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.
Tuesday 31 January 2017
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.
Spin
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.
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.
Antimatter
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'.
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'.
Friday 27 January 2017
Entanglement
We've talked about individual particles each having a wave, but quantum theory includes the possibility that the waves of similar particles combine to form multi-particle waves. In some ways the idea is straightforward, as we've already talked about the fact that overlapping waves simply add-up to make another hybrid wave of a more complicated shape. However, in quantum theory the idea of particle waves combining leads to yet more strange results.
You will remember from earlier posts that a particle's wave can suddenly change its shape when a property of the particle is measured. Imagine, then, a wave that is the combination of the individual waves of two particles. According to quantum theory, if you make a measurement on one of the two particles, the measurement can cause the combined wave to suddenly change shape- that means the measurement affects both the component of the wave belonging to the particle you are measuring and the other component belonging to the particle you are not measuring. So, making a measurement on one particle can affect the wave belonging to another particle. To see why this is weird, imagine the two particles start close together but then spread apart. The particles could be millions of miles apart, but, according to quantum theory, if their waves are entangled then making a measurement on one of the particles can have an effect on the other, and the effect is immediate.
You will remember from earlier posts that a particle's wave can suddenly change its shape when a property of the particle is measured. Imagine, then, a wave that is the combination of the individual waves of two particles. According to quantum theory, if you make a measurement on one of the two particles, the measurement can cause the combined wave to suddenly change shape- that means the measurement affects both the component of the wave belonging to the particle you are measuring and the other component belonging to the particle you are not measuring. So, making a measurement on one particle can affect the wave belonging to another particle. To see why this is weird, imagine the two particles start close together but then spread apart. The particles could be millions of miles apart, but, according to quantum theory, if their waves are entangled then making a measurement on one of the particles can have an effect on the other, and the effect is immediate.
The meaning of measurement, and decoherence
Quantum theory says that a particle has an associated wave, and that the wave can change shape suddenly when a measurement takes place. For example, when you measure the position of a particle that has a widely spread-out wave, its wave suddenly changes to a single tall spike where the particle has appeared.
There has been a lot of debate about why these changes happen when a 'measurement' occurs. Some quite famous physicists have even speculated that measurement is somehow linked to observation by a sentient being, and that somehow consciousness makes the fundamental particles jump from vague states into certain ones.
Nowadays it is hard to find any mainstream physicist who supports such mystical interpretations. My view is that by 'measurement' we just mean an interaction between the particle and whatever lumps of matter constitute the measuring device. It seems that the particle is most likely to have a wavelike behaviour when it is moving in free space or interacting with another quantum particle, but behaves in a more pointlike way when it interacts something much larger (like the screen behind the two slits).
A related question is why big objects don't behave in wavy quantum ways. Humans are made out of protons and electrons which behave in weird quantum ways individually, so why does the weirdness disappear when large numbers of particles are brought together to make big objects?
One train of thought that is being investigated is an idea called 'decoherence'. When you have a couple of waves, they can interfere with each other to give noticeable interference effects. In an earlier post we talked about the example of two separate circular waves on an otherwise glassy pond. Where the waves cross you will see very clear patterns. But as more and more waves get added to the surface of the pond you end up with a random choppy mess with no discernible pattern to it. Crudely, that's the idea behind decoherence. When lots of quantum particles are brought together their waves overlap is such a variety of different ways that any detectable quantum patterns are overwritten by noise.
There has been a lot of debate about why these changes happen when a 'measurement' occurs. Some quite famous physicists have even speculated that measurement is somehow linked to observation by a sentient being, and that somehow consciousness makes the fundamental particles jump from vague states into certain ones.
Nowadays it is hard to find any mainstream physicist who supports such mystical interpretations. My view is that by 'measurement' we just mean an interaction between the particle and whatever lumps of matter constitute the measuring device. It seems that the particle is most likely to have a wavelike behaviour when it is moving in free space or interacting with another quantum particle, but behaves in a more pointlike way when it interacts something much larger (like the screen behind the two slits).
A related question is why big objects don't behave in wavy quantum ways. Humans are made out of protons and electrons which behave in weird quantum ways individually, so why does the weirdness disappear when large numbers of particles are brought together to make big objects?
One train of thought that is being investigated is an idea called 'decoherence'. When you have a couple of waves, they can interfere with each other to give noticeable interference effects. In an earlier post we talked about the example of two separate circular waves on an otherwise glassy pond. Where the waves cross you will see very clear patterns. But as more and more waves get added to the surface of the pond you end up with a random choppy mess with no discernible pattern to it. Crudely, that's the idea behind decoherence. When lots of quantum particles are brought together their waves overlap is such a variety of different ways that any detectable quantum patterns are overwritten by noise.
Two slits
Imagine shining a beam of light through a tall narrow slit onto a screen. What you see on the screen is a tall narrow bar of light on the screen opposite the slit. It is a pretty easy thing to imagine- the light passes through the slit in a straight line and shines on the screen making a bright vertical bar.
Now imagine closing that slit and opening a second one just shifted a bit to the side of the first. Now you will see another vertical bar of light on the screen opposite the new slit. The vertical bar of light you see now is just like the first, but it is shifted to the side a bit because the slit has moved to one side.
Now, let's open both slits. You might now expect to see two vertical bars of light, one opposite each slit, but you don't. Instead what you find is a pattern of light and dark vertical stripes!
The stripes- which are called interference fringes- are easily explained by assuming that the light coming through the slits is a wave.
You remember that when two waves overlap they add together. Where a peak or trough meets another peak or trough you get a doubly tall peak or doubly deep trough, but when a peak meets a trough they cancel each other out and you get nothing.
When the light comes through two slits it is two separate waves, one coming from each slit. At certain points on the screen the two waves arrive with their peaks and troughs aligned, and you get brightness on the screen. At other places the waves arrive with the troughs of one lined up with the peaks of the other, and they cancel each other out, so you get a dark spot. There are hundreds of visual explanations of this on the Internet, so check out any of them if you can't imagine my written description.
The key thing to remember, however, is that when waves come through two slits they interfere with each other, reinforcing themselves in some places and cancelling each other out in others so you get alternating bars of light and dark where the waves meet.
Famously, the two slits experiment has been performed with fundamental particles, including electrons, protons, atoms, etc. And what is found is that the distribution of particles hitting the screen beyond the slits forms interference fringes. There are vertical areas where lots of particles land alternating with vertical areas where no particles land. The interference pattern happens even if the particles are fired through the slits one at a time. The only theory (currently) that accounts for this is that the particle has wave-like property, and the particle's wave goes through both slits, and the two component of the particle's wave interfere with each other when they recombine on the other side of the slits.
