Quantum Mechanics

[Dragar]

Introduction

I don't know how many posts I've got the drive for, but I'm hoping I can put together some sort of introduction for this subject for anyone who cares to try understand it. This is partly to sharpen my science-writing skills, partly to sharpen my understanding, partly because I like talking about it (and like hearing myself talk about it), and partly because I think people might find it interesting. I'll leave it to you to work out which of those are the main reasons.

There will, I'm afraid, be two areas that will be very lacking in this.

The first will be mathematics. I'm intending avoiding any and all mathematics. This will be difficult, and I may end up resorting to graphs and diagrams at various points. But I'm hoping to provide at least a vague sort of picture of this theory for people who can't stand the sight of an equation.

To those who enjoy mathematics, I apologise, but upon demand I can point you to a number of wonderful books or websites which can explain the concepts here alongside the mathematics which is (if I am to be honest) fundamental to the subject.

The second will be a historical basis for the theory. I can provide rough names and dates (and I will throw bits I know in there) but I am no historian of science. If anyone wishes to

I hope if I should make a mistake, or fail to explain something clearly, I will have this pointed out to me so I may correct myself.

Finally, I think I can probably make a stab at what most people who haven't studied the subject in depth will already know:


Quantum mechanics is a theory that looks at the behaviour of the smallest parts of nature.
The theory is 'strange', and produces counter-intuitive results.
There are a number of different interpretations of the theory, some more bizarre than others.


I'm not going to assume anything more. I'm going to explain things right from the ground up, and start in the same way most authors on the subject seems to start. I'll begin with the double slit experiment.

The Double Slit Experiment


Part 1 - Bullets

The experiment is a remarkably simple one - and you may be surprised it has caused us so much trouble!

The set up is rather trivial. We have a screen at the back of the room. Some distance away, we place a barrier of some sort, but cut two, small slits in it, a little distance apart.

Then, we fire things at the slits, and see where they land on the screen at the back of the room. Sounds like fun, doesn't it? We have to make sure the barrier cannot be penetrated by whatever we're throwing at it (or at least be as sure as we can), as well as make sure stuff can't go around it. But the people carrying out the experiments know what they're doing; so we can assume this has been done.

Now, first of all we need to decide what we're going to fire out the slits. We're going to start with bullets - not because they do anything mysterious, but because it's good to compare our later results with objects we're fairly familiar with. Bullets are a fairly good picture of how we used to view physics. This old picture was called 'classical physics', and involved particles - a bit like bullets, but obviously much, much smaller.

So, here's what our set up looks like:

http://homepage.ntlworld.com/stephen.hessey/gun1.gif

A few assumptions, first. We're going to pretend that the bullets are magic. They don't shatter on impact, or bash through the screen, or do anything real bullets do. We're also going to pretend we're able to count all the bullet holes perfectly, and note where they appear on the screen. We are idealising the scenario - pretending we're perfect experimenters with perfect equipment.

We're also going to make sure we fire them one after the other - not lots at a time. This is to make sure they don't bump into each other along the way. And the final assumption is that the bullets fly randomly toward the barrier. Our gunner is a pretty terrible shot - he could well be drunk!

Okay. Let's look at what happens if we close the lower slit, and let rip!

http://homepage.ntlworld.com/stephen.hessey/gun2.gif

What you can see on the right hand side there is just an image that shows the distribution of how all the bullets have landed - the 'higher' (or in this case, the further to the right) the line is, the more bullets have struck at the point. Where it is peaked, we find the most bullet holes. You can see that it's peaked directly opposite the slit, and gradually tapers off at the sides. This just shows that directly opposite the slit, we find the most bullets holes. And as we move further away from the point directly opposite the slit, we find fewer and fewer holes.

We find a pretty similar thing if we cover up the upper slit, and do the same again.

http://homepage.ntlworld.com/stephen.hessey/gun3.gif

Only this time, I hope you'll note, we find the most bullet holes have appeared directly opposite the lower slit - the one that was open this time.

Finally, we'll run the experiment with both slits open:

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/DoubleSlit/gun4.gif

Well, this shouldn't be too surprising, either. All we've found is that the the distribution of the bullet holes appears to be the sum of each of the single-slit experiment distributions. We just add the heights together - the dashed green and red curves show you how this works.

So, to summarise that:


With just one slit open, the distribution of bullet holes is peaked directly opposite the slit.
With both slits open, the distributions just add together.

[Dragar]

Part 2 - Water Waves


We're going to continue now, and take a look at how water waves react to this situation. First of all, let's take a peek at our new set-up.

http://homepage.ntlworld.com/stephen.hessey/water1.gif

Instead of a machine gun, we now have something tapping the surface of the water at regular intervals. That's the black object you can see in the middle of the concentric circles. We call that the 'source'.

This produces 'ripples' - or water waves. We know what they are; they're just disturbances in the height of the water. On the diagram, you can see them as the concentric circles, spreading out from the source. You'll have seen the same thing if you've ever dropped a pebble into a pond.

The two slits are the same, as before. But the detector is obviously different. Instead of counting bullet holes, we have a cork, which will bob up and down as the waves arrive (remember that the waves are disturbances in the height of the water). We can measure the wave-energy - how big a disturbance - arrives at a point by watching how the cork bobs up and down at that point.

