As you start journeying down the rabbit-hole into quantum mechanics, you are presented with the famous "double-slit" experiment, in which you must confront what it means when a single particle, say, an electron, is given the option of two slits to choose from and yet seemingly does not choose at all, but rather acts as if it went through both slits at the same time. (If you are scratching your head right now, see if this helps.)
Madness! Luckily, great minds are here to explain, in order of increasingly accurate personality approximation:
Newton: No no no, you must have done it wrong, this is impossible! The electron must pass through one slit or the other, as everyone knows it is nonsense to think a particle can dividing itself or otherwise being in two places at once.
De Broglie: But actually the electron is a wave, so this is not a problem.
Einstein: God does not play dice! This experiment reveals something about the electron of which we are utterly unaware and perhaps unable to determine.
Born: No, see, this is a result of an altered probability of where the electron will land, a probability which is determined by it's wave-function, which interfered with itself in the double slit.
Feynman: Forget that. The electron really did go through both slits at once, and also went to the moon, and also up your nose, and also out to lunch and back. This violates possibly every law of physics that we know including the inability to go faster than light speed and also the inability to suddenly be two places at once--but you've got to accept that nobody understand quantum mechanics.
Greene: If you go even more deeply then we'll ever be able to prove then you'll find that the electron and slits both are really a vibrating string and by the way isn't the universe so incredible!
Physics is at the point where oftentimes "what actually physically happened" becomes a meaningless concept, or at least very difficult to tease out from the models we impose to understand things.
Showing posts with label quantum mechanics. Show all posts
Showing posts with label quantum mechanics. Show all posts
Monday, May 2, 2011
Sunday, January 9, 2011
Quantum Psychology
This link seems pretty high on the "woo woo stuff claiming to be science" scale--but that's the predjudice we physicists have toward psychologists anyway.
(That was a joke, you can stop throwing tomatoes now.)
Actually, I say why not, if other psychologists find it up to snuff, I see no reason why it's not an interesting and plausible study that is worth more research. To wit:
"One of the most respected, senior and widely published professors of psychology, Daryl Bem of Cornell, has just published an article that suggests that people — ordinary people — can be altered by experiences they haven't had yet. Time, he suggests, is leaking. The Future has slipped, unannounced, into the Present. And he thinks he can prove it."
Without claiming that this explains it, I will say that the framework of quantum mechanics would permit something like this to be within the realm of the possible, and our understanding of quantum mechanics is getting so good that I would not be terribly surprised if we did begin to see quantum effects appearing in realms of science even very removed from pure physics. That being said, actually linking the effects credibly would be beyond the scope of either branch of science individually and pretty damn difficult all around.
When I was an undergrad I performed a lab experiment called the "quantum eraser," in which I took the extraordinarily simple case of a beam of light emitted by a laser (thus, all the little light particles were suitably similar to each other as is unique to how lasers, masers, and the like emit light), and made the photons act a certain way based on what I did after the experiment was over.
It was called the "quantum eraser" because of what the laser light looked like on the wall, after I shone the laser beam through a screen with two slits. Usually, the light pattern looks something like this, which is notably different from the single dot of light your cat likes to chase across the floor. This laser dot looks like a prison window grill instead because when it passed through two slits, it interfered with itself. Bring up the usual analogy to dropping two rocks in a pond and watching how the ripple patterns change when they intersect each other: some places get extra big ripples, in some places the ripple goes away, and that's the beauty of things that are waves. Light is a wave, and it's also a particle--welcome to your very first step on the journey down the rabbit hole of quantum mechanics.
When I added in polarizers after each slit, so that light passing through one of the two slits could be distinguished from the light passing through the other one, I stopped seeing the prison-grill pattern and started seeing a normal dot of light again--I erased the pattern. But why should simply distinguishing the light from each beam do that? There is no reason that it should, except that quantum mechanics illogically says that in a situation such as each little light photon was supposed to have faced, whereby it arrives at Door #1 and Door #2 and has to pick one, but I, the observer, have no way of telling which one it actually picked, then it doesn't pick. It just behaves as if it went through both. Since it behaves that way, it can interfere with itself and make a light pattern that shows interference.
When I changed the experiment so that I could tell which slit the photons went through, then I observed what would happen if half had gone through one and half had gone through the other: just a dot, no interference. Furthermore the polarizers were located behind the slits: the light had already passed through by the time it reached them, yet being there, giving me the ability to tell what had happened, was enough. Even weirder, when I added in another polarizer downstream that erased the information the first polarizers had gathered--I'd see the stripes again, instead of the dot. All done after the light had gone and passed through the slits.
