Quantum Mechanics

[kazza]

Quantum Mechanics, while taking a little while to learn, is by far the most extraordinary theory in science. It completely flies in the face of everything you thought you knew about reality, calling into question some of our most basic assumptions about the universe, assumptions like cause and effect, determinacy, locality, and the existence of an objective reality regardless of whether or not anyone is observing it.

So, since the science forum has been so quiet I figured I would explain it to anyone that is willing to listen. This is prety much what I do all day at uni (and some other topics as well), and probably what I will be getting involved in for the rest of my life. Feel free to ask me any questions along the way, I really love telling people about quantum mechanics, it's especially nice on those nights when you're sitting out under the stars, having a few beers and pondering the meaning of life, but I guess an internet forum is the next best thing.

I should warn you there is a bit of learning involved, but no maths or anything like that. I'll try to keep it as interesting as possible, so don't be scared by any long posts, I'm going to try to condense down a year or so worth of physics into something fit for an internet forum, and while I'm only bringing up the really interesting ideas, for you to really understand just how amazing they are you need a little bit of an understanding of the concepts that underpin them. So like I said, feel free to ask me to clarify anything that you don't understand.

[kazza]

It all begins in pre-20th century physics, with debates over what exactly light is. In our world things can be split into two classifications – particles (like a tennis ball, a car, a plane, a mote of dust, or a crocodile) and waves (like the waves on an ocean or sound).

Particles are pretty straightforward, but waves perhaps deserve a second to think about. A wave is simply a transfer of ****** without a transfer of particles. When a sound wave travels through the air molecules vibrate back and forth in order to propogate the wave, but the particle itself ends up back in the same place it started. The image below illustrates this:

Note that if you watch an individual particle it never goes anywhere, even though there is clearly a wave travelling from left to right. Two waves can travel through the same space at the same time, they form a superposition of waves, which is the same idea just a more complex pattern. This is why two people can talk to one another at the same time and the sound waves pass right through one another.

Waves also demonstrate a property called interference, and this is crucial to an understanding of some of the experiments that I'll mention later. If you have two waves coming from opposite directions and they hit one another then one of two things can happen – they can cancel each other out or they can reinforce each other (called destructive and constructive interference respectively). Consider the image above – the wave can be thought of as dense regions and sparse regions travelling from left to right. If another wave is coming from right to left (you'll need to use your imagination) and impacting this one, then at some points a dense region from the left will hit a dense region from the right and you will get a super dense region, at other points a dense region will hit a sparse region and they will cancel out.



See in this diagram one ball is in a position where the two waves always cancel out, the other is in a position where the two waves always reinforce one another.


Ok, moving on, pre-20th century people weren't sure whether light was a wave or a particle. Most people thought that it was a wave because it demonstrated many of the properties of a wave like interference, and some other things like diffraction (the bending of waves around a barrier, which is why you can hear someone even if there is something directly between you). However, there were a few niggling things that couldn't be explained: blackbody radiation and the photoelectric effect. I won't go into detail on these unless you want me to, but Planck and Einstein found that these phenomena led to certain conclusions:

1-The blackbody effect required that things that emit light do so in discrete steps, that is to say that you can't just emit light with whatever frequency you like, but it has to be at certain intervals. It would be like me saying you are only allowed to produce sound at 20Hz, 40Hz, 60Hz etc.. but not at 23Hz or 41.713Hz.

2-Light is a particle. This comes from the photoelectric effect and was such an incredible discovery that it earned Einstein the Nobel prize. It is important but the explanation isn't really necessary if you trust me when I say that this proved that light has to come in discrete packets, just like a particle.

[kazza]

So now we have two competing, seemingly incoompatible ideas – light is a wave, and light is a particle. A great deal of work was done on this, and I'll just summarise some of the key things that were discovered in the early part of last century.

1-Light definitely comes in discrete packets, this was proved by Einstein, and we call these discrete packets “photons”.

2-Photons have momentum, which is another way of saying inertia. So when a light beam hits you it actually pushes on you, albeit very weakly.

3-The amount of ****** each individual photon has is given by the formula “******=H*frequency”, where the letter H is just a number that gets the units right.

Despite this, the particle theory of light still could not explain why light demonstrates things like interference and diffraction, which are exclusively properties of waves.

Now the astute reader may have realised something a little strange in what I posted above. Namely that the ****** of a photon is given by “******=H*Frequency”. What's strange about that? Well, I've just said that the ****** of the particle depends on the frequency of the wave.... This conundrum won't be resolved for a little while, but now its time to delve into the very heart of Quantum Mechanics with the most famous experiment in all of physics, Young's Double Slit Experiment.

It's a very simple experiment, you take a light, shine it on a wall that has two very small slits in it and then project the image of those slits onto a screen. The results you get should tell you whether or not light is a particle or a wave, because a wave will diffract as it goes through the slits and in some areas you will get constructive interference and in others you will get destructive interference. The end result, if it is a wave, should be a band of light and dark patches, like is illustrated below (The bright patches are where you have constructive interference, the dark patches are where there is destructive interference).
[IMG]http://www.blacklight*****.com/theory/DoubleSlit/Fig_37-2_Two_Slit_Waves.jpg[/IMG]

If light is a partcle then the result should be similar to shooting a machine gun at the two slits, and the pattern you get will be very different:
[IMG]http://www.blacklight*****.com/theory/DoubleSlit/Fig_37-1_Two_Slit_Particles.jpg[/IMG]

So Young did this experiment, and these were the results:

(Different pictures are due to different slit spacings)

A pretty clear indication that light is a wave no? Well then what about all that stuff Einstein did that proved light is a particle... well... Some people said that light is a particle, but when there are lots of them all travelling at once they can behave like a wave, so Young decided to repeat the experiment, but he would use a light source so weak that he could be certain only 1 photon would be in the experiment at a time (This took months for him, but it can be done better nowadays, I've even done this myself). The results are as follows: (Image a is after only a few photons have been fired, b after more etc...)