Well, let's begin. We'll do the same as before, and close off one slit.

http://homepage.ntlworld.com/stephen.hessey/water2.gif

While it's not identical to the distribution we found for bullets, I hope you can see that it's pretty similar. As a point of interest, note how the waves start spreading out from the slit as they come through. You can see the same thing at the coast, if you watch waves coming through a narrow gap.

And for the other slit?

http://homepage.ntlworld.com/stephen.hessey/water3.gif

Just as before, the distribution of wave energy is pretty similar to the distribution of bullets.

But what about if we open both slits?

http://homepage.ntlworld.com/stephen.hessey/water4.gif

Whoa! What's happened here? The distributions are not just adding any more. In some places, the wave-energy is much greater than the sum. In other, the wave-energy has gone far lower than the height it was with just a single slit open!

This effect - which happens with all sorts of waves - is called interference. We can even explain it - but it requires to look at how waves add together.

As the waves leave each slit, they travel toward the screen. But, with the exception of right in the middle of the screen, the distance each wave travels to reach the screen is different. This means that their 'waving' may not be in sync any more.

If the two waves arrive completely out of sync with each other (or, if you want to use the words physicists use, out of 'phase'), we find the following:

http://homepage.ntlworld.com/stephen.hessey/destruct.gif

The low part of the first wave 'cancels out' the high part of the second wave - and vice-versa. The result is that you have no wave at all! It cancels out! This is what happens when we find a far reduced wave-energy at a point. We call this a minima.

What about if they are 'in step' with each other (or in phase):

http://homepage.ntlworld.com/stephen.hessey/construct.gif

In this case, the high parts add to each other and the low parts do too - making an even bigger wave! This is what happens when we find a 'peak' in our distribution (like the one at the centre). We call this a maxima.

Depending on how in phase the waves are with each other at the point they reach the screen, you can end up with 'in between' situations, where they partially add or partially cancel.

So, to summarise waves:


With just one slit open, the distribution looks pretty much like that of particles.
With both slits open, the distributions display 'interference' - the waves coming from each slit can get out of step with each other. When they're out of step, we find a minima, and where they're in step we find a maxima.



Now, I've prattled enough. I'll talk later about doing this with electrons, but first - are there any questions? And - for the more knowledgeable among you - have I made any mistakes?

Images (as well as a lot of the format for the descriptions, though I've re-worded things in a way more suited to a general audience - I hope) taken from here (http://www.upscale.utoronto.ca/GeneralInterest/Harrison/DoubleSlit/DoubleSlit.html).

I suppose it's not really fair for me to leave you with such a cliff hanger. So, if you really want to know what happens when we run the experiment for electrons, you can perform it yourself (http://phys.educ.ksu.edu/vqm/html/doubleslit/). Or, at least, see the results appearing. I hope the instructions are fairly self explanatory on this site.

[justaman]

Keep it coming, baby.
:popcorn:

[Crumb]

For what it is worth my physics minor says you are doing great. :super:

[livius drusus]

Fanfreakingtastic, Dragar. I'm someone who couldn't get through A Brief History of Time, so believe me it means a lot that I find your intro eminently understandable. I love the illustrations too. :bow:

Just as an aside, you can float images to the right or left of text, if you feel the need, but using the float code (http://www.freethought-forum.com/forum/misc.php?do=bbcode#float).

[viscousmemories]

Excellent intro, Dragar. Thanks. :)

Just FYI the link to where you found the images is broken.

[LadyShea]

Can we copy this into an article?

[livius drusus]

Definitely, Shea. I was thinking that myself. We should wait until after we upgrade our current article hack, though. That'll give Dragar a chance to finish too.

[Dragar]

Thanks all; I'm glad you're all following. Thanks for the code tip too, livius. I searched around the net for the best images in terms of explanatory power, and I thought they were the best. Link is corrected, VM. Thanks for pointing it out.

I guess we're ready to continue!

Part 3 Electrons

Last time, I finished explaining the double slit experiment, and the differences of behaviour we note when we compare particles (like bullets) to waves (like water waves). This time, I'm going to talk about electrons.

Depending on how much or how little you know about atoms, you may or may not know what an electron is. So we'll begin with a little bit about that first. To do that, I have to talk about atoms.

The idea was first proposed by Democritus, an ancient Greek, who lived over 2000 years ago. While the idea didn't really take hold back then, it reappeared in physics as an explanation for various experiments, and turned out to one of the most important discoveries in our understanding of Nature.


http://homepage.ntlworld.com/stephen.hessey/atom.jpg

To the right is a diagram of the classical (pre-quantum mechanics) theory of the atom. In the centre, we find the nucleus. This is composed of two particles. These are protons, which have a positive charge, and neutrons, which have no charge. The number of protons determins the type of atom. Hydrogen atoms just have a single proton, while carbon has six. The number of neutrons has a little bit of flexibility, but is usually around the same number of protons. Hydrogen usually doesn't have any (but sometimes can aquire one), while carbon usually has six (the same number as it has protons), but again, sometimes can aquire an extra one, or even two!

It was thought electrons orbited the nucleus, just as the planets orbit the Sun. But while gravity was what kept the planets in orbit, what held the electrons in orbit? There needed to be some attractive force, and it turned out there was. Protons have a positive charge, and electrons have a negative charge. And the two key points about charge can be summarised as 'opposite charges attract, identical charges repel'.