This experiment works even when the distance between the slits is much greater than the size the particle appears to have when it manifests itself as a particle. This seems to suggest to some physicists that the particle is in its wavelike form when it is passing through the slits, and manifests in its much smaller point-like form when it hits the screen the other side of the slits. What 'really' happens is still anyone's guess.
Now imagine closing that slit and opening a second one just shifted a bit to the side of the first. Now you will see another vertical bar of light on the screen opposite the new slit. The vertical bar of light you see now is just like the first, but it is shifted to the side a bit because the slit has moved to one side.
Now, let's open both slits. You might now expect to see two vertical bars of light, one opposite each slit, but you don't. Instead what you find is a pattern of light and dark vertical stripes!
The stripes- which are called interference fringes- are easily explained by assuming that the light coming through the slits is a wave.
You remember that when two waves overlap they add together. Where a peak or trough meets another peak or trough you get a doubly tall peak or doubly deep trough, but when a peak meets a trough they cancel each other out and you get nothing.
When the light comes through two slits it is two separate waves, one coming from each slit. At certain points on the screen the two waves arrive with their peaks and troughs aligned, and you get brightness on the screen. At other places the waves arrive with the troughs of one lined up with the peaks of the other, and they cancel each other out, so you get a dark spot. There are hundreds of visual explanations of this on the Internet, so check out any of them if you can't imagine my written description.
The key thing to remember, however, is that when waves come through two slits they interfere with each other, reinforcing themselves in some places and cancelling each other out in others so you get alternating bars of light and dark where the waves meet.
Famously, the two slits experiment has been performed with fundamental particles, including electrons, protons, atoms, etc. And what is found is that the distribution of particles hitting the screen beyond the slits forms interference fringes. There are vertical areas where lots of particles land alternating with vertical areas where no particles land. The interference pattern happens even if the particles are fired through the slits one at a time. The only theory (currently) that accounts for this is that the particle has wave-like property, and the particle's wave goes through both slits, and the two component of the particle's wave interfere with each other when they recombine on the other side of the slits.
This experiment works even when the distance between the slits is much greater than the size the particle appears to have when it manifests itself as a particle. This seems to suggest to some physicists that the particle is in its wavelike form when it is passing through the slits, and manifests in its much smaller point-like form when it hits the screen the other side of the slits. What 'really' happens is still anyone's guess.
Wonderful Wonderful Copenhagen
The explanation of quantum theory I've given in this blog follows (roughly!) some ideas first thrashed out by famous physicists working in Copenhagen in the 1920s. Such explanations are loosely categorised as examples of the 'Copenhagen interpretation' of quantum mechanics.
There are other interpretations of quantum theory, which differ mainly in how they interpret the significance of the particle wave and the discontinuous jumps in the shape of the wave that happen when you make measurements on the particle.
The fact that there are so many completing interpretations indicates how uncertain we are about what it 'really' means.
I am pretty open minded about the different interpretations. I have adopted the Copenhagen model here because it's the one I grew up with and have thought most about. My gut feeling is that none of the current interpretations is completely right.
Some people think the particle wave doesn't have any physical manifestation- it is just an abstract mathematical value that is varying in time and space. Others (including me) think there must be something wavy going on with fundamental particles, which brings us to the next post and the famous 'two slits' phenomenon...
There are other interpretations of quantum theory, which differ mainly in how they interpret the significance of the particle wave and the discontinuous jumps in the shape of the wave that happen when you make measurements on the particle.
The fact that there are so many completing interpretations indicates how uncertain we are about what it 'really' means.
I am pretty open minded about the different interpretations. I have adopted the Copenhagen model here because it's the one I grew up with and have thought most about. My gut feeling is that none of the current interpretations is completely right.
Some people think the particle wave doesn't have any physical manifestation- it is just an abstract mathematical value that is varying in time and space. Others (including me) think there must be something wavy going on with fundamental particles, which brings us to the next post and the famous 'two slits' phenomenon...
Measurement probabilities
In the last post we mentioned that waves on a guitar string could be 'pure' waves- each with a fixed number of equal vibrating segments and a specific rate of vibration (or frequency)- or, more generally, the string could vibrate as a more-complicated jumble of pure states.
The overall shape of a jumbled wave depends on exactly what combination of pure states make it up. If you think of the pure states as being ingredients in a recipe for a complicated wave, then we can adjust the relative quantities of each of the ingredients to make different hybrid waves.
You can actually hear this effect if you pluck a guitar string in different places along its length. If you pluck the string near one end you will get a more' twangy' sound than if you pluck it in the middle. The reason is that plucking it near the end sets off more of the higher frequency pure waves, thus changing the recipe of the sound somewhat.
The same holds true in quantum theory for the particle waves. Depending on the surroundings the particle finds itself in, its wave can be any mix of pure energy states (or eigenstates). In such a jumbled state the particle doesn't have a well-defined energy. As we mentioned in the last post, when you measure the particle's energy you do always find that the particle's jumbled wave does suddenly change into one of the pure energy waves, and the energy you measure is the energy associated with that pure wave.
The weird thing is that you can never tell which of the pure energy waves the particle's jumbled wave will switch into, and so you can never be sure what result you will get if you try to measure the energy of a particle in a jumbled state. Quantum theory contains an essential degree of uncertainty.
However, quantum theory does tell you the probability of getting a particular result when you measure the energy of a particle with a jumbled wave. It turns out that the probability of measuring a particular energy value is proportional to how much of that value's associated pure energy wave was in the recipe for the jumbled wave.
That's a difficult idea to express without maths, so let's go over it again in a different way. Let's suppose that 10% of particular pure energy wave was one of the ingredients for a jumbled wave of a particle. If you measure that particle's energy, there's a 10% chance that the answer you get will be the energy of that pure wave. If you have a particle with a different jumbled energy wave, which includes 50% of a certain pure energy wave, then there will be a 50% chance that the result of measuring the particle's energy will be the energy associated with that pure energy wave.
So, to recap, particles can have 'pure' energy waves, each of which has a specific energy associated with it. Conversely a particle can have a jumbled wave, which is made up of a bit of one pure wave, and a bit of some other pure wave, and so on. When you measure the energy of a particle, the particle's jumbled wave switches at random to become one of the pure waves in its recipe, and the probability of it switching to a particular pure wave depend upon how much of that pure wave is in the recipe.
As an analogy, imagine you made a jar of mixed spices with some pepper, some mace, some ginger, some coriander, some cumin, etc. Then you asked someone to taste it and say which single spice it tastes like. You won't know for sure which answer you will get, but probably the chance of getting one particular answer- ginger say,- will depend on how much ginger is in the mixture compared with anything else.
