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.
Friday, 27 January 2017
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.
Subscribe to:
Posts (Atom)