A proton, with a positive charge, would therefore attract the electrons, which had a negative charge, and keeps them in orbit.

How big was the atom thought to be?

Well, if you imagine a football stadium, and you pretend the electrons are whizzing around the outside of the stadium, the nucleus would be about the size of an ant. In other words, these things were tiny tiny tiny! And most of the atom was empty space!

What about the electrons themselves? Well, if you take the mass of a proton - and you divide it by two thousand - that's the mass of an electron. Of course, this doesn't really give you an idea of how big they are, but due to this tiny mass, they were thought to be pretty small - so small they could be thought of as occupying a single 'point' in space. Despite this dramatic difference in mass, electrons carry the same amount of charge as a proton (though negative instead of positive).

There are some problems with this picture, which I'll return to in a little while. And I haven't covered half the details (why do the protons in the nucleus, which all have the same charge, not repel one another?). But it's worth to keep in mind this old model.

Well, now we know what electrons are, we can run our double slit experiment with them! So let's do that!

Unless you decided to sneak a peak at the results of the experiment with electrons, you won't know what happens. So I'll describe it, bit by bit.

We have same set up as before: a screen at the back of the room, and a barrier with a pair of slits in front of it. This time, though, we don't have a machine gun or a source of water ripples. Instead we have an electron gun. Don't look at me funny, it really is called that! Using this device, we fire a stream of electrons at the slits.

Of course, we can't count bullet holes any more, or use a cork to bob up and down. Instead, we cover our screen in something that reacts with the electrons. We get a little white spot every time an electron hits.

What do we get out?

First of all, we send them through with one slit closed. This looks pretty much like before - just a distribution that peaks opposite the open slit, falls away either side. And we get the same with the other slit open. No surprises there.

But what about with both slits open?


http://homepage.ntlworld.com/stephen.hessey/electron1.jpg


To begin with, it just looks like a random distribution. There's nothing really remarkable here yet. The image shows what the screen looks like after the first 100 electrons have been sent through.



But as we put more and more electrons down, something interesting appears.









http://homepage.ntlworld.com/stephen.hessey/electron2.jpg







After about 3000 electrons, we see the beginnings of a pattern emerging.







http://homepage.ntlworld.com/stephen.hessey/electron3.jpg





And after 70,000 electrons, there is no denying it - what we have found is an interference pattern for electrons!










"Well", you might say, "Could it be that the electrons are somehow interfering with each other as we send them through?"

And, being good scientists, we check this. We do the experiment by sending electrons through one at a time. Same results!

So, let us now think about what this means. How can it be that something like an electron - which, as we know from the point on the screen, exists at a 'point' - somehow have the probability of where it ends up determined as if it were a wave?

This is the mystery. This is, at it's heart, what quantum mechanics is about - predicting the behaviour of the smallest constituents of Nature. We cannot call them waves - because they appear when we detect them to be at specific points. They come in particular 'packets', with a certain energy, mass, charge - they are quantized.

But we can neither call them particles - because particles do not produce the dramatic interference pattern we have found.

"But what can I call it? If I say they behave like particles I give the wrong impression; also if I say they behave like waves. They behave in their own inimitable way, which technically could be called a quantum mechanical way. They behave in a way that is like nothing that you have ever seen before." - Richard Feynman

The experiment has been repeated with other particles - neutrons, protons, photons (light), and all with the same results. They come in 'packets', existing at a 'point' when they interact with matter - but their behaviour follows a sort of wave-like pattern to it.

To summarise:


the tiny constituents of Nature always appear to be particles when they're observed (like as dots on the screen).
But their behaviour between observations has a wave-like character (resulting in an interference pattern building up over a great many observations).




Next time, I'll talk a little bit about the development of a new theory to explain these results, the Schrödinger equation, and the psi-function. But I promise, no maths will be involved.



Image of the atom taken from Your Dictionary (http://www.yourdictionary.com/ahd/a/a0501900.html)

Image for the electron double slit experiment results came from Brock University website (http://www.physics.brocku.ca/courses/1p22/)

[Crumb]

The plot thickens... :comfy:

[Ensign Steve]

more more more!

:popcorn:

[beyelzu]

I have read a brief history of time, in search of schoedinger's cat and other books about the subject, but I must say dragar that your explanation is particularly badass.

[livius drusus]

:cough:

[Dragar]

Not forgotten, just been a bit busy with studies, which included revising for a test on this subject, which took place today!

in search of schoedinger's cat

I've read the sequel, Schrödinger's Kittens (actually, I read it the other day when I found it in the university library, and devoured it in an afternoon). Well worth digging out, for anyone who's interested in this stuff.

[Shaguar]

Dragar, I spend a large amount of my time building and supplying Collimators to Rutherfors, ANSTO, Oak Ridge, Cern et al, this thread is home from home :D

[Dragar]

The Strangest Theory You Have Ever Seen

Part 1 - A Little History

Last time, I'd finished telling you about the Double Slit Experiment - which was just one of the many experiments that laid the groundwork for quantum mechanics. But there were a great many other experimental and theoretical reasons for the theory, including the photo-electric (http://www.physlink.com/Education/AskExperts/ae24.cfm) effect, problems with black bodies predicted to emit 'infinite energies' (called the 'ultraviolet catostraphe' (http://en.wikipedia.org/wiki/Ultraviolet_catastrophe)), which are two of the most interesting examples. But all of these things seemed to be pointing to a strange mix of properties in the smallest constituents of matter, a combination that we had never seen before.