The weird thing about quantum theory, however, is that once you've measured the energy of a particle with a jumbled wave, its wave switches to a single pure energy wave. It's as if your bottle of mixed spice turns into pure ginger if someone tastes it and thinks it is most like ginger!
he answer you get is one of the eigenvalues ButA particle wavemiddle you'll get a more mellow sound than if you
The overall shape of a jumbled wave depends on exactly what combination of pure states make it up. If you think of the pure states as being ingredients in a recipe for a complicated wave, then we can adjust the relative quantities of each of the ingredients to make different hybrid waves.
You can actually hear this effect if you pluck a guitar string in different places along its length. If you pluck the string near one end you will get a more' twangy' sound than if you pluck it in the middle. The reason is that plucking it near the end sets off more of the higher frequency pure waves, thus changing the recipe of the sound somewhat.
The same holds true in quantum theory for the particle waves. Depending on the surroundings the particle finds itself in, its wave can be any mix of pure energy states (or eigenstates). In such a jumbled state the particle doesn't have a well-defined energy. As we mentioned in the last post, when you measure the particle's energy you do always find that the particle's jumbled wave does suddenly change into one of the pure energy waves, and the energy you measure is the energy associated with that pure wave.
The weird thing is that you can never tell which of the pure energy waves the particle's jumbled wave will switch into, and so you can never be sure what result you will get if you try to measure the energy of a particle in a jumbled state. Quantum theory contains an essential degree of uncertainty.
However, quantum theory does tell you the probability of getting a particular result when you measure the energy of a particle with a jumbled wave. It turns out that the probability of measuring a particular energy value is proportional to how much of that value's associated pure energy wave was in the recipe for the jumbled wave.
That's a difficult idea to express without maths, so let's go over it again in a different way. Let's suppose that 10% of particular pure energy wave was one of the ingredients for a jumbled wave of a particle. If you measure that particle's energy, there's a 10% chance that the answer you get will be the energy of that pure wave. If you have a particle with a different jumbled energy wave, which includes 50% of a certain pure energy wave, then there will be a 50% chance that the result of measuring the particle's energy will be the energy associated with that pure energy wave.
So, to recap, particles can have 'pure' energy waves, each of which has a specific energy associated with it. Conversely a particle can have a jumbled wave, which is made up of a bit of one pure wave, and a bit of some other pure wave, and so on. When you measure the energy of a particle, the particle's jumbled wave switches at random to become one of the pure waves in its recipe, and the probability of it switching to a particular pure wave depend upon how much of that pure wave is in the recipe.
As an analogy, imagine you made a jar of mixed spices with some pepper, some mace, some ginger, some coriander, some cumin, etc. Then you asked someone to taste it and say which single spice it tastes like. You won't know for sure which answer you will get, but probably the chance of getting one particular answer- ginger say,- will depend on how much ginger is in the mixture compared with anything else.
The weird thing about quantum theory, however, is that once you've measured the energy of a particle with a jumbled wave, its wave switches to a single pure energy wave. It's as if your bottle of mixed spice turns into pure ginger if someone tastes it and thinks it is most like ginger!
he answer you get is one of the eigenvalues ButA particle wavemiddle you'll get a more mellow sound than if you
Thursday 26 January 2017
Eigenstates and Eigenvalues
When we discussed a standing wave on a string we saw that it
was possible to set up waves with any number of vibrating segments of equal
length. Here’s the picture showing four possible ways to make a string vibrate. You can imagine what the other ways look like- just keep increasing the number of segments.
Each of the four wave patterns has its own rate of vibration, or frequency, which is proportional to the number of vibrating segments. A wave with three segments has a frequency that is three times higher than a wave with one segment.
When you make a string vibrate in one of these patterns it is called a pure frequency state, or an 'eigenstate' of frequency. The frequency of the eigenstate is called the frequency 'eigenvalue'. The prefix 'eigen' is a German term meaning intrinsic.
In real life if you pluck a guitar string what actually happens is that some mixture of all these pure waves, or eigenstates, start vibrating together, and the overall pattern of movement can be very complicated. If you ask what is the frequency of the resulting hybrid wave, the answer is that it no longer has a single frequency- it is vibrating with a mix of frequencies all mingled together.
A very similar effect occurs in quantum theory. In the same way that a guitar string can vibrate in one of many distinctive eigenstates, each of which has a well-defined frequency, so the wave of a particle can be one of several eigenstates of the particle, each with a well-defined energy. But just as a guitar string can also vibrate in a hybrid way as a jumble of eigenstates with no well-defined frequency (in fact that's how guitar strings usually vibrate), so the wave of a particle can be a jumble of energy eigenstates with no well-defined energy.
What happens when you measure the energy of a particle in a jumbled state is that the wave of the particle switches from being a jumble of different eigenstates into being a single eigenstate with a single energy. It is exactly as if you had roughly plucked a guitar string to make it vibrate in a jumbled way, and when you tried to measure the frequency of the jumbled vibration the guitar string instantly started to vibrate in one of the pure frequency states rather than vibrating in a jumbled way!
Non-commuting operators
The title of this post is a technical term that describes another perplexing aspect of quantum theory. In everyday life we get used to the idea that physical objects have very definite characteristics. Your laptop has a definite width, position, weight etc, and it has all these properties at the same time. According to quantum theory, some characteristics of particles are fundamentally incompatible with each other. An electron, for example, can't have a definite energy and a definite location at the same time.
Again, there is nothing in everyday life that is like this implication of quantum theory. The best we can do is to give a rough idea of what is happening.
Suppose you had a particular sum of money. It is possible to express that sum of money in many different ways which are financially exactly the same but physically different. For example, you could express it in dollars or in euros or in pounds, and whatever currency it was in, you could have it in different combinations of notes and coins.
In real life you can have your money in US coins, for example, and you could take one of the coins- say a quarter- and measure both its width and its thickness. Quantum theory doesn't allow that. It is as if quantum theory says that if you want to measure coin widths then your money will always be in US coins, but if you want to measure coin thicknesses your money will always be in UK coins. So let's suppose you start with a UK penny and measure it's thickness. When you try to measure its width it changes to a US quarter and no longer has the thickness you measured when it was a penny. And having measured the width of the US quarter, when you try to measure its thickness it turns into a UK two pound coin and no longer has the width you've just measured, and so on. In this crazy analogy, you can pin down one of the dimensions at a time, but when you try to pin down the next the previous one changes.
Quantum theory says that electron energy and position are just like that. When the electron has a definite energy it no longer has a definite position, and when it has a definite position it no longer has a definite energy. However, perhaps the most amazing aspect of quantum theory is the way in which it predicts probabilities of experimental measurements of such incompatible quantities. We will look at this in the next post.
Again, there is nothing in everyday life that is like this implication of quantum theory. The best we can do is to give a rough idea of what is happening.
Suppose you had a particular sum of money. It is possible to express that sum of money in many different ways which are financially exactly the same but physically different. For example, you could express it in dollars or in euros or in pounds, and whatever currency it was in, you could have it in different combinations of notes and coins.