This is called delayed choice. One interpretation is that I can choose the outcome of the experiment based on what I want to observe, and it doesn't matter that I make that choice after the event I am observing actually happens. It doesn't necessarily make sense with how we usually interact with the world but it does make sense with the statistically-driven rules of quantum mechanics, and certain situation can be manipulated to reveal that weirdness.*
Of course, delayed choice doesn't have to be interpreted to mean that the future affects something that happened in the past. It could for instance mean that something you do in the present is only clarifying the past, or, as some of the people who first came up with the idea for delayed choice said, "the past has no existence except as is recorded in the present,"--which removes the time-travel-esque aspect. You could interpret it just as an example of probability threading it's way into real life--which is indeed how many people choose to interpret all of quantum's weirdness. Sometimes a roomful of physicists will not all agree on how to interpret how the heck subtle quantum mechanics things actually happens in reality, and physics is a field that is running into a wall on that whole "absolute objective reality" idea anyway--despite the fact that the math and general theories behind quantum mechanics are basically a settled science and research into how to use QM to make a better computer and help your GPS find itself, not to mention transmit information in space-age ways, are well underway.
So if a psychologist finds statistical significance to the idea that something we do in the present seems to inform something we've already done in the past--or however you want to interpret it--and if he has a carefully controlled and eventually repeated experiment, then hey, I'll call that an interesting stop in the scientific process.
*I have to put a disclaimer here that since photons are the only known and theoretically allowed things to move at the speed of light, interpreting as an actual example of delayed choice the experiment I did with just photons runs into some problems with special relativity. Ah well, the big problem in physics these days is that relativity and quantum mechanics don't mix. The entire setup, minus the lab components, is equally valid for electrons except that it costs a great deal more than the lab practical budget for an undergraduate physics department, in which case you have to admit you're in spooky-land because electrons, as clear particles, are then nonetheless doing some suspect wave-like things. And then it turns out that they've even passed fullerine molecules through double slits.
(That was a joke, you can stop throwing tomatoes now.)
Actually, I say why not, if other psychologists find it up to snuff, I see no reason why it's not an interesting and plausible study that is worth more research. To wit:
"One of the most respected, senior and widely published professors of psychology, Daryl Bem of Cornell, has just published an article that suggests that people — ordinary people — can be altered by experiences they haven't had yet. Time, he suggests, is leaking. The Future has slipped, unannounced, into the Present. And he thinks he can prove it."
Without claiming that this explains it, I will say that the framework of quantum mechanics would permit something like this to be within the realm of the possible, and our understanding of quantum mechanics is getting so good that I would not be terribly surprised if we did begin to see quantum effects appearing in realms of science even very removed from pure physics. That being said, actually linking the effects credibly would be beyond the scope of either branch of science individually and pretty damn difficult all around.
When I was an undergrad I performed a lab experiment called the "quantum eraser," in which I took the extraordinarily simple case of a beam of light emitted by a laser (thus, all the little light particles were suitably similar to each other as is unique to how lasers, masers, and the like emit light), and made the photons act a certain way based on what I did after the experiment was over.
It was called the "quantum eraser" because of what the laser light looked like on the wall, after I shone the laser beam through a screen with two slits. Usually, the light pattern looks something like this, which is notably different from the single dot of light your cat likes to chase across the floor. This laser dot looks like a prison window grill instead because when it passed through two slits, it interfered with itself. Bring up the usual analogy to dropping two rocks in a pond and watching how the ripple patterns change when they intersect each other: some places get extra big ripples, in some places the ripple goes away, and that's the beauty of things that are waves. Light is a wave, and it's also a particle--welcome to your very first step on the journey down the rabbit hole of quantum mechanics.
When I added in polarizers after each slit, so that light passing through one of the two slits could be distinguished from the light passing through the other one, I stopped seeing the prison-grill pattern and started seeing a normal dot of light again--I erased the pattern. But why should simply distinguishing the light from each beam do that? There is no reason that it should, except that quantum mechanics illogically says that in a situation such as each little light photon was supposed to have faced, whereby it arrives at Door #1 and Door #2 and has to pick one, but I, the observer, have no way of telling which one it actually picked, then it doesn't pick. It just behaves as if it went through both. Since it behaves that way, it can interfere with itself and make a light pattern that shows interference.
When I changed the experiment so that I could tell which slit the photons went through, then I observed what would happen if half had gone through one and half had gone through the other: just a dot, no interference. Furthermore the polarizers were located behind the slits: the light had already passed through by the time it reached them, yet being there, giving me the ability to tell what had happened, was enough. Even weirder, when I added in another polarizer downstream that erased the information the first polarizers had gathered--I'd see the stripes again, instead of the dot. All done after the light had gone and passed through the slits.