All the things we had classically thought were particles - because we always found them in 'lumps' or quanta - seemed to wander around like waves. In the same way, light - which we had very good reasons to think of as a wave - turned out to also arrive in individual quanta of energy (which we call photons).

You may remember, when I first introduced you to the electron, that I told you the picture I presented - of an electron orbiting the nucleus as the planets orbit the sun - was not quite right. In fact, it's a lot worse than 'not quite right' - it's completely wrong!

At the time, the current theory of electromagnetism - the theory of how electric and magnetic fields worked - predicted that an electron orbiting an atom in such a way would be terribly unstable. In far, far less than a thousandth of a second, the electron was predicted to crash into the nucleus, emitting light (electromagnetic waves) as it did so.

Many people tried different ideas for resolving this, and a very clever physicist named Böhr had a bright idea. He suggested that the electron really was a sort of 'wave', and only certain orbits were around. Why were only certain orbits allowed?

Well, if you think back to when we discussed waves, I talked about interference, and how sometimes waves cancel out, and other times they add up. If you set up a wave going around in a circle, you'll find that only certain waves will survive - most waves just 'cancel out' due to interfering with itself. This meant only certain orbits were 'allowed' - ones where a whole number of wavelengths (the length of a wave, before it starts to repeat) filled the circle. All the others would just cancel out!

Each orbit corresponded to a different, discrete value of energy, called an 'energy level' - and these were the only allowed energy values of the electron.

Böhr's model worked for hydrogen - it gave us the right predictions for the energy levels - but gave us the wrong results entirely for helium, and many other atoms.

It was clear a new theory was needed.

"Once at the end of a colloquium I heard Debye saying something like: Schrödinger, you are not working right now on very important problems...why don't you tell us some time about that thesis of de Brogle which seems to have attracted some attention?" So, in one of the next colloquia, Schrödinger gave a beautifully clear account of how de Broglie associated a wave with a particle...

When he had finished, Debye casually remarked that he thought this way of talking was rather childish...To deal properly with waves, on had to have a wave equation."

- Felix Bloch, Address to the American Physical Society (1976)


http://homepage.ntlworld.com/stephen.hessey/erwin.gif

Erwin Schrödinger (pictured right) and Werner Heisenberg (not pictured)
independantly came up with two, new, seperate theories. Heisenberg's worked with highly abstract mathematical devices called matrices, and was called 'matrix mechanics', while Schrödinger had a 'wave equation', and so was called 'wave mechanics'. Not surprisingly, the two weren't overly fond of each other's theories - especially since they both gave the same results!

Schrödinger - along with much of the physics community - found Heisenburg's methods far too abstract and unusual to visualize.

I knew of [Heisenberg's] theory, of course, but I felt discouraged, not to say repelled, by the methods of transcendental algebra, which appeared difficult to me, and by the lack of visualizability.

-Schrödinger in 1926

Schrödinger, on the other hand, appeared to have a theory that resolved these difficulties - though even with his theory, there were some difficulties in the interpretation, as Heisenberg commented...

The more I think about the physical portion of Schrödinger's theory, the more repulsive I find it...What Schrödinger writes about the visualizability of his theory 'is probably not quite right,' in other words it's crap.

--Heisenberg, writing to Pauli, 1926

Later that year, it was shown the two theories were mathematically equivalent. Both were just different mathematical descriptions of the same theory! But Schrödinger's description tended to be favoured over Heisenberg's. After all, it appeared to actually have an interpretation attached to it - that of particles moving around as waves.

But was it really that simple?

I'll continue the discussion of the Schrödinger equation next time.

[Ensign Steve]

:super:

[Crumb]

This is good stuff Dragar. More than worth the wait. :yup:

[livius drusus]

I loudly second that. This is my favorite episode yet. It's an actual cliffhanger. :appl:

[davidm]

:think:

So, the particle behaves like a wave when it isn't observed or detected, and it behaves like a particle when it is.

This means something about the act of observation reduces the wave to a point. But what? What could possibly do that, and how would that work?

Also, what is the electron interfering with when it passes through the slits to make a wave pattern? Itself? How?

It almost sounds as if there must be different versions of the particle passing through the slits -- hidden doppelgangers attending the path of each electron; hidden, that is, until their existence was revealed by the double-slit experiment.

And, since all big things, like people, are made of particles, it seems to follow that there must be hidden doppelgangers of ... uh ...:mirror:

:shocked:

But perhaps this is too big a leap. Carry on. Fascinating. :popcorn:

[Ensign Steve]

:cough:

:yeahthat:

[The Lone Ranger]

Dragar:

Excellent stuff!

I've read Hawking, Murray Gell-Mann, and Victor Stenger, among others, but seldom if ever have I read such a clear and accessible overview of QM theory. I heartily second (third?) the suggestion that this be made into an article or two!

Just a nitpick, but the collective peaks and troughs are maxima and minima; an individual peak is a maximum and a trough is a minimum, if I'm not mistaken.