In real life you can have your money in US coins, for example, and you could take one of the coins- say a quarter- and measure both its width and its thickness. Quantum theory doesn't allow that. It is as if quantum theory says that if you want to measure coin widths then your money will always be in US coins, but if you want to measure coin thicknesses your money will always be in UK coins. So let's suppose you start with a UK penny and measure it's thickness. When you try to measure its width it changes to a US quarter and no longer has the thickness you measured when it was a penny. And having measured the width of the US quarter, when you try to measure its thickness it turns into a UK two pound coin and no longer has the width you've just measured, and so on. In this crazy analogy, you can pin down one of the dimensions at a time, but when you try to pin down the next the previous one changes.
Quantum theory says that electron energy and position are just like that. When the electron has a definite energy it no longer has a definite position, and when it has a definite position it no longer has a definite energy. However, perhaps the most amazing aspect of quantum theory is the way in which it predicts probabilities of experimental measurements of such incompatible quantities. We will look at this in the next post.
Collapsing waves
One of the conceptual difficulties of quantum theory- for some physicists at least- is to do with the change that a particle's wave undergoes when the particle manifests itself through an interaction with other matter. This is often given the name 'the collapse of the wave packet'. We'll look into it in a bit more detail now.
There is nothing in principle in quantum theory that stops a particle's wave from spreading out very widely. If you remember that the height of the wave at any given point represents the probability of the particle appearing there, then a widely spread-out wave means that the particle could appear anywhere over a wide area.
When the particle does actually make its presence felt by interacting with other matter then we know much more precisely where it is. The particle's wave suddenly contracts to become a single tall ripple where the particle has appeared.
For some physicists, the idea that the wave suddenly shrinks is a problem. In theory, at least, the wave could have spread enormously beforehand, so to assume it shrinks to a point in an instant conflicts with other principles in physics that say that sudden changes, or 'discontinuities' are unnatural, so any physical theory that includes or implies them must be wrong.
You only have to read a physics forum on the Internet to see that there are lots of conflicting views on this subject from within the physics community. Some argue that the particle wave doesn't really have any physical existence- it's just a measure of our uncertainty about where the particle is. There are other mathematical formalisms of quantum theory in which the particle waves don't explicitly appear.
Another complexity that we have ignored so far is that particle waves overlap and 'interfere' with each other. Imagine the surface of a glassy pond upon which you cause a single ripple to spread-out in a beautifully pure circular wave, the evolution of which you can clearly see. Suppose someone else on the pond causes another ripple in a different place. That too spreads out smoothly. But where the waves meet they create a more choppy pattern. Where the peak of one wave coincides with the peak of another they add together to give a doubly tall peak. Likewise the troughs. And when the peak of one wave coincides with the trough of another they cancel each other out. The resulting pattern changes all the time as the waves cross each other. Now imagine more and more waves get started by people all over the pond, some being gentle waves from the flick of a finger, others being big waves where someone had dropped a huge boulder. As more and more waves coincide and overlap the surface gets more and more chaotic and you can no longer recognise the existence of the individual waves, even though mathematically they are all still there, and the random chaos you are seeing is just the sum of thousands of beautifully symmetric waves.
The real universe is more like the choppy chaotic surface of the ocean than the ideal glassy pond with a single ripple. Many of the calculations that physicists perform using quantum theory ignore the individual interactions between a particle and the millions of other individual particles that make up real physical objects. Instead the physicists adopt simplified models in order to make the calculations easier. It is amazing that in spite of the simplifying assumptions the models can still make very precise predictions of the physical properties of matter.
There is nothing in principle in quantum theory that stops a particle's wave from spreading out very widely. If you remember that the height of the wave at any given point represents the probability of the particle appearing there, then a widely spread-out wave means that the particle could appear anywhere over a wide area.
When the particle does actually make its presence felt by interacting with other matter then we know much more precisely where it is. The particle's wave suddenly contracts to become a single tall ripple where the particle has appeared.
For some physicists, the idea that the wave suddenly shrinks is a problem. In theory, at least, the wave could have spread enormously beforehand, so to assume it shrinks to a point in an instant conflicts with other principles in physics that say that sudden changes, or 'discontinuities' are unnatural, so any physical theory that includes or implies them must be wrong.
You only have to read a physics forum on the Internet to see that there are lots of conflicting views on this subject from within the physics community. Some argue that the particle wave doesn't really have any physical existence- it's just a measure of our uncertainty about where the particle is. There are other mathematical formalisms of quantum theory in which the particle waves don't explicitly appear.
Another complexity that we have ignored so far is that particle waves overlap and 'interfere' with each other. Imagine the surface of a glassy pond upon which you cause a single ripple to spread-out in a beautifully pure circular wave, the evolution of which you can clearly see. Suppose someone else on the pond causes another ripple in a different place. That too spreads out smoothly. But where the waves meet they create a more choppy pattern. Where the peak of one wave coincides with the peak of another they add together to give a doubly tall peak. Likewise the troughs. And when the peak of one wave coincides with the trough of another they cancel each other out. The resulting pattern changes all the time as the waves cross each other. Now imagine more and more waves get started by people all over the pond, some being gentle waves from the flick of a finger, others being big waves where someone had dropped a huge boulder. As more and more waves coincide and overlap the surface gets more and more chaotic and you can no longer recognise the existence of the individual waves, even though mathematically they are all still there, and the random chaos you are seeing is just the sum of thousands of beautifully symmetric waves.
The real universe is more like the choppy chaotic surface of the ocean than the ideal glassy pond with a single ripple. Many of the calculations that physicists perform using quantum theory ignore the individual interactions between a particle and the millions of other individual particles that make up real physical objects. Instead the physicists adopt simplified models in order to make the calculations easier. It is amazing that in spite of the simplifying assumptions the models can still make very precise predictions of the physical properties of matter.
The meaning of the particle waves
So far we've explained that according to quantum theory every particle has an associated wave. The shape of the wave depends upon several factors, and determines- among other things- the energy of the particle.
There have been lots and lots and lots of experiments that have proved that the idea of a particle having an associated wave is a good way to model the way in which nature works. The predictions of quantum theory agree with experiments to an astonishing degree of precision.
What is less clear is what the wave actually is, and how it is associated with the particle. Indeed, it is possible that the wave and the particle are two manifestations of the same thing, and that the thing acts like a particle in some circumstances and like a wave in others. There is nothing in everyday life that is a good analogy of this mixture of a wave and a particle, and that is one of the reasons why quantum theory is mysterious.