This is called delayed choice. One interpretation is that I can choose the outcome of the experiment based on what I want to observe, and it doesn't matter that I make that choice after the event I am observing actually happens. It doesn't necessarily make sense with how we usually interact with the world but it does make sense with the statistically-driven rules of quantum mechanics, and certain situation can be manipulated to reveal that weirdness.*
Of course, delayed choice doesn't have to be interpreted to mean that the future affects something that happened in the past. It could for instance mean that something you do in the present is only clarifying the past, or, as some of the people who first came up with the idea for delayed choice said, "the past has no existence except as is recorded in the present,"--which removes the time-travel-esque aspect. You could interpret it just as an example of probability threading it's way into real life--which is indeed how many people choose to interpret all of quantum's weirdness. Sometimes a roomful of physicists will not all agree on how to interpret how the heck subtle quantum mechanics things actually happens in reality, and physics is a field that is running into a wall on that whole "absolute objective reality" idea anyway--despite the fact that the math and general theories behind quantum mechanics are basically a settled science and research into how to use QM to make a better computer and help your GPS find itself, not to mention transmit information in space-age ways, are well underway.
So if a psychologist finds statistical significance to the idea that something we do in the present seems to inform something we've already done in the past--or however you want to interpret it--and if he has a carefully controlled and eventually repeated experiment, then hey, I'll call that an interesting stop in the scientific process.
*I have to put a disclaimer here that since photons are the only known and theoretically allowed things to move at the speed of light, interpreting as an actual example of delayed choice the experiment I did with just photons runs into some problems with special relativity. Ah well, the big problem in physics these days is that relativity and quantum mechanics don't mix. The entire setup, minus the lab components, is equally valid for electrons except that it costs a great deal more than the lab practical budget for an undergraduate physics department, in which case you have to admit you're in spooky-land because electrons, as clear particles, are then nonetheless doing some suspect wave-like things. And then it turns out that they've even passed fullerine molecules through double slits.
Tuesday, April 13, 2010
Some Things to Know About Quantum Mechanics
Sometimes throwing the word "quantum" around makes you look smart.
In the right company, of course, it has the opposite effect, at least if you use it incorrectly. See my rant on the book, The Secret.
Quantum mechanics, the branch of physics that deals with things on a small scale--but not to be confused with particle physics, which deals with things on an even smaller scale--has the property of seeming out-there, highly subjective, and sometimes even mystical, because the set of rules and physical processes being described are really better understood by staring at math equations than by trying to use declarative sentences. As a result, different physicists might describe the same thing with disparate analogies, yet when they get into the nerdy details together, they both understand what's going on.
So, to make your life a little more interesting at parties, I'm going to share some insights into the magical, mystical world of quantum, with the disclaimer that to really understand, you have to have had a fair amount of training in calculus and linear algebra.
And if you think you're too un-smart to understand calculus and linear algebra, I say you might not be, you should try it some time. Some people would call mathematics a precisely reasoned logic system. I would call it a very precise language, which describes the universe quite elegantly.
First, quantum mechanics is, like any branch of physics, a set of rules.
Now, a general set of rules is something we are all well familiar with. The rules governing things here on Earth are things we know intuitively because we can see them and experience them: force, mass, inertia, gravity, etc and the rules of their relationships are referred to by we physics types as classical mechanics.
Well, it turns out that quantum mechanics--dealing principally with the protons, neutrons, and electrons that all classical objects are made up of when you get down to it--works here on Earth as well as anywhere else, and a good thing too, but we can't see quantum mechanics working because all of those protons, neutrons, and electrons doing their thing together aggregates into what we actually observe, which is classical mechanics. If you never bothered to delve deeper you'd never know that classical mechanics wasn't the only set of rules in play, and for thousands of years, we didn't know, because we didn't have the technology and hadn't done the thought experiments.
So quantum mechanics--one set of rules, and classical mechanics--another, don't necessarily conflict, so much as provide different frameworks to understand what is happening. Each has it's appropriate place to be used, just like you'd look through sunglasses when your outside, and normal glasses, (if you need glasses) when you're inside. Both are correct when used in the right place. So while quantum may seem magical and mystical, it is in fact happening right now, right in front of your nose--you just don't have the tools to see it, and odds are to accomplish most things you don't need to anyway.
In the case of classical mechanics, as I said, everybody knows about it, because everybody can see it. Even if you've never taken a physics class, you intuitively use classical mechanics every time you do most anything, expecting a certain result. We developed math and things like Newton's Laws, to describe what was always expect to happen and called it 'physics', and tortured students with what they already knew, darn it, for years. Oh, and refining of our knowledge of it helped us build things like cars, skyscrapers, rockets, and satellites.
Quantum mechanics was sort of the opposite. On a very generalized level, the math that describes quantum mechanics, namely, calculus (nasty integrals, and symbolically unsolvable differential equations, to be precise) and linear algebra, already existed. Scientists noticed a number of things that no longer made any sense if you just used classical mechanics to look at them, and as it turned out most of these things had to do with the behavior of the sub-atomic particles involved. Smart people began piecing these puzzles together. Math had proven a useful too thus far, so they used math to create a set of rules, describing how really tiny things act. The set of rules was refined to matched reality well, and with more evidence better, and better still. We don't know that they are true, and we probably never can. But the rules work so well in so many cases, that the likelihood of their truth is no longer in dispute by most scientists. (Some few are still proponents of the 'hidden variables' explanation, something Einstein himself proposed, not liking the un-pin-down-able nature of quantum mechanics.)