By the way, do you plan to discuss the EPR Paradox and Bell's Inequality? I'd dearly love to read a clearly-written, well-informed discussion of those.

Cheers,

Michael

[Dragar]

I've read Hawking, Murray Gell-Mann, and Victor Stenger, among others, but seldom if ever have I read such a clear and accessible overview of QM theory. I heartily second (third?) the suggestion that this be made into an article or two!

Thank you! When I finish this (I fully intend to) I'd be happy for that to happen, provided I have a chance to go through and correct typos and mistakes.

Just a nitpick, but the collective peaks and troughs are maxima and minima; an individual peak is a maximum and a trough is a minimum, if I'm not mistaken.

You could well be correct.

By the way, do you plan to discuss the EPR Paradox and Bell's Inequality? I'd dearly love to read a clearly-written, well-informed discussion of those.

I do. They're one of the best bits. ;)

Once my exams are finished, I'll pick up from where I left off. I haven't forgotten this thread.

[livius drusus]

Once my exams are finished, I'll pick up from where I left off. I haven't forgotten this thread.

:woohoo:

That's awesome because I totally want to see you snap up Degenerate Hieroglyph's bait as well as cover The Lone Ranger's requests.

[godfry n. glad]

I'm, like, ready any time.

Are we there yet?

:waiting:

[Crumb]

Genius takes time.

[beyelzu]

yeah anyone can dash off 2000+ posts in a few months but dragar's quality posts take longer

[Crumb]

Exactly. :innocent2:

[Godwhacker]

New to the forum, and I have some background in physics, but I gotta say that you should really teach this stuff! :bow:

I always wanted to set up my own basement physics lab and actually do some of these experiments. I remember doing some of them back in HS in AP Physics. This thread brings back memories...I used to do a comic strip for the school newspaper called "physics kills" (because, when you really think about it, physics kills everybody eventually)

Relativity was more my forte. When this thread wraps up, perhaps you could take us on that journey of knowledge as well?

[lpetrich]

Notice in the two-slit experiment that the light is spread out between the slits, yet shrinks drastically when it interacts with the film. This is the "collapse of the wavefunction", and this has been a difficult and controversial part of quantum mechanics.

A spectacular version of it is the paradox of Schrödinger's Cat. It is in a box with a jar of cyanide that can be smashed open with an electrically-operated hammer that is triggered by a radioactive decay.

Is the cat alive or dead? Because that decaying atom is in a mixed quantum state, part intact and part decayed, the cat must also be in a mixed state, part-alive and part-dead. But when one looks inside the box, the cat is either all-alive or all-dead.

There are several solutions offered:

* The Bohr or Copenhagen solution, to the effect that observation causes the wavefunction collapse. But what is an observation?

A common variant of it is that consciousness has the ability to collapse wavefunctions, something that leads into the "mystic physics" hypotheses that some New Agers advocate.

* Everett's many-worlds solution, in which the Universe is actually many parallel Universes, each one representing a different outcome of wavefunction collapses. Thus, the live cat and the dead cat are in two different Universes.

* Being macroscopic forces the collapse, because of its numerous overlapping individual-particle wavefunctions. This is a relatively recent theoretical development, and it is still somewhat speculative, but it does avoid the conundrum of trying to work out what an observation is. That cat stays in a well-defined state because it and its killing mechanism are macroscopic -- it is composed of about 4*10^26 atoms; and its killing mechanism has similar bulk.

[beyelzu]

I read in search of shroedinger's cat but I do not remember the macroscopic forcing wave function collapse, but the first two explanations were familiar. I find that answer to be far more satisfying from a science point of view. on the other hand, it makes me sad from a believing in conciousness point of view.

[Crumb]

I just read an article in Discover about Roger Penrose's explanation of the collapsing wave function. His explanation was that gravity forces the collapse and he thought this explained why macroscopic objects stayed in such quantum state for much sorter periods of time then the very small things. The larger the mass of the object the quicker gravity would collapse the quantum state. I haven't the expertise to comment on its usefulness or veracity. :dunno:

[Godwhacker]

I just read an article in Discover about Roger Penrose's explanation of the collapsing wave function. His explanation was that gravity forces the collapse and he thought this explained why macroscopic objects stayed in such quantum state for much sorter periods of time then the very small things. The larger the mass of the object the quicker gravity would collapse the quantum state. I haven't the expertise to comment on its usefulness or veracity. :dunno:

Please forgive if this is dumb, I'm no expert, but I dabble in reading some of this stuff. I always wondered if the nuclear weak force is gravity, but since its in a quantum (small as it gets) level, it looks and acts different. Since quantum level objects are so small, the quantum state does not collapse (or at least doesnt for a very very long time), and hence you can never tell where exactly a particle at the quantum level is, because, like the cat, it can be in two states (two places? two times?) at once? Does gravity, or the nuclear weak forces, keep the quantum level particles/waves from getting too too far away from each other?

Did I make any sense?

[Dragar]

Is there anyone who would be willing to host some images for me? I've got another part to add to this, but I cannot access my webspace to get them hosted until I return to my university dwellings.

Beware; it's a complicated one. Trying to explain second order partial differential equations without using mathematics, to people who don't know any mathematics, is harder than I thought...