The mainstream interpretation of quantum theory is that the wave represents the probability of the particle materialising in a given place, and there is a lot of experimental evidence to support this. If you imagine the example of the wave that you send down a long rope by twitching one end of the rope. At any given time there are parts of the rope either side of the moving ripple that are completely undisturbed, and the ripple itself gets gradually larger, reaches a maximum, then gets smaller. If the ripple on the rope were the matter wave of an electron say, what that would mean is that there was no chance whatsoever of the electron appearing either side of the ripple. The chances of the electron appearing would gradually increase as you approach the maximum peak of the ripple then gradually decreased as you passed downs the other side of it.
What is very important to understand is that this interpretation does not assume that the electron even has a position until it materialises by interacting with other matter. The interpretation assumes that the electron is somehow smeared out until it materialises. We will consider this in more detail in the next post.
There have been lots and lots and lots of experiments that have proved that the idea of a particle having an associated wave is a good way to model the way in which nature works. The predictions of quantum theory agree with experiments to an astonishing degree of precision.
What is less clear is what the wave actually is, and how it is associated with the particle. Indeed, it is possible that the wave and the particle are two manifestations of the same thing, and that the thing acts like a particle in some circumstances and like a wave in others. There is nothing in everyday life that is a good analogy of this mixture of a wave and a particle, and that is one of the reasons why quantum theory is mysterious.
The mainstream interpretation of quantum theory is that the wave represents the probability of the particle materialising in a given place, and there is a lot of experimental evidence to support this. If you imagine the example of the wave that you send down a long rope by twitching one end of the rope. At any given time there are parts of the rope either side of the moving ripple that are completely undisturbed, and the ripple itself gets gradually larger, reaches a maximum, then gets smaller. If the ripple on the rope were the matter wave of an electron say, what that would mean is that there was no chance whatsoever of the electron appearing either side of the ripple. The chances of the electron appearing would gradually increase as you approach the maximum peak of the ripple then gradually decreased as you passed downs the other side of it.
What is very important to understand is that this interpretation does not assume that the electron even has a position until it materialises by interacting with other matter. The interpretation assumes that the electron is somehow smeared out until it materialises. We will consider this in more detail in the next post.
Wednesday 25 January 2017
Schrodinger's equation
The idea of a wave naturally explains the properties of electrons in atoms. If you remember, we said that an atom is made of heavy protons at its centre, somehow surrounded by electrons, but it wasn't clear why the electrons weren't just pulled into the centre by the electromagnetic attraction from the protons.
At the heart of quantum theory is an equation formulated in the 1920s by a physicist called Schrodinger. If you could understand the mathematics behind it (which is hard to explain) you would be astonished by the elegance of the idea it expresses.
The equation explains the way in which the shape of a particle's wave is determined by the forces on the particle, and how the energy of a particle is determined by the shape of the wave. When you apply the equation to an electron in an atom the equation predicts that the energy of an electron can only have certain values which go up or down in steps of different sizes, rather than being smoothly variable. This is the origin of the famous term a 'quantum leap' or a 'quantum jump'. An electron's energy can go up or down the steps, but can never take an intermediate value.
An electron can 'jump' to a higher energy by absorbing a photon of the right frequency. Do you remember that photons were particles of light, and their energy depended on their frequency? If a photon passes an electron in an atom and has just the right energy then the electron can absorb it and use its energy to jump to a higher energy step.
The process works in reverse. An electron on a higher energy step can jump to a lower step by emitting a photon- the photon carries away the energy that the electron has discarded by jumping down.
Do you remember when we spoke of standing waves on strings or hoops we saw that the wave could have multiple vibrating segments, and the more segments it had the higher the frequency at which it vibrated and the higher the energy it had? That principle underlies the behaviour of electrons. Schrodinger's equation says that the wave of an electron in an atom can have a (whole) number of vibrating segments. When an electron absorbs a photon to gain energy, the electron's wave around the atom changes to get more vibrating segments. Conversely, when the electron emits a photon, the wave changes to have fewer vibrating segments.
At the heart of quantum theory is an equation formulated in the 1920s by a physicist called Schrodinger. If you could understand the mathematics behind it (which is hard to explain) you would be astonished by the elegance of the idea it expresses.
The equation explains the way in which the shape of a particle's wave is determined by the forces on the particle, and how the energy of a particle is determined by the shape of the wave. When you apply the equation to an electron in an atom the equation predicts that the energy of an electron can only have certain values which go up or down in steps of different sizes, rather than being smoothly variable. This is the origin of the famous term a 'quantum leap' or a 'quantum jump'. An electron's energy can go up or down the steps, but can never take an intermediate value.
An electron can 'jump' to a higher energy by absorbing a photon of the right frequency. Do you remember that photons were particles of light, and their energy depended on their frequency? If a photon passes an electron in an atom and has just the right energy then the electron can absorb it and use its energy to jump to a higher energy step.
The process works in reverse. An electron on a higher energy step can jump to a lower step by emitting a photon- the photon carries away the energy that the electron has discarded by jumping down.
Do you remember when we spoke of standing waves on strings or hoops we saw that the wave could have multiple vibrating segments, and the more segments it had the higher the frequency at which it vibrated and the higher the energy it had? That principle underlies the behaviour of electrons. Schrodinger's equation says that the wave of an electron in an atom can have a (whole) number of vibrating segments. When an electron absorbs a photon to gain energy, the electron's wave around the atom changes to get more vibrating segments. Conversely, when the electron emits a photon, the wave changes to have fewer vibrating segments.
The wave-particle duality
According to quantum theory, the fundamental particles from which everything is made combine the properties of a particle and of a wave. This idea is called the wave-particle duality, and is one of the confusing aspects of the theory.
More specifically, the theory says that every particle has a wave associated with it. Quite what is waving we don't know, but the idea leads to extremely accurate predictions of the properties of atoms (as we will see in the next post).
The idea is confusing because although the waves can be quite spread-out, whenever a particle interacts with a large number of other particles it always seems to do so at a very localised point, a point much smaller than the spread of its wave.
Imagine the following idea. Suppose there was a completely flat lake, and you dropped a stone into it. You would make a wave which would ripple out in a circle from where you had dropped the stone. If you had people in boats on the lake measuring the height of the surface of the water they would all be able to record when the wave went past them. Quantum theory says that the waves associated with fundamental particles behave in a different way. They seem to spread out like the wave on the water does, but the first time they are detected they seem to contract to a point at the detector. It is almost as if the wave on the water suddenly contracted to a spike at the first boat it made contact with.
Photons- particles of light
Light is made up of tiny particles called photons.
A photon is like an isolated ripple of electromagnetism. If you remember back to an earlier post, we imagined a long rope tethered at one end. If you get the other end of the rope and twitch it once, you can send a single ripple travelling down the rope to the other end. If you continue twitching the rope you can send a continuous wave down the rope, with one ripple following after another. What you can't do, however, is to send a fraction of a ripple- vibrations don't work that way. It's the same with light. A photon is the smallest unit of light that can be made.