Now it is clear from photo A that only individual photons are being fired through the apparatus, but given enough time they eventually build up into the same pattern that Young expected to see for a wave. Somehow, even if there is only one photon travelling through the experiment it knows that there are two slits, and it knows what the interference pattern should look like. This diagram illustrates what is happening:

[IMG]http://www.blacklight*****.com/theory/DoubleSlit/Fig_37-3_Two_Slit_Experiment.jpg[/IMG]

So, I will finish up for now by asking a question... Of the individual photons you see in photo A, pick one and tell me, did that photon pass through the slit on the right or the slit on the left?

[Qi123]

it passed through both?

[kazza]

Originally Posted by Qi123

it passed through both?

Very good! Here's a smiley :).

(In another sense though, it didn't pass through either and it was a nonsensical question for me to ask, but that is all interpretation)

It turned out that our way of viewing the world was wrong, the world doesn't consist of particles and waves, it consists of things that experience "wave-particle duality" (we don't have a word for it, though some have been suggested. Things don't behave like waves, and they don't behave like particles, they just behave like they behave). Now the magnitude of these effects decreases as things get larger, which is why you don't notice it in everyday life. If you calculate the wavelength of a human being it is much smaller than the width of a single atom.

So, physicists had found that something that was once considered to be a wave had both particle and wave characteristics, the next step was to ask if something that was a particle experienced the same effects.

The experiment was repeated with electrons, firing single electrons at a time at the double slit, and the exact same pattern emerged. Then it was tried with neutrons, protons, even entire atoms. The most recent thing to have been tested is an atom called a "Bucky Ball" which consists of 60 carbon atoms. It is, relatively speaking, a very large atom. Now what will be really interesting, and what will probably give the philosophers a headache, will be when we are able to send something living through this experiment, like a virus.

This whole situation started to get people a bit worried. What exactly was going on? They could predict what would happen based on a few simple ideas relating to waves, but other than that there wasn't really any idea of what was going on with these atoms. Does an electron really exist? How can it possibly pass through two slits at the same time? What happens if you watch the slits to see which one it passes through?

This last question was one they could answer. Why not just set up a device that measures whether or not an electron passes through a slit, it should be very easy to do, and so they did it. Here's where things get (more) interesting, this is the result they got when they didn't check which slit it went through:
[IMG]http://www.blacklight*****.com/theory/DoubleSlit/Fig_37-3_Two_Slit_Experiment.jpg[/IMG]
And this is the result they got when they did check which slit it went through:
[IMG]http://www.blacklight*****.com/theory/DoubleSlit/Fig_37-1_Two_Slit_Particles.jpg[/IMG]
Apparantely the electron not only knows whether or not someone is watching it, but if someone is watching it then it changes it's behaviour! This really was (and still is) extraordinary! It doesn't matter how careful you are to make sure that your measurement doesn't interfere with the electron as it goes through, even in principle (and we'll get to why this is true) you can not ever know which slit the electron went through and still obtain the diffraction pattern.

This is a point that bears repeating, because it is a break from a very long held assumption. There is absolutely no way that you can observe the electron during it's voyage without making it revert back to it's particle nature. The act of observing changes reality.

(What precisely can be considered an observation is still a debate that is taking place, is subject to various interpretations and is also the focus of many experiments.)

[pancho]

Apparantely the electron not only knows whether or not someone is watching it, but if someone is watching it then it changes it's behaviour!

Seems as if we dont have the language to really describe this.
As how can an electron KNOW?
It could react to observation perhaps?

[dr poormouth]

We all get a little self-conscious when someone's watching us.

(sigh) Centuries of the scientific method leading us away from attributing human characteristics to non-human life and matter, and here comes QM, practically demanding that we anthropomorphise every smeggin' wavicle.

[kazza]

Originally Posted by pancho

Seems as if we dont have the language to really describe this.
As how can an electron KNOW?
It could react to observation perhaps?

Yep, it's bad wording on my part, an electron can't 'know' anything.

What is actually happening (and I'll get into all of this in a bit) is that when an electron isn't being watched it doesn't have a definite position. We are used to thinking that an object like a billiard ball has a position, but on a small scale there is a 'fuzziness' about the world. Things aren't in one position at a particular time, but they are sort of spread out all over the place. This is part of the reason the electron can go through two different slits at once.

[kazza]

Originally Posted by dr poormouth

We all get a little self-conscious when someone's watching us.

(sigh) Centuries of the scientific method leading us away from attributing human characteristics to non-human life and matter, and here comes QM, practically demanding that we anthropomorphise every smeggin' wavicle.

QM can be interpreted in an anthropocentric way, but there are other interpretations too. In fact, the most recent research is pointing towards it not being anthropocentric at all, but that an observation is merely an interaction between a microscopic system and a macroscopic system (but no one is quite sure where you draw the line between the two).

Anyway, for the purpose of trying to understand the theory, it is probably best if you just think of an observation as the sort of thing we do in the laboratory. We use a ruler to measure position, or we use some other tool to measure velocity etc... Until you get into some really deep stuff it doesn't make much difference what is considered an observation and what isn't.

[bogie]

Busy for awhile today but I'll get to your "assignment" as soon as I can. Never liked school so much. Good on ya Kaz.

[lexx]

Originally Posted by pancho

Seems as if we dont have the language to really describe this.
As how can an electron KNOW?
It could react to observation perhaps?

who cares about electrons!?what i want to know is!?how can YOU know ANYTHING!?(me too!?)but/and of course 1 cannot ignore the IDEA of RELATIVITY!?hehe!!......just askn..

[lexx]

Originally Posted by bogie

Busy for awhile today but I'll get to your "assignment" as soon as I can. Never liked school so much. Good on ya Kaz.

define "good"!?and do you think his status as super senior member implies super senior or super member and if so what's super about senior and what's super about member!?and no....dont go there!?hehe!!.....just askn...

[SubJunk]

Great thread! I think I'll sticky it even, if the lessons continue :)

[bogie]

Originally Posted by lexx

define "good"!?and do you think his status as super senior member implies super senior or super member and if so what's super about senior and what's super about member!?and no....dont go there!?hehe!!.....just askn...

Kazza is a young pup. It's my member that is senior. :cool:

[bogie]

Ok, I finally got time to go through it and I'm ashamed to say that for the things that I don't quite understand, I can't even formulate an intelligent question to ask. But I'll keep reading it over and over till I do.