[livius drusus]

Are you kidding? Yes, Dragar. The Freethought Forum would be delighted to host some images for you. E-mail them to us (or attach them in a post here if they fit) and we'll pop them right up.

[Ensign Steve]

:wriggle:

[Dragar]

Are you kidding? Yes, Dragar. The Freethought Forum would be delighted to host some images for you. E-mail them to us (or attach them in a post here if they fit) and we'll pop them right up.

Great!

Er...what's the email address I send to?

[livius drusus]

Our addy (admins@freethought-forum.com). :)

[Dragar]

Cool. Now I have to find or steal some.

[Dragar]

Huge email coming your way, livius. Thanks for hosting these.

[Dragar]

The Strangest Theory You Have Ever Seen

Part 2 - Using The Schrödinger Equation

I'm not actually going to show you the equation, despite this entire post being about it. Instead, I'm going to try and explain what we do with this equation, and why it's so important, and where the real weirdness in quantum mechanics begins. Some of you may find this very dry, some of you may find it interesting, some of you may find it frustrating. But this is likely as close as you can come to actually doing very simple quantum mechanics without knowing any mathematics. (Mathematicians, please do not read this post - it will make you cry.)

So, let's start.

We have the Schrödinger Equation, that Schrödinger wrote down all those years ago. What does it comprise of? How is it so different to classical mechanics? What makes it special? If equations do not frighten you, feel free to take a peek (http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/seq1.gif) before we begin.

You might expect that since the Schrödinger Equation governs how the smallest constituents of matter behave, it should somehow include a few things at the least.

The most obvious should be time - that should be mixed up in there, since we're still pretty sure that behaviour really needs to include changing over time. And it is, conventionally using the symbol ‘t’.

Another obvious one should be the mass of an object. If we think back to Newton, we know that the more massive an object (the more mass it has), the more force we need to impart to get it to go place to place. An electron (small mass) will be more easily influenced by forces than a neutron (larger mass). And do we find the mass in there? We do, and it uses the symbol ‘m’.

What else might we find in there? Well, what about those forces? We need some way to account for the attraction between a positive and a negative charge, for instance. And we find a term to represent the various forces acting on the quantum object in there, too. It is usually called the ‘potential’, and has the symbol ‘U’ or ‘V’ typically. (Mathematicians should note that the Schrödinger Equation bears some resemblance to the Hamilton-Jacobi formulation of Newtonian mechanics).

One I have missed out - one that really should strike you as odd that I have missed out - is the position of the thing. If this equation is about how quantum things move about, surely we should have where quantum things are? And, you might even expect, we should have some idea of how fast they are moving about (their speed), and in what direction? The two combined (speed and direction) we like to call velocity.

The curious thing about the Schrödinger Equation is that, on the face of it, these things are missing. What we have instead is a new variable. This variable uses the Greek letter psi. Sometimes it is called just that, other times it is known as the 'wave function', and a whole slew of other names.


http://www.freethought-forum.com/images/dragar/psi.jpg

But I'm calling it psi for now.

There's something important you should know about psi. There's a trick, that I'll come to in a bit, that let will let us extract information (like position and velocity) from psi. So you can think of this variable as being our key for getting at that information.

There are also a few other constants mixed up in there - we'll hopefully be returning to them later.

I may have given the impression psi is just a 'number' that gets bigger or smaller over time. That's not quite true. Psi is actually a number for every point in space. And worse, that number at each point in space changes over time! And somehow, we have to work out what this huge array of numbers is, and then we can have a clue about how our quantum object behaves. Quite a task!

Let's go over this again, so it's clear in our minds.

The Schrödinger Equation describes how this mysterious 'psi', which is somehow related to the position and velocity of our quantum object, is influenced by forces acting upon our object, as well as its mass, and also governs how psi evolves (a fancy word physicists use for ‘changes’) over time.

Now, clearly if we want to see how our quantum objects behave, we want to know what psi is. That's our key to getting out information on the position and the velocity. We know the mass of the object, and we should know what forces are acting on the quantum object. If we know all these conditions, how will the object move and behave?

We need to work out what psi is, and that process is called solving the equation. The Schrödinger Equation is quite a complicated one, but there are some situations where it can be solved precisely. A lot of the time, the situation means that we can only approximate an answer. Fortunately computers mean we can get a very, very close answer in simple situations, and at least a rough one in more complicated situations.

This is likely sounding very abstract right now. So I'm going to give you a concrete, if simple, example of how we actually do quantum mechanics, and how we work out what psi is. It has pictures, too. And you won’t see a single equation.

An Example: A Particle in a 1-Dimensional Box

For those more mathematically inclined, feel free to visit this (http://en.wikipedia.org/wiki/Particle_in_a_box) more mathematical explanation of this example.

Let's imagine we have a box. To keep things simple, let's make it a box in one dimension - our quantum thing is confined to moving left to right. It cannot move back to forth or up and down.


http://www.freethought-forum.com/images/dragar/squarewell.gif

The start of the box is at 0 on the bottom axis, which is called the x-axis. We call that x=0, because at that point the x-axis is equal to 0.

The end of the box is at another point, that I've labelled L. Don't be afraid of that, it's just a label. We call that point x=L, because at that point the 'x-value' is equal to L.