Do you remember earlier we talked about frequency and energy? We said that a rapid vibration contains more energy than a slow one (all other things being equal). That is true of photons too. The higher the frequency of the photon, the more energy it has.
The energy of a photon can be calculated by multiplying its frequency (ie the number of times it vibrates in a second), by a number called Planck's constant. It is named after Max Planxk who was a famous physicist.
Planck's constant is a tiny number- it is just under seven divided by ten thousand million million million million million, so the energy in most photons is tiny. That said, elecrttrons are tiny too, so a single photon can make an electron move.
A photon is like an isolated ripple of electromagnetism. If you remember back to an earlier post, we imagined a long rope tethered at one end. If you get the other end of the rope and twitch it once, you can send a single ripple travelling down the rope to the other end. If you continue twitching the rope you can send a continuous wave down the rope, with one ripple following after another. What you can't do, however, is to send a fraction of a ripple- vibrations don't work that way. It's the same with light. A photon is the smallest unit of light that can be made.
Do you remember earlier we talked about frequency and energy? We said that a rapid vibration contains more energy than a slow one (all other things being equal). That is true of photons too. The higher the frequency of the photon, the more energy it has.
The energy of a photon can be calculated by multiplying its frequency (ie the number of times it vibrates in a second), by a number called Planck's constant. It is named after Max Planxk who was a famous physicist.
Planck's constant is a tiny number- it is just under seven divided by ten thousand million million million million million, so the energy in most photons is tiny. That said, elecrttrons are tiny too, so a single photon can make an electron move.
Light
The effect of electromagnetic vibrations depends upon their
frequency. If the force vibrates around 430 million million times a second then
it will interact with your eyes and you
will see it as red light; as it vibrates faster it appears as the other colours
of the spectrum- orange, then yellow, green, blue, and indigo- until about 770 million million times a second when it
appears violet. When it vibrates faster than that you can’t see it. We then
give it names such as ultraviolet, x-rays, and gamma rays as the frequency of
vibration increases (the names are a hang-over from the days before we realised
that they were all electromagnetic vibrations differing only by the rate of the
vibration).
When the vibration is less than 430 million million times,
the waves again become invisible, and we give them the following names broadly
according to how slowly they vibrate: infrared, microwaves, UHF, VHF, short
wave, medium wave and long wave.
Broadcast TV signals, for example, are just electromagnetic
vibrations that vibrate about a million times less often than visible light.
When the waves pass your TV aerial they make electrons in the aerial move back
and forth, and that movement is ultimately detected and decoded by your TV.
Electromagnetism
If you are reading this you are using electricity. It
turns out that electricity and magnetism are two symptoms of a single
underlying thing. There is a force which acts between protons and electrons,
which we call electromagnetism. This force is attractive been a proton and an
electron, but repulsive between electrons or between protons. When we talk
about an electric current flowing in a circuit, we actually mean that electrons
are being forced to move around the circuit by an electromagnetic force.
Crudely you can think of a battery as a container in which protons and
electrons start evenly spread out. When you charge the battery, the charger
applies an electromagnetic force that makes the electrons pile-up in one part
of the battery, leaving too few electrons in the other part. When you then use
the battery, all the electrons that have been piled up rush back (via your
phone, or torch, or car starter motor, or whatever) to the other part of the
battery. When there’s no longer any build-up of electrons the battery is flat.
If you want a nice analogy to understand batteries, try
this. Imagine a tank half filled with water. The water level is the same
throughout the tank. Now imagine you put a divider in the middle of the tank so
water can’t flow from one side to the other. Then imagine using a bucket or
pump to lift water from one side to the other over the divider. You’ll end up
with the tank nearly full on one side and nearly empty on the other. The water
on the full side would like to drop down into the empty half of the tank but it
can’t because the divider is in the way. Now suppose you connect the two halves
of the tank at the bottom with a pipe. The water from the full half of the tank
will rush through the pipe to the empty half. Gradually the level on one side
of the tank will fall, and the level on the other side will rise until the
level is the same throughout, then the water will stop flowing in the pipe. In
this analogy the water tank is the battery and the pipe is the electric
circuit.
Tuesday 24 January 2017
Waves on a circle
In the last couple of posts we've imagine the different ways in which a guitar string vibrates, and we've seen that each of the different ways contains a different number of vibrating segments, and there is always a whole number of segments.
The next idea is tricky, but imagine if the guitar string was instead a hoop of steel. It turns out that even for a hoop, the rules about vibrations are the same as for a straight string. You can get a hoop of steel to vibrate in different ways, but there is always a whole number of vibrating segments on the hoop.
You might be able to visualise this by imagining that the guitar string was actually a stiff rod of springy steel. It would still vibrate like the guitar string does, so if you bend the stiff rod into a hoop it still vibrates in the same way.
The next idea is tricky, but imagine if the guitar string was instead a hoop of steel. It turns out that even for a hoop, the rules about vibrations are the same as for a straight string. You can get a hoop of steel to vibrate in different ways, but there is always a whole number of vibrating segments on the hoop.
You might be able to visualise this by imagining that the guitar string was actually a stiff rod of springy steel. It would still vibrate like the guitar string does, so if you bend the stiff rod into a hoop it still vibrates in the same way.
Quantization
In our last post we saw how it was possible for a guitar string to vibrate in a number of different ways.
In the picture we saw the string vibrating first with just one segment (ie the whole string moves from side to side), then with two segments (each half the length of the string), then with three segments, and the with four. In theory it is possible for a string to vibrate with any number of segments (although it practice it gets harder to set-up the vibration as the number of segments increases).
An import point to realise is that the different types of vibration always include a whole number of segments. You can make the string vibrate with one segment or with two, but not with any fractional number in between.
The different ways in which a string vibrates are an example of a physical process that is 'quantized', or in whole number steps. We'll see later how the idea of whole number steps led to the name quantum theory.
In the picture we saw the string vibrating first with just one segment (ie the whole string moves from side to side), then with two segments (each half the length of the string), then with three segments, and the with four. In theory it is possible for a string to vibrate with any number of segments (although it practice it gets harder to set-up the vibration as the number of segments increases).
An import point to realise is that the different types of vibration always include a whole number of segments. You can make the string vibrate with one segment or with two, but not with any fractional number in between.
The different ways in which a string vibrates are an example of a physical process that is 'quantized', or in whole number steps. We'll see later how the idea of whole number steps led to the name quantum theory.
More about waves
Imagine the guitar string we plucked in the middle in an earlier post. it vibrates from side to side. The amplitude, or size of the vibration is widest at the middle and tails off towards the end of a string.