[lexx]

please continue with the omitted black body photoelectric stuff!?hehe!!....just askn...

[kazza]

Originally Posted by bogie

Ok, I finally got time to go through it and I'm ashamed to say that for the things that I don't quite understand, I can't even formulate an intelligent question to ask. But I'll keep reading it over and over till I do.

Thats understandable. Not only does quantum mechanics does not naturally lend itself to expression via english, but the human brain has not *****ed to deal with the concepts presented in it. The natural language of the universe is maths, and in that QM is incredibly elegant, but hopefully things will become clearer as we go on.

I'm glad people are interested, I'll keep going with it tonight. :)

[kazza]

Before we go on, I just want to preface this by saying that it will be hard to understand. Don't be discouraged though, quantum mechanics is a complete break from the reality you are used to and I will go over things a couple of times and from a couple of different perspectives. Again, please ask me to clarify anything, and trust me, there are no stupid questions when it comes to QM.

When it comes to classical mechanics Isaac Newton discovered a few laws, you may be familiar with these – every action has an equal and opposite reaction, F=ma etc.. There was no “deeper meaning” behind these laws, but for some reason that is just the way the universe works. If you believe in a god then when god decided to create the universe he also decided that Force = Mass * Acceleration, and that every action has an equal and opposite reaction. These laws just exist, and they describe the way things in the universe behave. You just have to accept it.

The most basic problem in physics is this: You know the state of a system and you want to predict the state of a system at some later stage. If I know that two billiard balls are heading towards one another (and I know all the angles and velocities and masses), then I can use Newton's laws to predict how that system will change in the future. There may also be boundary conditions for the problem, for example the balls may be restricted to the surface of a table, and that table may have edges that the balls can't get over (these sorts of things are called boundary conditions - you can think of them as a description of the environment).

So if I know the state of a system, I can apply Newton's laws and then I know the state of the system in the future. *

In Quantum Mechanics there is something called the Schrodinger Equation. It has the same function as Newton's laws, and like Newton's laws it just exists for some reason and things in the universe follow it. This is the Schrodinger Equation:

Don't worry, I don't expect you to understand it, instead I'm just going to tell you what it does in English.

If we have a system, say an electron in an atom, and we know the state of that system at some point in time, the Schrodinger equation allows us to find out the state of that system at any later point in time. All we have to do is input the information, like how the electron and the proton interact, how many protons etc... and what the state of the system is intially, and the Schrodinger equation will spit out what the system is going to look like at some later time. (In practice, this is almost impossible for realistic systems. The most *****ful supercomputers in the world can't solve the Schrodinger equation for any molecule with more than 3 atoms.)

What is important about the SE (Schrodinger Equation) though, is that it where Newton's laws described the motion of particles, the SE describes the motion of a wave. Here is an example of the SE in action:
http://www.benfold.com/sse/timedep.html
When you open the page, there is a hump on the left, that hump represents something of interest, say an electron. The blue and red lines are the boundary conditions, so they are like the environment and the electron is going to interact with them in some manner. Now we know what the initial state of the system is (there is a big hump on the left) and when you click “start” the applet will simulate what happens to the system. So go ahead and click “start”.

The SE is what allows us the predict how that wave behaves. It tells us how it interacts with things like other electrons, atoms, walls etc... and also how it acts in the absence of these things.

Ok, so all the SE does is describe the motion of a wave. The next logical question is, what does that wave have to do with reality?

The simple answer is this, and this is perhaps the most important postulate in quantum mechanics:

The size of the wave at some point represents the probability of finding the particle at that point.

So in the applet given above, the fact that there was a big hump on the left meant that we had measured the system, and we knew the particle was on the left hand side. Since no experiment is perfect we weren't sure exactly where it was, which is why the wave is spread out over an area. Where the wave is highest is where the most likely place to find the particle is, where the wave is is small there is very little chance of finding the particle.

The Schrodinger Equation describes the evolution of a wave of probability. That wave of probability has all the characteristics of a normal wave – it can have constructive and destructive interference, it can diffract around barriers etc... all the things I mentioned earlier. So when the electron is fired at a double slit as in the experiments I mentioned above, this is what happens:

-The electron is described by a probability wave.
-The wave heads towards the barrier
-The wave passes through both parts of the barrier and so you end up with a diffraction pattern (the bands of light and dark)
-When the wave impacts the screen, that is a measurement, and so all of a sudden instead of being a wave of probability something actually happens and the electron is forced to be in a single place instead of being spread out. It could have appeared anywhere, and the probability of where it appears depends on what the wave looks like at that point.
-If the wave is big (which is the same as bright band when we were looking at light waves) then the electron is very likely to turn up at that spot.
-If the wave is small (which is the same as a dark band when we were looking at light waves) then the electron is highly unlikely to turn up at that point.

We call this the collapse of the wave function and this is why when we send individual electrons at the double slit, one at a time, we end up with exactly the same pattern as if we had sent a wave.



*This caused philosophers all sorts of headaches. Imagine if there was some all *****ful demon (traditionally called the Laplace Demon), and this demon simultaneously knew the position and velocity of every single subatomic particle in the universe. He could then sit down and do a whole lot of maths and predict the state of the universe at any time in the future. If this is the case then how can there be such a thing as free will? Luckily quantum mechanics changes all this. (Another way to say this is that classical physics is determinate, and quantum mechanics is indeterminate).

[kazza]

Now depending on how you have interpreted what I've wrote above, this may or may not sound all that exciting. If you have interpreted it the same way Einstein did then it probably doesn't.

Let's summarise what I've told you:

We start off knowing that a particle is roughly in some position, but since we don't know exactly what that position is we describe it by a wave. We can then apply a whole bunch of maths and we get a new wave. This new wave then describes the probability of obtaining a certain outcome when I measure the system.

So as an example lets consider an electron in a potential well (this is analogous to a ball bouncing back and forth between two walls, it can move around between the two walls but it can't escape). The wave predicted by the SE looks like this.