In order to confine it inside the box, we say that the walls of the box are infinitely high. If you like, you can imagine instead there are some infinitely strong forces confining it to that little box.

Now we start looking for what psi might be. Remember I said that psi is actually a number for every point in space? The particular form of the equation puts some restraints on what those numbers can be. In particular it puts a restriction on how quickly those numbers change as we move through space (the rate of change of psi, with respect to x - sometimes called the gradient). The numbers can't be just any old numbers, they have to satisfy the restrictions of the equation. Luckily, this has already been worked out, so we don't need to know anything about the details to see the answer.

There is a particular mathematical function which you may remember from school, called sine. A function is a technical word, but you can think of it as meaning that you plug one number into it, and you get another number out.

Imagine if psi is equal to this sine function. This means you plug a number for the x-value into this psi function (or sine function, as we are saying they are the same), and you get a number out. If you do this for every number along the x-axis, we now know what psi is at every point in space! If sine satisfies the equation, we have found our psi!

Here's what sine looks like:

http://www.freethought-forum.com/images/dragar/sine.jpg

Note that it looks like wave. Psi often has this sort of appearance. Now you can start to understand why people call psi the wavefunction sometimes.

But does this satisfy the equation? It does! We have our psi specified! But there is something of a problem remaining.

If you recall, I said psi is our key to knowing about the position and the velocity of our quantum object. I also said that the object was trapped in the box - it cannot get out. While it may seem like jumping ahead of the game a little bit, I'm going to tell you a rule. The rule is this:

If psi is zero at a particular point in space, the quantum object cannot be there.

Mathematically, why this is so I will explain next post. For now, I hope you can accept it as true. At the very least, humour me.

If the object cannot be found outside the box, this means that psi must be zero everywhere outside of the box. Luckily, zero is another solution of the Schrödinger Equation, so we are allowed to use it.

But we know psi cannot be zero inside the box, because our quantum object must be in there - psi must be non-zero within. These sorts of conditions and restraints upon the form psi takes are known as boundary conditions.

But now we have a problem. How do we resolve this dilemma of keeping the particle inside the box (using a sine wave solution), but not outside of the box (using our 'zero' solution)?

The answer, daft as it may appear, is to glue them together.

So we actually have two parts to our psi function, 'stapled together'. Psi is zero everywhere, apart from inside the box. Inside the box, it takes the form of a sine wave - and it looks like this:

http://www.freethought-forum.com/images/dragar/groundstate.jpg

(When we 'staple' two different solutions together like this, there is another rule that we have discovered that says where the two solutions touch they have to be the same value. So it is important the sine wave drops to zero at each end of the box (when x=0 and x=L, psi = 0 too) - otherwise they would be different values where they touch.

(You will find lots of rules like this as we go along. Some have mathematical justifications. Some...well, we just made them up and kept them because they seem to work. Welcome to quantum mechanics.)

This is not the only solution. Couldn't we have another sine wave at twice the frequency? The ends would still go to zero at each ends of the box, and a sine wave still satisfies the Schrödinger equation, right?

Right. Here's another solution:

http://www.freethought-forum.com/images/dragar/excitedstate.gif

We can even go further than this – an infinite number of solutions are in fact possible, each a sine wave with a faster and faster frequency - a more squashed wave in the box.

Something to think upon while I try to write up the next instalment (and don't rush; I have no idea when I wil do it): quantum mechanics was dreamed up to incorporate energy coming in packets or quanta. Perhaps you might like to take a guess at how this relates to the distinct, separate allowed solutions we find in the example above, each corresponding to a different frequency of sine wave? This is something that very simply drops out of the mathematics of the Schrödinger Equation and the problem posed.

That pretty much wraps up this post, and the mathematics are out of the way again (for now). Next time, I'll be answering the half-question I posed above, and talking about how we use psi to find positions and velocities (sort of) and explain some other differences that even these simple examples can reveal. If I have time, I’ll even start talking about how some of those differences between quantum mechanics and classical mechanics helped our understanding of the reactions taking place within the sun.




---------------------------

Does anyone have any questions? Please, please ask them. This was a very complicated one, because it’s really explaining mathematical ideas without mathematics. There are certainly bits that I have glossed over or not explained very well.

I hope you enjoyed actually seeing (in a very hand-wavey sort of way) what quantum mechanics actually involves in a very simple case of actually doing it. I'm also hoping this sort of foundation will bring you a bit more insight and understanding when I get onto the more nebulous aspects of the theory.

Would you like me to try to add more mathematics in, or is this very hand wavy explanation alright for you guys? Would you like me to be less mathematical and just gloss over the details of how we actually do quantum mechanics?

Feedback is needed, peeps.

[Farren]

Dragar I just chanced on this thread and read it from beginning to end. Its brilliant

[Farren]

Although, in all honesty I think you should have a few more analogies to aid understanding in the last part :) Other than that, flawless

:appl: :appl: :appl:

[viscousmemories]

Yeah I have to be honest, Dragar. This latest entry doesn't make any sense to me at all. :(

When you say "doing QM" do you mean like doing calculations on a computer, or is there some kind of physical 'doing' involved. I'm really confused.