So, what are the rules?
Unfortunately the "real" rules are pretty much inseparable from the math. In this case, the math makes it way easier to get a handle on what's happening. One line for each, verses the paragraphs I go into below.
But they go something like this. I'm going to state them out of order, because that is the way that they make the most sense.
Rule 3: A system can be represented by a state function, called a wave-funciton.
What? In math, remember, a function is an equation of two variables. It describes the relationship between those variables. Example: y=x. If you plotted that, you'd get a diagonal line, because the x and y values of that line, no matter what they are, are always the same.
Rule 3 says that there exists a function that describes every system. An electron floating around has a function that describes it. An atom has a function. A person, consisting of millions and millions of atoms, does too. Are those functions hella complicated? Yeah. They're called wave-functions because they usually always have a sine in them somewhere, and if you've ever looked at a graph of y=sin(x), you'll see a line that goes up and down in a repeated pattern. That's a wave. There are some implications, of course, to everything being described as a wave, and I might explain some of those in a later entry.
Furthermore, any system at any instant in time has a wave-function, and the information describing the system is contained in that function. This is a pretty profound rule. It says that things are describable by math.
Rules number 1 and 2 are closely related.
Rule 1: For an observable value, say, momentum of a particle, you can do math on the wave-function that results in that wave-function being multiplied by that observable value.
Okay, I told you this is just too math-y to use good old declarative sentences. Basically, say the energy of an electron is equal to 2 Joules. I just made that number up, I don't think electrons ever have that much energy. But in my example the electron has a certain energy at a certain instant in time, and that is 2 Joules. From Rule 3, we know that the electron at that moment in time can be described by a function, probably a complicated one. Whatever it is, the objective of Rule 1 is to obtain the result: (electron function, whatever it is) multiplied by 2. That would go on one side of the equals sign. On the other, you would Do Math, but the result of the math you are doing would give you the electron's Rule 3 function multiplied by 2, and 2 is the energy of the electron at that point in time.
What math do you do? That depends. You do certain math for each observable value. There's certain math you do for momentum, certain math you do for energy, position, etc. So rule number 1 is stating that Math Exists that you can do, for any observable you want to measure. Once you've done that math, you will have the wave-function multiplied by the value of the thing you want to measure. Momentum Math on the wave-function gives you the wave-function multiplied by the momentum of the system. Position Math gives you the wave-function multiplied by the position of the system.
The implication of this is that you can find out these things about the system, not by measuring the energy, the momentum, the position, but by doing math. And it gives you an exact result. (Assuming you know what the wave-function is, of course.) In the case of our example, applying Rule 1 told us that the electron has 2 joules. Not 2.5, not 2.01, not 2.001, not 2 and some error inherent in measuring. 2. Exactly. "Quantum" comes form "Quantized", or "quanta", meaning, discrete, exact value. Not a range, but a yes or a no.
Rule 2 Is closely related. It says that if you do decide to measure an observable value, again, something like energy, momentum, or position, of a system, then the wave-function of that system, at the instant you take a measurement, will be equal to the wave-function required to make Rule # 1 work for whatever value you get from your measurement.
I'll use a tiny bit of algebra to demonstrate, because you can do algebra with numbers, but you can also do algebra with functions.
y=4x. We know what y is from Rule 1, that's the Math that Exists that you Do, to get the value of the observable multiplied by the wave-function. Say 4 is value of the observable, and x represents the wave-function, whatever hella complicated thing it is. Y is something that doesn't change, remember that there is certain math to measure energy, certain math to measure momentum, certain math to measure each thing that you want to measure. So if we are trying to measure the energy of an electron at a specific moment in time, then y is the Energy Math, and 4 is what we got when we took the measurement. Just for simplicity, I will say y is equal to 2. You know two parts of the equation, and two are all you need to solve it:
2=4x. If this is so, we can figure out what x is. x is equal to 1/2.
So you can figure out the wave-function. It has to be the wave-function that make the algebraic expression above work. That is the only wave-function it could be at that instant.
Rule 2 two helps us understand a last important thing about quantum mechanics, which is that even though the rules are set it up so that things are exact, the applications tend to be probabilistic. This is because these wave-functions are not simple numbers, like 1/2, but very complicated beasts, consisting of multiple variables. Rules 1 and 2 always apply: if you Do the Math that Exists, you get an exact value and a matching wave-function. But often instead of something simple, like an electron floating around in space, there are complications, like, an electron within an atom, a bunch of atoms together in a molecule. In these case, there are many possible states. An electron within at atom could be in one orbit, or another. There may be two electrons, and the proton matters too. In this case, the wave-function is often what we call a superposition of possibilities. Superposition is a term describing how waves interact with each other: unlike matter, waves can occupy the same space at once, they are superimposed on each other. The wave-function for a complex system is often a superposition of all the many, many allowed states that the system could be in.