So, as you should realise, the size of the wave tells us how likely we are to find the particle at that point – hence the most likely place to find the particle is right in the middle, and its very unlikely we would find it near the edges. If I now make a measurement on the particle, I'm going to find it somewhere in the box, but the SE can't tell me where I will find it, only the probabilities.

Many scientists didn't like this (Einstein included). They felt that quantum mechanics was missing something. Clearly the particle is at some point whether you measure it or not. The fact that the SE can't tell you where just means that the SE is not a very good way to predict the position of the particle. If this is the case then there is nothing very special about quantum mechanics at all, all we have done is find a new way of predicting probabilities. I'm sure you can guess though, that it turns out this isn't the case.

There are various ways to think of quantum mechanics, but I will present only the two most important ones, and we will use the picture above as an example. Lets say I measure the particle and I find it somewhere over on the left hand side, lets call the place where I find it “point A”.

1) The realist: The particle was always at point A.
This is the most obvious choice, the one that most people would make, the one that many people made when quantum mechanics first came about. If the particle really was at point A then quantum mechanics is incomplete because the particle was at point A the whole time and quantum mechanics just couldn't tell us that. There must be more information available about the system, it's just that we don't have that information. The position of the particle was never indeterminate, it was just unknown to the experimenter.

2) The orthodox (or Cophenhagen) interpretation: The particle wasn't really anywhere.
It wasn't until we measured the particle that it appeared in a definite place. In the words of Jordan “Observations not only disturb what is to be measured, they produce it... We compel the particle to assume a definite position.” Prior to measurement the particle did not have a position, in fact the wave drawn in the picture above represents everything there is to know about the particle. Even in theory, an all knowing omniprescent god could not tell you anything more about the particle than that it exists as a probality wave with the shape of the wave in that picture.

Well, the two ideas presented here are very different, and the question needs to be asked how on Earth can you tell the difference between them? How do I know what the state of the particle is before I measure it, since I can't measure it to find out? The answer to this question didn't come until the 70's when Bell came up with an ingenious answer. The solution to this question is very elegant, but a little complicated, but there is actually an experiment that can tell the difference between the two positions I outlined.

If anyone wants me to go over this experiment I'll do it in another thread, because it will take a little bit of explaining, otherwise you will just have to trust me that it has been proven, almost beyond a shadow of a doubt, that the second position is correct. Prior to measuring the particle, it does not have a position, instead all that exists are probabilities.

[Qi123]

..............

At the subatomic level, matter does not exist with certaintity at definite places, but rather shows 'tendancies to exist', and atomic events do not occur with certainty at definite times and definite ways, but rather show 'tendancies to occur'. In the formalism of quantum theory, these tendancies are expressed as probabilities and are associated with mathematical quantities which take the form of waves.

This is why particles can be waves at same time. They are not 'real' three-dimensional waves like sound or ***** waves. They are 'probability waves', abstract mathematical quantities with all the characteristic properties of waves which are related to the probabilities of finding the particles at particular points in space and at particular times. All the laws of atomic physics are expressed in ter ms of these probabilities. We can never predict an atomic event with certainity; we can only say how likely it is to happen.

[SubJunk]

More, more!

[kazza]

Originally Posted by SubJunk

More, more!

Tonight :)

(Yay, stickied!)

[yossarian]

Originally Posted by kazza

Tonight :)

(Yay, stickied!)

You got stickied!?

Meh.

Brown-noser. :D

Seriously interesting stuff... I look forward to reading more, even if my scant background in science prevents me from comprehending anything more than than the basic, surface-level aspects of QM.

[SubJunk]

Originally Posted by yossarian

You got stickied!?

Meh.

Brown-noser. :D

Seriously interesting stuff... I look forward to reading more, even if my scant background in science prevents me from comprehending anything more than than the basic, surface-level aspects of QM.

Hehe, he deserves the sticky, this is a great thread.
If you write a lesson for another science topic you might get stuck too :)

[kazza]

Ok, rather than going on and introducing new concepts, I think I'll go over a couple of example situations. Hopefully they will help illustrate the ideas I've presented above.

The first thing we will consider is the infinite well:

The infinite well is analgous to a ball bouncing back and forth between two walls. If the world were pefectly frictionless then the ball would keep bouncing back and forth forever (on the quantum scale, everything is essentially frictionless).

Like this :)

Lets imagine I put the ball in the well, start it bouncing, and then walk away from it. I come back some time later, and I decide to do a simple experiment – I'm going to check the position of the ball (told you it was simple). When I have a look in the well, every position is equally likely to contain the ball. That is to say that the ball spends an equal amount of time in every position within the well. So if I divided the well into 10 parts, and labelled them A to J, then on average I would find the ball is in part A 10% of the time, it's in part B 10% of the time, it's in part C 10% of the time....

Now lets replace this with a quantum particle, say an electron in an infinite square well. I've already shown you what this looks like, but here it is again. This is the probability wave for an electron in an infinite square well:

Now already you should be able to see that this is different to the ball bouncing between two walls. For starters there isn't an equal distribution of probabilities. The odds of finding the ball in the center are much higher than the odds of finding it near the edges.

Now, I haven't given you the full picture just yet, but here it is. There are actually many possible waves that can describe an electron in an infinite well, the picture above is the one for when the electron has very little ******. If I add a little bit more ****** the wave looks like this:

(sorry about the inconsistencies in pictures, I have to find them on the net)
Notice that now the chances of finding the electron at the center of the well is zero, and the most likely places to find it are off to the left and right.

If I add a little bit more ****** the wave looks like this:

You may be starting to see a pattern here.

Now if I add a lot of ******, the wave is going to look like this:
[IMG]http://img399.imageshack.us/img399/6356/high******ta9.jpg[/IMG]
Lets consider what is happening here. As I add more and more ****** the distribution of probabilties is becoming more and more evened out. If I had an infinite amount of ****** then the wave would just be a flat line, which would mean I am equally likely to find the electron anywhere – exactly the same situation as our ball bouncing between two walls. It's not as easy to explain why, but the same thing will happen if I make the electron really, really heavy, or if I make the well really, really big. In other words, as the scale of this experiment starts to resemble something in our everyday world, the results also start to resemble everyday results.

