quantum tunnelling

Leaving aside the controversy about whether tunnelling should have only one 'l', quantum tunnelling is a real effect predicted by quantum theory which is impossible to explain with classical physics. It is another example of the way in which the wierd ideas underlying quantum theory are validated by strange effects in the real world.

According to classical physics, an object can't get over a barrier unless it has at least a certain amount of energy. The higher the barrier, the more energy is needed. The rule is black and white- if the object has enough energy it can get over the barrier- if it doesn't it can't. Think of throwing a ball over a wall. Unless you throw it hard enough it won't get over the top.

Quantum theory, on the other hand, says there's always a chance an object will get over a barrier no matter what energy it has. The chances depend on the shape of the object's quantum wave and on the height and thickness of the barrier.

To get an idea of why this happens, remember that every object has an associated wave, and there is a chance that the object could be anywhere within the area covered by its wave- the chance of an object being at a particular point depending upon the height of its wave at that point. Now, one general characteristic of waves is that they never come to an abrupt stop- they always gradually fade away at their edges. When the quantum wave of an object meets the surface of a barrier the wave doesn't suddenly drop to nothing- instead the wave gradually fades away through the space occupied by the barrier. If the barrier is a tall one, and the object doesn't have much energy, then the wave reduces in height very rapidly. On the other hand, if the barrier isn't too high compared with the energy of the object, then the wave will reduce over a greater area. If the barrier is thinner than the area over which the wave reduces to nothing then the wave will continue on the other side of the barrier, meaning that there is a chance that its object can appear there.

Quantum tunnelling explains the rates at which certain radioactive substances emit radiation. Particles at the centre of an atom of radioactive material are held in place by forces that act as a barrier to their escape. They don't have enough energy, in classical terms, to pass over the barrier, but their quantum waves do just manage to extend beyond the effective width of the barrier, so there is a small chance that the particles will appear outside the barrier that is holding them in the atom. The mathematics of quantum theory can predict the decay rates of radioactive materials very precisely.

Quantum tunnelling is routinely exploited in the design of modern electronics, and explains certain effects in biology, such as certain types of DNA mutation.

In theory quantum tunnelling applies to all objects and all barriers, but once objects get much larger than a few atoms the chances of tunnelling get very small. Since humans are trillions of times bigger than atoms, the chances of us tunnelling through barriers are vanishingly small.


Not long after Schrodinger developed his famous equation in the 1920s, Paul Dirac made a more accurate version that included the recently discovered principles of relativity. Dirac’s version of Schrodinger’s equation predicted the existence of particles that were a mirror image of electrons, and that if one of these particles and an electron were to come into contact they would destroy each other.

 At the time physicists thought this proved there was something wrong with Dirac’s equation, but in 1932 Carl Anderson discovered these ‘anti-electrons’ in experiments with ‘cosmic rays’- high energy photons that arrive at the earth from the Sun.

Since then, quantum theory has developed to suggest that every type of particle has a corresponding anti-particle, and physicists have managed to combine anti-electrons with anti-protons and anti-neutrons to form anti-atoms of anti-elements. It seems that the particles of anti-matter behave in exactly the same way as ordinary matter, except that matter and anti-matter annihilate each other.

When a particle and an anti-particle interact, they vanish, and their mass is converted into pure energy (in the form of very energetic photons). The exchange rate between mass and energy is huge- about a hundred thousand million million- so when even a tiny amount of mass is destroyed a huge amount of energy is released.

One of the many big mysteries in physics is why the universe seems to be made almost entirely of matter, with hardly any anti-matter.

The Uncertainty Principle

The Uncertainty Principle was first identified by a physicist called Heisenberg in the 1920s. It is a fundamental consequence of quantum theory which means that a particle's physical properties are never all fixed at the same time. If you measure one of them precisely you will just make another of them more uncertain.

To understand this, remember that pure energy waves, which have an exact energy associated with them, are very spread out in space. This means that when a particle has an exact energy, its wave is one of the spread-out pure energy waves, which means that the particle could be anywhere within the spread-out wave.

Since a particle can be anywhere within its wave, for a particle to have a clear position, its wave must be a tall narrow ripple. The taller and narrower the single ripple becomes, the more precise is the particle's position.

You can make a tall narrow wave by combining portions of pure energy waves with different energies, but you need lots of them. In fact the more energy waves you add to the recipe, the narrower the ripple you can make with them. But that means you have no idea what energy the particle has when it has a narrow wave, because its narrow wave is made up of lots and lots and lots of small parts of different energy waves, each with a different energy.

The example we've just worked through explains why a particle can't have a well-defined energy and a well-defined position - it can have one or the other. The uncertainty principle applies to other combinations of physical properties too. In the language of quantum theory, when two properties can't be precisely held by a particle at the same time, they are said to 'not commute'.