In this case the wave-function does not tell us which possibility is reality, it just tells us the probability of each one. When you make the measurement (assuming, in a situation like this, that you could), you get one of these possibilities, and the wave-function becomes the function which describes that reality over any other. But until you make the measurement, there may be multiple possibilities for the state of the system, with different probabilities, and the wave-function will reflect that.
So A, B, and C, may all be options. Because of the Rules, we know that "sorta A sorta B", is not an option, or A.5, or however you want to look at it. It will be A, B, or C, when you take your measurement. Until you do, the wave-function, could it be determined, would tell you something like A is 50% probable, B is 45%, and C is 5%. If you take 100 measurements, you will measure A 50 times, B 45 times, and C 5 times.
And this is how it works in the world of the small. There are only certain ways to be. But there are also probabilities of being there, which govern how everything actually is.
What do these rules enables us to do and say about the universe? There are so many implications, and so many successful applications of quantum mechanics. The periodic table, all that stuff you might have learned and hated in chemistry about the strange shapes of electron orbits--all of that can be solved precisely with quantum mechanics. I will take the remainder of your reading attention to explain just one, particularly important example.
We owe a functioning sun, and in fact all stars, to quantum mechanics. It turns out that even inside the very hot sun, the collisions of all the protons are not energetic enough to enact the heating mechanism of the sun, in which two protons get close enough to overcome the Electrostatic force that usually repels like charges, are sucked together by the stronger but shorter-distance Strong Nuclear Force, and start the quark-swapping chain of events known as nuclear fusion. It just isn't hot enough. Yet we know for a fact that fusion does happen, and it's a pretty good thing for us that it does.
There happens to be a quantum behavior, called tunneling, in which--even though the energy required to create fusion is much higher than the energy of protons colliding at 15 million Kelvins--because of the probabilistic, rather than deterministic, rules of QM, there is a very, very small probability that the protons will "tunnel through the energy barrier", or, despite not having enough energy to fuse, will fuse anyway. This is kind of like how quantum mechanics says that if you throw yourself at the wall enough times, there IS a tiny probability that you will go through it. But go throw yourself into the wall, and I guarantee that even if you did it your entire life, you will not go through it, because that probability is so incredibly tiny. You can't take enough measurements to make that wave-function the reality.
The probability of two protons fusing anyway is also very, very small. Thing is, in the sun, there are, let us just say, a lot of protons. The sun is, after all, 300,000 times bigger than the Earth, and a proton is inconceivably small even when compared to an ant. Furthermore, it's darn hot in the center of the sun, meaning the protons there have loads of kinetic energy and are zipping around at incredible speeds. Lots of things in a a finite space moving very quickly means that those things are going to run into each other. Many millions of collisions per second, in fact. And so even if the probability for a single collision resulting in fusion is tiny, there are so many collisions happening, enough that yes, fusion does occur, the sun does shine, and life on Earth does reap considerable benefit.
So yeah. Try that out at a party sometime. "Did you know that fusion is due to the improbable possibility of proton tunneling within the sun?"
In the right company, of course, it has the opposite effect, at least if you use it incorrectly. See my rant on the book, The Secret.
Quantum mechanics, the branch of physics that deals with things on a small scale--but not to be confused with particle physics, which deals with things on an even smaller scale--has the property of seeming out-there, highly subjective, and sometimes even mystical, because the set of rules and physical processes being described are really better understood by staring at math equations than by trying to use declarative sentences. As a result, different physicists might describe the same thing with disparate analogies, yet when they get into the nerdy details together, they both understand what's going on.
So, to make your life a little more interesting at parties, I'm going to share some insights into the magical, mystical world of quantum, with the disclaimer that to really understand, you have to have had a fair amount of training in calculus and linear algebra.
And if you think you're too un-smart to understand calculus and linear algebra, I say you might not be, you should try it some time. Some people would call mathematics a precisely reasoned logic system. I would call it a very precise language, which describes the universe quite elegantly.
First, quantum mechanics is, like any branch of physics, a set of rules.
Now, a general set of rules is something we are all well familiar with. The rules governing things here on Earth are things we know intuitively because we can see them and experience them: force, mass, inertia, gravity, etc and the rules of their relationships are referred to by we physics types as classical mechanics.
Well, it turns out that quantum mechanics--dealing principally with the protons, neutrons, and electrons that all classical objects are made up of when you get down to it--works here on Earth as well as anywhere else, and a good thing too, but we can't see quantum mechanics working because all of those protons, neutrons, and electrons doing their thing together aggregates into what we actually observe, which is classical mechanics. If you never bothered to delve deeper you'd never know that classical mechanics wasn't the only set of rules in play, and for thousands of years, we didn't know, because we didn't have the technology and hadn't done the thought experiments.