If I were to calculate the wave that describes the ball bouncing between two walls, it would be almost indistinguishable from a flat line; if the ball is 1kg, the walls are 1m apart, and it has a reasonable amount of ******, the wave describing the ball would have roughly 1 billion billion billion billion billion billion billion peaks. This shouldn't be surprising, if quantum mechanics didn't agree with our everyday experiences it wouldn't be a very good theory.

As an aside, consider the second graph I presented above, the one with two peaks. There's a 50% chance that I will find the electron on the left, and a 50% chance that I will find it on the right, but there is zero chance that I will ever find it in the middle. If this is the case, how can the electron get from left to right without ever being in the middle? (There isn't really an answer to this question, just something cool to think about).

[kazza]

Now let's consider the case again where the particle is in the infinite square well. Now what we want to consider is what happens when I make a measurement. Again, this is what the probability wave looks like before I measure it (I'm assuming it has very little ******, just because it makes things simpler).

Let's say I open up my box, I have a look at my electron, and I find that it is in the region labelled C. We can never get an exact position, there is always some error (and this is fact is going to become very important later on), but we know that it is within some finite region.

What has happened to the wave function?

Well, we know with absolute certainty that the particle is in the region called C, and so the wave function now looks like this:

Evidently something has happened. The fact that we know where the electron is has changed the state of the electron. Prior to us making an observation the electron was quite happy just to exist as a wave. Even if the situation were more complicated, it would still have existed as a wave with the motion of that wave described by Schrodinger's Equation. All of a sudden, though things have changed.

This is the collapse of the wave function, that I mentioned earlier. It happens instantaneously when we measure the particle. It doesn't matter how careful we are not to disturb the system when we make the measurement, it still collapses. Measuring the system changes it irreversibly.

Lets again just quickly go over the two possible explanations. The first is that the particle was in the region C all along, it was there whether we measured it or not, we just didn't know about it until we measured it. The second is that the particle didn't have a position until we measured it, prior to measurement it was smeared out across the whole region and by making a measurement we have forced it to have a position. The second explanation is the correct one

So there are two things that happen in quantum mechanics. The first is that things progress along nicely as waves, with the motion of that wave described by Schrodinger's equation. The second thing that can happen is that the wave function collapses because someone measured the system. There is still no consensus as to what actually happens when the wave function collapses, or why it does so, the universe just seems to work this way.

[kazza]

Now lets look at one more example.

This time we are going to consider a barrier. The classical analogy would be this, imagine a ball rolling towards a hill. When it reaches that hill one of two things is going to happen. Either it has enough ****** to roll over the hill and keep going on the other side, or it doesn't have enough ****** and it will roll part way up the hill, then roll back down and go back the way it came. It doesn't matter how many times you repeat this, if you roll the same ball at the same hill at the same speed you will always get the exact same result.

The image below represents the quantum situation:

Time for an explanation....
On the left hand side there is an electron approaching a barrier. For reasons we won't go into, a free electron travelling through space is represented by a wave like that you see on the left hand side, but the correct way to interpret that wave is as an electron travelling from left to right. The shaded region in the middle is a barrier, and the electron does not have enough ****** to overcome the barrier, HOWEVER, you will notice that there is also a wave on the right hand side.

When we solve the Schrodinger Equation, we find that something odd occurs. The wave that we end up with looks like that in the picture above. It has a portion on the right hand side as well. What this means is that there is a probability that the electron will go straight through the barrier! The wider the barrier the less chance that it will go through, but if it is small then there is a decent chance that the electron will just keep going on it's way as if there were nothing there.

Think about this using the ball and hill analogy I gave at the start of this post. You roll the ball at the hill and it doesn't have enough ****** to go over, so it rolls up part of the way then comes back down again. You do this again and the same thing happens. You keep doing this and then on one occasion you roll the ball at the hill and the ball goes straight through! If you pick a bigger hill then it's less likely that the ball will go through, but there is always a small chance of it happening. In fact, if you run at a wall there is a vanishingly small, but finite, chance that you will go straight through it as if it didn't exist(don't try this at home).

This is actually a very big problem in computers. So far computers have been getting faster and faster and faster, but now the parts are so small that quantum mechanical effects are noticeable. You can think of an electron in a wire as being like the situation above, except with a barrier on either side and the electron trapped in the middle. As we make computer chips smaller and smaller the widths of the barriers become smaller and smaller and there is a chance that an electron will just jump right out of the wire. When this happens we lose data, or an operator doesn't give the right answer, or some other bad thing occurs that we don't want to happen in a computer chip. We are already reaching this limit, and so computer manufacturers are looking to new technologies, like devices that use light instead of electricity.

Also, this is the same phenomenon that allows us to use Scanning Transmission Microscopes, the microscopes that are responsible for the really, really, high *****ed images that you may have seen. This is an example of one:

The raw data from the microscope is just a series of numbers, so it has to be put into a computer to produce an image. This was done by IBM (duh..) and each of those spikes is an individual atom!

Now so far we have been considering this probability wave, and the size of the wave tells you how likely it is that you will find a particle in a particular location. Now if we measure the location of the particle many, many, many times and plot a graph of our results, the graph should look the same as the wave. This is precisely what has been done with a scanning tunnelling electron microscope, and these are the results:

The big spikes represent atoms. You can think of the circle of atoms being like an infinte well, and trapped inside that well is a single electron. The electron is demonstrating precisely the wave behaviour that we thought it would (except this time it is two dimensional). If it had more ****** there would be a greater number of peaks between the center of the circle and the edge. If it had less ****** there would be fewer. I would also like to reiterate, this is not a computer simulation, this is actual data from a microscope (it's just that we use a computer to produce a visual representation)

Just to clarify, this is actually a 2D situation, not a 3D one. Where there is a high point on the wave it doesn't mean that the electron is actually far from the surface, what it means is that it is very likely to be found there. Where there is a low point it means that the electron is very unlikely to be found there.

[bogie]

I doubt that what I'm learning here Kazza will really help me put together many intelligent explanations to others that I end up in debates with. But it does provide me enough knowledge to help me illustrate to others that quantum mechanics defies our familiar ways of veiwing our surroundings. I'll never be able to explain in any real detail the probabilities that you talk about but at least I'll be able to explain that quantum mechanics deals with varying degrees of probability and the Schrodinger equation proves it. At least mathematically. I assume also that all objects or particles obey the rules of quantum mechanics. Am I on the right track?

