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?"

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