Now, if you place your finger lightly at the middle of a guitar string, and pluck it gently about three quarters of the way along it, you will find a completely different pattern of vibration. Now the middle of the string doesn't move at all. Instead the two halves of the string vibrate. The size of the vibration is biggest at the mid-points of the two halves of the string. We probably need a picture to explain this clearly, so I've added one below. If you are careful you can make the guitar string vibrate in three or even four separate sections.
The picture below shows a string vibrating in four separate ways. Each has one more vibrating segment than the one above it.
Now, if you place your finger lightly at the middle of a guitar string, and pluck it gently about three quarters of the way along it, you will find a completely different pattern of vibration. Now the middle of the string doesn't move at all. Instead the two halves of the string vibrate. The size of the vibration is biggest at the mid-points of the two halves of the string. We probably need a picture to explain this clearly, so I've added one below. If you are careful you can make the guitar string vibrate in three or even four separate sections.
The picture below shows a string vibrating in four separate ways. Each has one more vibrating segment than the one above it.
Introducing waves
The idea of waves plays a big part in quantum theory. We'll see later that fundamental particles seem sometimes to behave like waves instead of points of matter. Now we need to get more familiar with the notion of a wave.
Roughly speaking, a wave is a spread-out vibration or periodic change in some quantity. Imagine a guitar string. If you pluck it, the string vibrates from side to side. The movement of the string is a type of wave.
The amount by which the string moves to the side is called the 'amplitude' of the vibration, and it varies along the length of the string. On a guitar string that has been carefully plucked at the middle, the amplitude is greatest at the middle and gradually gets smaller towards each end. The amplitude has to be zero at the ends because they are fixed in position.
The frequency of a vibration is the number of times it vibrates in a second. We'll see later that a typical wave is actually a combination of vibrating movements with different frequencies, but usually there is one vibration in a wave that is distinctly bigger than the others, so we can loosely talk about the frequency of that bigger vibration as being the frequency of the wave.
Some waves- like the waving of a guitar string that is carefully plucked in the middle- happen in a fixed position. They are called standing waves. Other waves- like the waves that move across the surface of the sea- are called travelling waves.(With the right equipment you can set-up standing waves on water).
Sound is a vibration. When you listen to a radio, the loudspeaker in the radio vibrates, and that makes the air in front of the speaker vibrate too. The vibrations spread through the air (it is a travelling wave) and enters your ear, where it makes the tiny parts in your ear vibrate. Your brain interprets the vibrating of your ear as sound.
When you speak your vocal chords vibrate and cause the air coming out of your mouth to vibrate too. For a typical human voice the frequency of the vibration ranges from around 200 vibrations a second to a few thousand. The faster the vibration the 'higher' the sound appears to the human ear.
Roughly speaking, a wave is a spread-out vibration or periodic change in some quantity. Imagine a guitar string. If you pluck it, the string vibrates from side to side. The movement of the string is a type of wave.
The amount by which the string moves to the side is called the 'amplitude' of the vibration, and it varies along the length of the string. On a guitar string that has been carefully plucked at the middle, the amplitude is greatest at the middle and gradually gets smaller towards each end. The amplitude has to be zero at the ends because they are fixed in position.
The frequency of a vibration is the number of times it vibrates in a second. We'll see later that a typical wave is actually a combination of vibrating movements with different frequencies, but usually there is one vibration in a wave that is distinctly bigger than the others, so we can loosely talk about the frequency of that bigger vibration as being the frequency of the wave.
Some waves- like the waving of a guitar string that is carefully plucked in the middle- happen in a fixed position. They are called standing waves. Other waves- like the waves that move across the surface of the sea- are called travelling waves.(With the right equipment you can set-up standing waves on water).
Sound is a vibration. When you listen to a radio, the loudspeaker in the radio vibrates, and that makes the air in front of the speaker vibrate too. The vibrations spread through the air (it is a travelling wave) and enters your ear, where it makes the tiny parts in your ear vibrate. Your brain interprets the vibrating of your ear as sound.
When you speak your vocal chords vibrate and cause the air coming out of your mouth to vibrate too. For a typical human voice the frequency of the vibration ranges from around 200 vibrations a second to a few thousand. The faster the vibration the 'higher' the sound appears to the human ear.
Saturday 21 January 2017
The need for a new theory
When physicists learned that everything seemed to be made of combinations of protons, neutrons and electrons they realised that the classical theories of physics were completely unable to explain how just three ingredients could make elements with such widely differing properties. What was worse, the classical theories predicted that the electrons would be pulled into the centre of the atoms by the much heavier protons, so the theories weren't just inadequate they were wrong.
Quantum theory came about when physicists tried to make sense of certain properties of atoms. But before we get into that, we need to talk about a new idea- waves.
Quantum theory came about when physicists tried to make sense of certain properties of atoms. But before we get into that, we need to talk about a new idea- waves.
More about particles- protons and electrons in atoms
One characteristic of the fundamental particles is that whenever we detect them individually they seem to be very tiny point-like objects. That is, they seem to be solid and compact like a billiard ball, rather than long and stretched out like a hose-pipe or a bed-sheet, or smeared like a cloud of steam. However, we'll see later that according to quantum theory the idea of the particles being like little points of matter is too simple, but that's the idea we will continue to work with for now.
Protons are about 2,000 times heavier than electrons. In atoms, the protons seems to be clustered tightly together at the centre- called the nucleus- while the electrons are more widely spread around the nucleus.
We used to think that the electrons circled around the nucleus of protons like planets orbiting the Sun, and for now that's a good image to have in your head.
What's surprising is that when atoms come together to form a solid- imagine a lump of gold say- there's still relatively huge gaps between their heavy centres. If the nucleus of an atom of gold was the size of a grapefruit, say, then the atoms in gold would be about a kilometre apart, with the electrons somehow buzzing around in the space between them. It's odd to consider that something that seems as solid and heavy as gold is actually mainly empty space!
Protons are about 2,000 times heavier than electrons. In atoms, the protons seems to be clustered tightly together at the centre- called the nucleus- while the electrons are more widely spread around the nucleus.
We used to think that the electrons circled around the nucleus of protons like planets orbiting the Sun, and for now that's a good image to have in your head.
What's surprising is that when atoms come together to form a solid- imagine a lump of gold say- there's still relatively huge gaps between their heavy centres. If the nucleus of an atom of gold was the size of a grapefruit, say, then the atoms in gold would be about a kilometre apart, with the electrons somehow buzzing around in the space between them. It's odd to consider that something that seems as solid and heavy as gold is actually mainly empty space!
Thursday 19 January 2017
Fundamental particles
In our story so far we've seen that everything seems to be made of protons, electrons and neutrons, which are bundled together to form atoms. The number of protons and electrons in an atom determines what element the atom forms. For example, an atom of copper contains twenty nine protons and twenty nine electrons.