[kazza]

Originally Posted by bogie

I doubt that what I'm learning here Kazza will really help me put together many intelligent explanations to others that I end up in debates with. But it does provide me enough knowledge to help me illustrate to others that quantum mechanics defies our familiar ways of veiwing our surroundings. I'll never be able to explain in any real detail the probabilities that you talk about but at least I'll be able to explain that quantum mechanics deals with varying degrees of probability and the Schrodinger equation proves it. At least mathematically. I assume also that all objects or particles obey the rules of quantum mechanics. Am I on the right track?

Yep, you're on the right track. As far as we know everything on a small scale follows the rules layed down by quantum mechanics. The Schrodinger Equation works for most of the particles you may be familiar with - electrons, protons, neutrons... as long as they aren't moving to fast. When they approach the speed of light you need the Dirac equation. Some other particles use the Klein Gordon equation, or the Breit equation or a whole bunch of others, but they are all essentially the same thing just for different types of particles.

The most shocking thing about quantum mechanics, and relativity for that matter, is just how wrong our general perceptions are. For example: Cause and Effect - every effect must be caused by something, things don't just happen without a reason. But consider this situation:

You have two uranium atoms, one of them decays and spits out a few neutrons and some radiation, the other one doesn't. Prior to the decay they were both described only by probabilities, and they were identical in every way. Like I've mentioned for the Schroedinger equation, these probabilities are all you can possibly know about the system, there are no other "hidden variables" that influence which atom decays and which doesn't, they really are absolutely identical in every way. And yet one atom decays and the other doesn't..... Nothing "caused" the atom to decay, it was simply probabilities, hence we have an effect without a cause.

edit: oh, and as long as you are finding it an interesting read that is all that I'm worried about; I've been studying this stuff for 3 years and I'm only just scratching the surface, I've got years more study to go before I would consider that I have a good grasp of it. Feynamn once said "Anyone that says they understand Quantum Mechanics does not understand Quantum Mechanics."

[bogie]

Originally Posted by kazza

edit: oh, and as long as you are finding it an interesting read that is all that I'm worried about;.

It's interesting enough that I read everything you write about a half dozen times before I begin to get it. :cool:

[lexx]

Originally Posted by bogie

It's interesting enough that I read everything you write about a half dozen times before I begin to get it. :cool:

define "it".....hehe!!....just askn...

[pancho]

Many thanks for this kazza.
Have great holiday

[bogie]

Originally Posted by lexx

define "it".....hehe!!....just askn...

"It" is a teentsy weentzy teeny tiny itty bitty little baby scrap of knowledge that sooner or later gets planted way back in my brain somewhere. Doesn't mean I'll ever be able to pass it along but it's there. :D

[kazza]

Ok, lets now take a look at another highly unusual aspect of quantum mechanics – Uncertainty.

In a practical sense, when we make a measurement on something we have to disturb it in some way. We can try as hard as possible to minimise that disturbance, but our measurement will always have some effect. Say for example, you want to find the position of an electron. How do you go about doing that? The best way is to try bouncing a photon (light beam) off it and look for the reflection, but remember that photons have momentum, and so when you bounce the photon off the electron the electron recoils from the impact. You may also be able to do it by using a magnetic field, but the magnetic field imparts a force to the electron and changes it's motion.

Another way to think about it is that in order for us to make a measurement, there has to be an effect for us to measure, and if the particle of interest has an effect on our measuring equipment, our measuring equipment must have an effect on the particle. Every action has an equal and opposite reaction. Because of this, and because we just simply can't measure anything 100% accurately, there will always be an “uncertainty” in our measurement.

Now, for technical reasons I won't go in to, if we measure the position of a particle really, really well, we can't know very much about its momentum (think speed and direction when I say momentum, ie. where it's headed). In the case of measuring the position of an electron with a photon, if we want to know the electron's position accurately we have to use really high ****** light beams, and that means that we will bump the electron really hard and it could go flying off in any direction and at any speed.

The result of this is that we have something called an uncertainty principle:

The triangle with a 'v' means the uncertainty in momentum, and the triangle with an 'x' means the uncertainty in position, h/m just represents a very small number. So what this equation is saying is that if you take the uncertainty in position and multiply it by the uncertainty in momentum, it will always be greater than some number. Hence there is a minimum uncertainty that we can never improve upon, this is called Heisenberg's Uncertainty Principle. If you know the position of a particle infinitely well (ie. with no uncertainty at all) then the uncertainty in the momentum has to be infinite, and vice versa.

Now, we can arrive at this equation through purely practical considerations like I've stated above - there are very real, practical reasons why we there is always some uncertainty in our measurements, however, there is also an entirely different way to arrive at this equation and it is far more meaningful to go via that route. The way to do this, like most things in quantum mechanics, is to use the Schrodinger Equation. The derivation is long and tricky (and I actually had to do it on my exam yesterday), so I'll skip it, but it's important to realise that from the very fundamental description of how particles behave you can arrive at the uncertainty principle.

The fact that the uncertainty principle can be derived from the Schrodinger equation is extremely important – it means that uncertainty is not just something that happens in the lab, but it is a property of nature itself. Hence, it is not just the case that if we measure the position of a particle really accurately then we can't know what it's momentum is, rather if we measure the position of a particle really accurately then it doesn't have a well defined momentum. I can make an even stronger statement of this, and say that if a particle has a well defined position then it does not have a well defined momentum.

Again, this is not merely due to some limitation on our measuring devices, I really want to hammer this point home. Even if you could bulid the most accurate device ever – even if you could manipulate individual atoms and build something on an atomic scale, even if you could shrink yourself to the size of an electron so you could examine things yourself – there would still be uncertainty in the measurement simply because fundamentally a particle does not have both a well defined position and a well defined momentum at the same time. When we measure a particle's position we force it to have a well defined position and so we also force it to have a very “fuzzy” momentum. When we measure a particle's momentum we force it to have have well defined momentum and so it's position becomes “fuzzy”.