So quantum mechanics--one set of rules, and classical mechanics--another, don't necessarily conflict, so much as provide different frameworks to understand what is happening. Each has it's appropriate place to be used, just like you'd look through sunglasses when your outside, and normal glasses, (if you need glasses) when you're inside. Both are correct when used in the right place. So while quantum may seem magical and mystical, it is in fact happening right now, right in front of your nose--you just don't have the tools to see it, and odds are to accomplish most things you don't need to anyway.
In the case of classical mechanics, as I said, everybody knows about it, because everybody can see it. Even if you've never taken a physics class, you intuitively use classical mechanics every time you do most anything, expecting a certain result. We developed math and things like Newton's Laws, to describe what was always expect to happen and called it 'physics', and tortured students with what they already knew, darn it, for years. Oh, and refining of our knowledge of it helped us build things like cars, skyscrapers, rockets, and satellites.
Quantum mechanics was sort of the opposite. On a very generalized level, the math that describes quantum mechanics, namely, calculus (nasty integrals, and symbolically unsolvable differential equations, to be precise) and linear algebra, already existed. Scientists noticed a number of things that no longer made any sense if you just used classical mechanics to look at them, and as it turned out most of these things had to do with the behavior of the sub-atomic particles involved. Smart people began piecing these puzzles together. Math had proven a useful too thus far, so they used math to create a set of rules, describing how really tiny things act. The set of rules was refined to matched reality well, and with more evidence better, and better still. We don't know that they are true, and we probably never can. But the rules work so well in so many cases, that the likelihood of their truth is no longer in dispute by most scientists. (Some few are still proponents of the 'hidden variables' explanation, something Einstein himself proposed, not liking the un-pin-down-able nature of quantum mechanics.)
So, what are the rules?
Unfortunately the "real" rules are pretty much inseparable from the math. In this case, the math makes it way easier to get a handle on what's happening. One line for each, verses the paragraphs I go into below.
But they go something like this. I'm going to state them out of order, because that is the way that they make the most sense.
Rule 3: A system can be represented by a state function, called a wave-funciton.
What? In math, remember, a function is an equation of two variables. It describes the relationship between those variables. Example: y=x. If you plotted that, you'd get a diagonal line, because the x and y values of that line, no matter what they are, are always the same.
Rule 3 says that there exists a function that describes every system. An electron floating around has a function that describes it. An atom has a function. A person, consisting of millions and millions of atoms, does too. Are those functions hella complicated? Yeah. They're called wave-functions because they usually always have a sine in them somewhere, and if you've ever looked at a graph of y=sin(x), you'll see a line that goes up and down in a repeated pattern. That's a wave. There are some implications, of course, to everything being described as a wave, and I might explain some of those in a later entry.
Furthermore, any system at any instant in time has a wave-function, and the information describing the system is contained in that function. This is a pretty profound rule. It says that things are describable by math.
Rules number 1 and 2 are closely related.
Rule 1: For an observable value, say, momentum of a particle, you can do math on the wave-function that results in that wave-function being multiplied by that observable value.
Okay, I told you this is just too math-y to use good old declarative sentences. Basically, say the energy of an electron is equal to 2 Joules. I just made that number up, I don't think electrons ever have that much energy. But in my example the electron has a certain energy at a certain instant in time, and that is 2 Joules. From Rule 3, we know that the electron at that moment in time can be described by a function, probably a complicated one. Whatever it is, the objective of Rule 1 is to obtain the result: (electron function, whatever it is) multiplied by 2. That would go on one side of the equals sign. On the other, you would Do Math, but the result of the math you are doing would give you the electron's Rule 3 function multiplied by 2, and 2 is the energy of the electron at that point in time.
What math do you do? That depends. You do certain math for each observable value. There's certain math you do for momentum, certain math you do for energy, position, etc. So rule number 1 is stating that Math Exists that you can do, for any observable you want to measure. Once you've done that math, you will have the wave-function multiplied by the value of the thing you want to measure. Momentum Math on the wave-function gives you the wave-function multiplied by the momentum of the system. Position Math gives you the wave-function multiplied by the position of the system.
The implication of this is that you can find out these things about the system, not by measuring the energy, the momentum, the position, but by doing math. And it gives you an exact result. (Assuming you know what the wave-function is, of course.) In the case of our example, applying Rule 1 told us that the electron has 2 joules. Not 2.5, not 2.01, not 2.001, not 2 and some error inherent in measuring. 2. Exactly. "Quantum" comes form "Quantized", or "quanta", meaning, discrete, exact value. Not a range, but a yes or a no.
Rule 2 Is closely related. It says that if you do decide to measure an observable value, again, something like energy, momentum, or position, of a system, then the wave-function of that system, at the instant you take a measurement, will be equal to the wave-function required to make Rule # 1 work for whatever value you get from your measurement.