However, to illustrate the principles of quantum theory we can ignore most of the other fundamental particles that have been found, and concentrate mainly on protons and electrons. There is one other important particle, called a photon, which is actually a particle of light. We will get to know more about photons later.
NEXT LESSON: protons and electrons in atoms
The idea that atoms were made of protons, electrons and neutrons was established early in the 20th century. Since then we've found that reality is more complicated. For example:
- Protons and neutrons seem to be made of even smaller particles called quarks.
- While protons, electrons and neutrons seems to last for ever (the ones in your body were made in the big bang thirteen billion years ago), there are particles that play a role in holding atoms together that are only last for about a million million million millionth of a second.
However, to illustrate the principles of quantum theory we can ignore most of the other fundamental particles that have been found, and concentrate mainly on protons and electrons. There is one other important particle, called a photon, which is actually a particle of light. We will get to know more about photons later.
NEXT LESSON: protons and electrons in atoms
Atoms and elements
Every material thing in the world (and probably the universe) is made from basic ingredients called elements. There are just over a hundred elements, many of which you're likely to know, such as gold, silver, lead, oxygen, hydrogen, sodium, carbon, iron, uranium and so on.
Here's a list of the main elements in some everyday things:
The physical properties of elements vary enormously. Some are gases, some are liquids. Some of the solid elements are extremely hard, and other are very soft. Some conduct electricity, others don't. Some interact violently with other elements, whereas others- like gold for example- hardly react at all. Most of them are opaque, but some transmit light (diamonds are made of carbon).
The smallest chunks of elements are called atoms. Atoms are tiny. There are about seven billion billion billion atoms in the average human body. If each atom was the size of a house brick, then a human body would be about half a million miles high.
People used to think that elements were completely unrelated from each other- that an atom of gold, say, was completely different to an atom of iron. Now we know that atoms of all elements are made of the same even-smaller building blocks, called protons and electrons. The elements only differ in the number of protons and electrons they contain. For example, hydrogen has one of each, helium has two, lithium has three, beryllium has four, boron has five, carbon has six, nitrogen has seven...every time you add another proton and another electron you get a new element.
Platinum, for instance, has seventy eight protons and seventy eight electrons. If you add another proton and another electron to an atom of platinum you get an atom of gold.
In fact, with the exception of atoms of hydrogen, all atoms have a third ingredient called a neutron, but the neutron is a bit of a distraction in our explanation of atoms, so we can leave it out for now.
NEXT LESSON: Fundamental particles
Here's a list of the main elements in some everyday things:
- Steel is mainly iron, usually with some carbon and possibly elements such as nickel, vanadium, and chromium.
- People are largely made of oxygen, carbon, hydrogen, nitrogen, calcium, and phosphorus (which account for about 99% of your body weight).
- Chocolate is made of carbon, hydrogen, nitrogen and oxygen.
- Water is oxygen and hydrogen.
- Glass is mainly silicon and oxygen.
The physical properties of elements vary enormously. Some are gases, some are liquids. Some of the solid elements are extremely hard, and other are very soft. Some conduct electricity, others don't. Some interact violently with other elements, whereas others- like gold for example- hardly react at all. Most of them are opaque, but some transmit light (diamonds are made of carbon).
The smallest chunks of elements are called atoms. Atoms are tiny. There are about seven billion billion billion atoms in the average human body. If each atom was the size of a house brick, then a human body would be about half a million miles high.
People used to think that elements were completely unrelated from each other- that an atom of gold, say, was completely different to an atom of iron. Now we know that atoms of all elements are made of the same even-smaller building blocks, called protons and electrons. The elements only differ in the number of protons and electrons they contain. For example, hydrogen has one of each, helium has two, lithium has three, beryllium has four, boron has five, carbon has six, nitrogen has seven...every time you add another proton and another electron you get a new element.
Platinum, for instance, has seventy eight protons and seventy eight electrons. If you add another proton and another electron to an atom of platinum you get an atom of gold.
In fact, with the exception of atoms of hydrogen, all atoms have a third ingredient called a neutron, but the neutron is a bit of a distraction in our explanation of atoms, so we can leave it out for now.
NEXT LESSON: Fundamental particles
Wednesday 18 January 2017
Why is quantum theory hard to understand?
There are two main reasons why quantum theory is hard for anyone to explain and understand.
One is that it includes a lot of complicated mathematics, so if you don't understand the mathematics you're stuck.
The other is that quantum theory conflicts with our everyday experience. People can usually understand a new idea through analogy- by likening it to something they already know. Much of quantum theory is unlike anything you already know, so understanding it by analogy isn't an option.
There's a third reason that makes it extra difficult for some people. Quantum theory is a branch of physics, and you need to be familiar with some of the ideas of physics, especially those to do with the tiny particles that make up the universe. In this blog we'll start completely from scratch, so even if you know nothing about physics you should be able to keep up.
NEXT LESSON: Atoms and elements
One is that it includes a lot of complicated mathematics, so if you don't understand the mathematics you're stuck.
The other is that quantum theory conflicts with our everyday experience. People can usually understand a new idea through analogy- by likening it to something they already know. Much of quantum theory is unlike anything you already know, so understanding it by analogy isn't an option.
There's a third reason that makes it extra difficult for some people. Quantum theory is a branch of physics, and you need to be familiar with some of the ideas of physics, especially those to do with the tiny particles that make up the universe. In this blog we'll start completely from scratch, so even if you know nothing about physics you should be able to keep up.
NEXT LESSON: Atoms and elements
What is quantum theory?
The posts in this blog explain the main concepts of quantum theory without expecting you to have any understanding of physics or maths. Each post will just take a few minutes to read.
Quantum theory (aka quantum physics or quantum mechanics) explains and predicts the interactions between the tiny particles that make up the universe. Quantum theory has become controversial because it shows that reality is weird and unpredictable.
The underlying ideas of quantum theory were conceived early in the 20th Century. The older ideas they displaced are called 'classical physics'.
Quantum theory provides an extremely accurate model of many physical effects, but it is famously enigmatic because physicists can't agree what it really means, even though they have been trying for a hundred years.
Click the link below for the next lesson.
NEXT LESSON: Why is quantum theory hard to understand?
Quantum theory (aka quantum physics or quantum mechanics) explains and predicts the interactions between the tiny particles that make up the universe. Quantum theory has become controversial because it shows that reality is weird and unpredictable.
The underlying ideas of quantum theory were conceived early in the 20th Century. The older ideas they displaced are called 'classical physics'.
Quantum theory provides an extremely accurate model of many physical effects, but it is famously enigmatic because physicists can't agree what it really means, even though they have been trying for a hundred years.
Click the link below for the next lesson.
NEXT LESSON: Why is quantum theory hard to understand?
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