Actually, I guess thats a good way to think about uncertainty in position. Rather than it just being the case that we don't know the position of the particle, it's the case that it's position is “fuzzy” - the particle is kind of smeared out across a wide range of positions, as if it's out of focus or something. As soon as we try to measure it's position though, it comes back into focus – but the position at which it comes back into focus could be anywhere within that “fuzzy” area.

Lets do an experiement to illustrate the uncertainty principle:

First we measure the position of a particle.

We now know the position of the particle, and by measuring the position we have made the momentum fuzzy.

We now decide to measure the momentum of the particle - In doing so we have made the position fuzzy.

But don't we now know both the position and momentum of the particle? Well, lets go back and check the position of the particle again.

Huh? That's strange! The position of the particle is different now to what it was when we first measured it!

So what has happened? When we measured the momentum of the particle we made the position fuzzy. Again, it is not simply the case that we don't know what the position of the particle is, it actually doesn't have a position. When we then go back to measure the position a second time, it could turn up anywhere within the “fuzzy” area, and it chooses a place randomly (or semi-randomly, it does so according to the Schrodinger Equation). So even though we measured the position and then measured the momentum, we actually didn't know the the position and momentum at the same time, because by measuring the momentum we had made the position measurement obsolete.

Ok, I think maybe if I keep going over this I'm going to confuse the issue more than I'm going to clarify it, so let's stop there with the uncertainty principle. It will come up later though, it's very important in quantum mechanics.

This is a huge break from the reality we are used to. It turns out things simply do not have a definite position, or a definite speed and direction.

[kazza]

Originally Posted by pancho

Many thanks for this kazza.
Have great holiday

Thankyou, I'll do my best :) And you're quite welcome for the thread.

[bogie]

Originally Posted by kazza

Ok, I think maybe if I keep going over this I'm going to confuse the issue more than I'm going to clarify it, so let's stop there with the uncertainty principle. It will come up later though, it's very important in quantum mechanics.

This is a huge break from the reality we are used to. It turns out things simply do not have a definite position, or a definite speed and direction.

Thank god it's Friday. I need a drink. :cool:

[kazza]

Originally Posted by bogie

Thank god it's Friday. I need a drink. :cool:

Lol. It's saturday morning here and I'm hungover :cool:

[kazza]

I'll just mention as well that quantum mechanics has some very cool real life applications. I'll go into these in a bit more detail later, but for example my university is part of an international team that is working on building a quantum computer. This isn't just a computer on a really small scale, but its a computer that cleverly exploits some of these things we are talking about. It uses the fact that things are fuzzy like this (not position, but another variable) to create "qubits" instead of the regular "bits" that you have in a computer. A qubit, rather than being a 0 or 1, can be part 0 and part 1, so whereas a series of 8 bits i8n a normal computer can be any number between 0 and 255, a series of 8 qubits is every number between 0 and 255 simultaneously!

This will eventually revolutionise computing, but there are still technical challenges (so far they've only built a 1 qubit, or maybe a 3 qubit, computer). It will also be able to crack the uncrackable encryption that is used around the world RSID. Convential computers require ever longer periods of time to crack the RSID code - I think a 512 bit key was cracked in about 7 months last year, but a 1024 bit code would probably require 10 times as long - so as computers get more *****ful you just increase the length of your RSID key and no one will ever be able to crack it. Since a quantum computer can represent every number simultaneously in its quabits, even a very small quantum computer (1000 qubits or so), can crack the RSID code almost instantaneously.

[Zilenna]

WOWWWW Wot a great interesting and exciting and very pleasurable reading you have presented to ~All Of Us~
Thank you very much...
psss l am extremely interested in using the minds waves from the particles of the 'physical mechanics' to 'travel' to do so would absolutely mean 'reducing' the 'garbage' that we have all been trained in accepting....practically reverting back to being a child of the Universal Mother...l can do this, yet wonder greatly this aloness, as l know from experiences of 'traveling' we are very much alone in this Quantum Journey, even tho there are many around us in different stages of this 'being', it is still quietly alone....maybe l can redefine this perserverence and just go with the flow of a river that has always been there waiting for me...Tah

[lexx]

Originally Posted by bogie

"It" is a teentsy weentzy teeny tiny itty bitty little baby scrap of knowledge that sooner or later gets planted way back in my brain somewhere. Doesn't mean I'll ever be able to pass it along but it's there. :D

ok,let me re-phrase the question.what do you GET from it!?or get FROM it!?....get it!?(quote).....hehe!!....just askn..

[lexx]

Originally Posted by pancho

Seems as if we dont have the language to really describe this.
As how can an electron KNOW?
It could react to observation perhaps?

sorta looks like a law of attraction!?wherever you are,(observe/measure!?)there i'll be/become!?why!?cause i dont/wont/cant..... know anything!?unless....i find you!?hehe!!.....just askn...

[bogie]

Finaly back in class. A lot of catch up reading to do. Sorry but I didn't bring an apple kaz.

[kazza]

Originally Posted by bogie

Finaly back in class. A lot of catch up reading to do. Sorry but I didn't bring an apple kaz.

This is the second time you've been late this week son. Once more and I'm going to have to call your parents. ;)

I'll post some more over the weekend, there is still mountains of really cool stuff that will blow your mind.

[bogie]

Originally Posted by kazza

This is the second time you've been late this week son. Once more and I'm going to have to call your parents. ;)

I'll post some more over the weekend, there is still mountains of really cool stuff that will blow your mind.

I'll just say that my dog ate my homework teach. If I told you the real reason I was gone you'd never believe it. :cool:

[yossarian]

Originally Posted by kazza

I'll just mention as well that quantum mechanics has some very cool real life applications. I'll go into these in a bit more detail later, but for example my university is part of an international team that is working on building a quantum computer. This isn't just a computer on a really small scale, but its a computer that cleverly exploits some of these things we are talking about. It uses the fact that things are fuzzy like this (not position, but another variable) to create "qubits" instead of the regular "bits" that you have in a computer. A qubit, rather than being a 0 or 1, can be part 0 and part 1, so whereas a series of 8 bits i8n a normal computer can be any number between 0 and 255, a series of 8 qubits is every number between 0 and 255 simultaneously!