I'll use a tiny bit of algebra to demonstrate, because you can do algebra with numbers, but you can also do algebra with functions.
y=4x. We know what y is from Rule 1, that's the Math that Exists that you Do, to get the value of the observable multiplied by the wave-function. Say 4 is value of the observable, and x represents the wave-function, whatever hella complicated thing it is. Y is something that doesn't change, remember that there is certain math to measure energy, certain math to measure momentum, certain math to measure each thing that you want to measure. So if we are trying to measure the energy of an electron at a specific moment in time, then y is the Energy Math, and 4 is what we got when we took the measurement. Just for simplicity, I will say y is equal to 2. You know two parts of the equation, and two are all you need to solve it:
2=4x. If this is so, we can figure out what x is. x is equal to 1/2.
So you can figure out the wave-function. It has to be the wave-function that make the algebraic expression above work. That is the only wave-function it could be at that instant.
Rule 2 two helps us understand a last important thing about quantum mechanics, which is that even though the rules are set it up so that things are exact, the applications tend to be probabilistic. This is because these wave-functions are not simple numbers, like 1/2, but very complicated beasts, consisting of multiple variables. Rules 1 and 2 always apply: if you Do the Math that Exists, you get an exact value and a matching wave-function. But often instead of something simple, like an electron floating around in space, there are complications, like, an electron within an atom, a bunch of atoms together in a molecule. In these case, there are many possible states. An electron within at atom could be in one orbit, or another. There may be two electrons, and the proton matters too. In this case, the wave-function is often what we call a superposition of possibilities. Superposition is a term describing how waves interact with each other: unlike matter, waves can occupy the same space at once, they are superimposed on each other. The wave-function for a complex system is often a superposition of all the many, many allowed states that the system could be in.
In this case the wave-function does not tell us which possibility is reality, it just tells us the probability of each one. When you make the measurement (assuming, in a situation like this, that you could), you get one of these possibilities, and the wave-function becomes the function which describes that reality over any other. But until you make the measurement, there may be multiple possibilities for the state of the system, with different probabilities, and the wave-function will reflect that.
So A, B, and C, may all be options. Because of the Rules, we know that "sorta A sorta B", is not an option, or A.5, or however you want to look at it. It will be A, B, or C, when you take your measurement. Until you do, the wave-function, could it be determined, would tell you something like A is 50% probable, B is 45%, and C is 5%. If you take 100 measurements, you will measure A 50 times, B 45 times, and C 5 times.
And this is how it works in the world of the small. There are only certain ways to be. But there are also probabilities of being there, which govern how everything actually is.
What do these rules enables us to do and say about the universe? There are so many implications, and so many successful applications of quantum mechanics. The periodic table, all that stuff you might have learned and hated in chemistry about the strange shapes of electron orbits--all of that can be solved precisely with quantum mechanics. I will take the remainder of your reading attention to explain just one, particularly important example.
We owe a functioning sun, and in fact all stars, to quantum mechanics. It turns out that even inside the very hot sun, the collisions of all the protons are not energetic enough to enact the heating mechanism of the sun, in which two protons get close enough to overcome the Electrostatic force that usually repels like charges, are sucked together by the stronger but shorter-distance Strong Nuclear Force, and start the quark-swapping chain of events known as nuclear fusion. It just isn't hot enough. Yet we know for a fact that fusion does happen, and it's a pretty good thing for us that it does.
There happens to be a quantum behavior, called tunneling, in which--even though the energy required to create fusion is much higher than the energy of protons colliding at 15 million Kelvins--because of the probabilistic, rather than deterministic, rules of QM, there is a very, very small probability that the protons will "tunnel through the energy barrier", or, despite not having enough energy to fuse, will fuse anyway. This is kind of like how quantum mechanics says that if you throw yourself at the wall enough times, there IS a tiny probability that you will go through it. But go throw yourself into the wall, and I guarantee that even if you did it your entire life, you will not go through it, because that probability is so incredibly tiny. You can't take enough measurements to make that wave-function the reality.
The probability of two protons fusing anyway is also very, very small. Thing is, in the sun, there are, let us just say, a lot of protons. The sun is, after all, 300,000 times bigger than the Earth, and a proton is inconceivably small even when compared to an ant. Furthermore, it's darn hot in the center of the sun, meaning the protons there have loads of kinetic energy and are zipping around at incredible speeds. Lots of things in a a finite space moving very quickly means that those things are going to run into each other. Many millions of collisions per second, in fact. And so even if the probability for a single collision resulting in fusion is tiny, there are so many collisions happening, enough that yes, fusion does occur, the sun does shine, and life on Earth does reap considerable benefit.
So yeah. Try that out at a party sometime. "Did you know that fusion is due to the improbable possibility of proton tunneling within the sun?"
Saturday, March 13, 2010
Quantum Kitty
Last night I had a super nerdy dream.
My parents' cat is not allowed outside. Usually she's not just gunning to go outside either, being, as cats go, a fairly timid creature, but every now and then she does look longingly out the window. I know it's against their nature and all that, take that rule up with my mom if you think locking her inside is cruel. At least the local bird, chipmunk, vole, mouse, squirrel population can be thankful for it.