This will eventually revolutionise computing, but there are still technical challenges (so far they've only built a 1 qubit, or maybe a 3 qubit, computer). It will also be able to crack the uncrackable encryption that is used around the world RSID. Convential computers require ever longer periods of time to crack the RSID code - I think a 512 bit key was cracked in about 7 months last year, but a 1024 bit code would probably require 10 times as long - so as computers get more *****ful you just increase the length of your RSID key and no one will ever be able to crack it. Since a quantum computer can represent every number simultaneously in its quabits, even a very small quantum computer (1000 qubits or so), can crack the RSID code almost instantaneously.

Question: Have you ever read Timeline by Michael Chrichton?

In the book, scientists utilize quantum computers to travel through time. While I understand the time-travel thing was pure science fiction, I'd be interested to know how accurate his description of the quantum computers was.

If you're ever in a bookstore, and wouldn't mind scanning the first few chapters -- the part explaining quantum computers is only about half a dozen pages long -- let me know how much he says is realistic and how much is a liberal use of the creative license.

[yossarian]

...oh, and I forgot to ask... as someone within the field, how long do you think it'll be before quantum computers will be ready for practical use? 15-20 years? Longer?

[kazza]

Originally Posted by yossarian

Question: Have you ever read Timeline by Michael Chrichton?

In the book, scientists utilize quantum computers to travel through time. While I understand the time-travel thing was pure science fiction, I'd be interested to know how accurate his description of the quantum computers was.

If you're ever in a bookstore, and wouldn't mind scanning the first few chapters -- the part explaining quantum computers is only about half a dozen pages long -- let me know how much he says is realistic and how much is a liberal use of the creative license.

No, I haven't read it, but if I get a chance I'll browse over it. I'll also write some stuff about quantum computing on this thread within the next week or so if you like (before I leave for the US), so you can work out for yourself how much of what he said was real and how much wasn't.

...oh, and I forgot to ask... as someone within the field, how long do you think it'll be before quantum computers will be ready for practical use? 15-20 years? Longer?

I don't know nearly enough to give my own estimate, but my lecturers are heavily involved with the project and they give pretty much that time frame, somewhere between 15 and 20 years.

Of course, there are rumours that the US government has built one and is using it to decypher encrypted transmissions, but I don't think there is much credibility to that. On the other hand, if someone were able to build one and no one else knew about it, they would have free access to every piece of information that is transmitted via the internet - bank accounts, personal details, medical records, national secrets...

[yossarian]

Originally Posted by kazaa

No, I haven't read it, but if I get a chance I'll browse over it. I'll also write some stuff about quantum computing on this thread within the next week or so if you like (before I leave for the US), so you can work out for yourself how much of what he said was real and how much wasn't.

Thanks, I'd appreciate that. Fascinating stuff.

Of course, there are rumours that the US government has built one and is using it to decypher encrypted transmissions, but I don't think there is much credibility to that. On the other hand, if someone were able to build one and no one else knew about it, they would have free access to every piece of information that is transmitted via the internet - bank accounts, personal details, medical records, national secrets...

That's not a comforting thought.

[kazza]

Ok, as a prelude to Quantum Computing we're going to need to look at two-state systems. In order to do that we need to introduce a concept called Spin

Despite the name, spin has nothing to do with whether or not an object is spinning. In fact, it is probably best at this stage if you don't worry about precisely what spin is, just worry about some of its properties. In short:
-Every particle has a well defined spin value, and this never changes. It's just a property of the particle, like mass, or charge, or size.
-For Quantum Computing we are only going to be interested in electrons, and electrons have a spin of ½.
-We can never measure the total spin of an electron, but we can measure how much of it points along one direction (*see below for an example of what I mean by this).
Now this is the important bit: If we measure the spin of an electron we will either get a value of +1/2 or -1/2. To make things clearer we call these up or down. So if I measure an electron's spin, I will either find it to be pointing up, or pointing down.

Also important: When we were looking at position we found that if we aren't looking at it a particle doesn't have a definite position. Something similar happens with spin:
-If I measure the spin of the electron and find that it is spin up then it will stay that way. If I measure it and I find that it is spin down it will stay that way as well.
-However, I can do something to it that will mean it is no longer well defined. Instead of being either up or down it will be part up and part down at the same time. This isn't to stay that instead of being either +1/2 or -1/2 it is now +1/5 or something. It can only ever have a value of +1/2 or -1/2, but it can be both at the same time (don't think about this too hard, you'll get a headache).

(Now I said that if I measure it and find it spin up it will stay that way. In fact it will only stay that way for a little while, and this is one of the big difficulties with actually building a quantum computer. It's why they have to be built at very low temperatures, because at low temperatures the electron will stay the way we want it to for longer.)


* This is what I mean by measuring how much of the spin points in a particular direction. Don't worry about the letters and numbers on this picture.

Imagine the blue arrow is the spin of the electron. It's pointing in some random direction. Now when I say “how much of that spin is pointing in the horizontal direction?” I'm talking about the length of the red arrow. When I say “how much of the spin is pointing in the vertical direction?” I'm talking about the length of the green arrow.

Spin is kind of like an arrow, it can point in any direction it likes, but when we measure it we only measure how much of it is pointing in a particular direction. So we can measure the length of the red arrow or the length of the green arrow, but not the blue arrow. Because of certain effects that you don't need to worry about for now, when we measure this, no matter which direction we choose, we either get +1/2 or -1/2.

[kazza]

Reading that back I may have made the issue more complicated than it needs to be.

Forget the + or - 1/2 bit, this is what you need to know:

If I measure an electron it will either be spin up or spin down. After I have measured the electron it will stay that way.

I can do something to the electron that will put it in a superposition of spin up and spin down. Another way of saying this is that it is part spin up and part spin down. In the previous examples I gave with the waves trapped in an infinite well, it's like saying the particle is partly in the left half of the box and partly in the right half. It's not simply the case that we don't know whether the electron is spin up or spin down, it really is part spin up and part spin down.


This is all you need to know about spin to be able to understand quantum computing.