In the dream, a door to a house that was definitely not my parents' house but nevermind that, opened, and my parents' cat rushed to it, eager to get out. I tried to shut the door quickly, but the fuzzy dream events occurred in such a way that I could not tell if I was keeping the cat in or letting it slip out, my dream-brain ignoring the entirely-too-classical possibility that I ws shutting the door on the cat.
In the end, it seemed like I had kept the cat inside. Yet as I watched, the cat walked through the solid door, and suddenly there were two cats, one inside, and one outside.
And in the dream I exclaimed, "She just tunneled through the door! She bifurcated! Now there's two of her. Mom, what do I do about it?"
Dream mom was unsure, looking to the physics major for answers. "Don't you know how to fix it?" she asked.
"I don't know," I answered. "I don't know how to put her wave-function back together." But I went outside and scooped outside-cat up, and brought her back in, next to inside-cat, hoping that, in the language of quantum mechanics, if they were together it would induce their wave-function to be an eigenstate of inside-cat only, rather than being a superposition of the eigenstates of inside-cat and outside-cat both.
Now, the quantum concepts I'm referring to are not exactly right, obviously because cats are well outside of the limit where quantum behavior comes into effect. Also, the phenomenon in the dream, wheras something that, faced with two possibilities, behaves as if it does both if you are not able to determine what happened, goes away as soon as you can determine it (although yes, it would not actually happen with a cat.)
If I couldn't tell what happened and so suddenly I saw two cats, maybe some other sense, like touch or sound, would have broken the time of no-information, so as soon as I picked up outside cat, inside cat should have gone away, or rather, the cat wave-function would stop being a superposition of outside cat and inside cat and start being just outside cat again.
Then again, it didn't exactly occur that way: I couldn't tell what happened while I was closing the door but afterwards I saw inside cat "tunnel" through the it, which is an incredibly improbable occurrence, but if it did happen would not have caused the two-cat problem. So my brain was just mixing up scattered and sensationalized quantum concepts.
I did do a lab experiment once which verified the photon-faced-with-two-options-behaves-as-if-it-takes-both-if-you-can't-tell phenomenon, but in general these concepts are all subtle and mathematical and obviously non-nonsensical when applied to our world.
But cats are pretty magical, after all.
My parents' cat is not allowed outside. Usually she's not just gunning to go outside either, being, as cats go, a fairly timid creature, but every now and then she does look longingly out the window. I know it's against their nature and all that, take that rule up with my mom if you think locking her inside is cruel. At least the local bird, chipmunk, vole, mouse, squirrel population can be thankful for it.
In the dream, a door to a house that was definitely not my parents' house but nevermind that, opened, and my parents' cat rushed to it, eager to get out. I tried to shut the door quickly, but the fuzzy dream events occurred in such a way that I could not tell if I was keeping the cat in or letting it slip out, my dream-brain ignoring the entirely-too-classical possibility that I ws shutting the door on the cat.
In the end, it seemed like I had kept the cat inside. Yet as I watched, the cat walked through the solid door, and suddenly there were two cats, one inside, and one outside.
And in the dream I exclaimed, "She just tunneled through the door! She bifurcated! Now there's two of her. Mom, what do I do about it?"
Dream mom was unsure, looking to the physics major for answers. "Don't you know how to fix it?" she asked.
"I don't know," I answered. "I don't know how to put her wave-function back together." But I went outside and scooped outside-cat up, and brought her back in, next to inside-cat, hoping that, in the language of quantum mechanics, if they were together it would induce their wave-function to be an eigenstate of inside-cat only, rather than being a superposition of the eigenstates of inside-cat and outside-cat both.
Now, the quantum concepts I'm referring to are not exactly right, obviously because cats are well outside of the limit where quantum behavior comes into effect. Also, the phenomenon in the dream, wheras something that, faced with two possibilities, behaves as if it does both if you are not able to determine what happened, goes away as soon as you can determine it (although yes, it would not actually happen with a cat.)
If I couldn't tell what happened and so suddenly I saw two cats, maybe some other sense, like touch or sound, would have broken the time of no-information, so as soon as I picked up outside cat, inside cat should have gone away, or rather, the cat wave-function would stop being a superposition of outside cat and inside cat and start being just outside cat again.
Then again, it didn't exactly occur that way: I couldn't tell what happened while I was closing the door but afterwards I saw inside cat "tunnel" through the it, which is an incredibly improbable occurrence, but if it did happen would not have caused the two-cat problem. So my brain was just mixing up scattered and sensationalized quantum concepts.
I did do a lab experiment once which verified the photon-faced-with-two-options-behaves-as-if-it-takes-both-if-you-can't-tell phenomenon, but in general these concepts are all subtle and mathematical and obviously non-nonsensical when applied to our world.
But cats are pretty magical, after all.
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