[fragment]

I'm guessing by "doing QM" Dragar means using the theory (and specifically the Schrödinger wave equation) to make predictions that can be experimentally verified. This should become clearer in the next post which apparently will cover using the psi function to make statements about the position and momentum of the particle.

vm, it might help Dragar if you can point to some specific places where you get lost. For example, are you wondering where the hell this strange psi thing came from, and what it represents in real terms?

Dragar, I've also just found this thread, and I think you're doing a great job. I only know a little about QM, so it's always good to read introductions from different writers taking different angles. While I'm copmfortable with mathematical material, I like your approach of avoiding it and trying to describe what's going on in plain English, with links to other articles covering the maths. Must be damn hard to do though...

It might be helpful for people if you could provide an example of a real situation that can be covered by the particle-in-a-box model, if there is such a thing. Perhaps this is something you intend to cover in the next post. Either that or some analogies, as Farrren suggests. Would a macroscopic example of a similar problem, such as a ball in a closed pipe, be too misleading? It could give people a mental picture to work with, even though they will have to discard it later.

[viscousmemories]

vm, it might help Dragar if you can point to some specific places where you get lost.
I tried, really I did. I think I'm just not in the right frame of mind today. I can't even articulate what I can't understand.

[Dragar]

Yeah I have to be honest, Dragar. This latest entry doesn't make any sense to me at all.

When you say "doing QM" do you mean like doing calculations on a computer, or is there some kind of physical 'doing' involved. I'm really confused.

No problem VM. I knew this part was going to be the most difficult part, because it really is mathematical - and I'm not using any mathematics.

A good example of this 'box' would be quantum 'wells' that can be produced on the atomic scale, using nano-technology. You can build up a bunch of atoms on either side, and then drop an electron inside. Using quantum mechanics (a rough approximation is what we've just done, in fact - albeit in a hand wavey sort of way), you can work out what psi - the wavefunction - will be inside that well, and the various properties of the electron within (energy, probability of finding it at different positions, etc.).

Perhaps I went a bit too deep into the mathematics of the thing? I can understand if not everyone shares me fascination. What I'm trying to give you us a glimpse at where all the weird stuff comes from and that requires some understanding of roughly what we're doing with the mathematics.

Maybe I could go through what I've said again, step by step with you, VM?

[viscousmemories]

That helps, Dragar. I can now visualize the atomic well with the electron in it, but I find myself wondering the same thing as when you talked about shooting a single electron into the screen for the double slit test. How does one go about coralling a single electron? Aren't they so tiny they're invisible to us?

[Dragar]

Farren, you're correct about needing more anologies. I'll probably rework that post soon and include some stuff about guitar strings. Perhaps I should just drop the mathematics and skip to the results?

[Dragar]

That helps, Dragar. I can now visualize the atomic well with the electron in it, but I find myself wondering the same thing as when you talked about shooting a single electron into the screen for the double slit test. How does one go about coralling a single electron? Aren't they so tiny they're invisible to us?

Er...

I confess to my lack of knowledge here. This is far too much of a useful application of physics for me to know!

[viscousmemories]

Well alrighty then! Maybe I'm not as lost as I thought. :laugh:

[Dragar]

A brief google search suggests tiny magnetic or electric fields, VM, are what are used to control electrons. Of course, it's very difficult, and relies on as much the calculations and computations of QM as anything else. As you pointed out, they're too small to see, so we have to use other methods to detect their position (firing photons and detecting the angle the bounce back, etc.).

[viscousmemories]

Cool, thanks again Dragar. I'm really profoundly ignorant about this stuff. I didn't even realize that I have a bunch of electron guns in the house! I found a little bit of info about them at How Stuff Works (http://electronics.howstuffworks.com/question694.htm). I even got to watch a video of a guy shooting up a computer monitor with a .22 rifle and tearing out the electron gun. :chuckle:

Anyway, sorry to interrupt... I was just having a really hard time visualizing loading an electron gun the way I know how to load a real gun, with a bullet so tiny it's invisible.

[Dragar]

You see, I never imagined that would ever cause you any problems imagining. This is half of way I'm doing this - I'm trying to learn what to do and what not to do when tailoring a piece to an audience of non-scientists.

Your feedback has been the most helpful so far in that regard, VM. :)

[Ensign Steve]

Would you like me to try to add more mathematics in, or is this very hand wavy explanation alright for you guys? Would you like me to be less mathematical and just gloss over the details of how we actually do quantum mechanics?

Feedback is needed, peeps.

I can't speak for others, but it is working for me. I know a bit of math myself and could probably handle a little bit more, but I am enjoying your spin on it without the math. I'm severely impressed by your ability to do that! Anybody can explain QM using math.

[beyelzu]

i would actually like to see a little more math please.

[D. Scarlatti]

This is one hell of a thread, dragar. Thank you.

I second the motion that it be memorialized in a special corner of the FF. On the other hand its presence in a regular forum facilitates discussion and the illuminating comments of the membership at large. For example, Ensign Steve's controversial sexuality is especially helpful in providing a more colloquial analogue to the wave/particle duality.

Seriously though, superb thread.

[Joshua Adams]

I'd appreciate more math. However, don't assume we can necessarily follow the math just because we had the gonads to read it. I think the best way would be to enclose math in spoiler tags, and include a verbal walkthrough to go along with it. After all, you are explaining mathematical concepts as is; it'd be perfectly functional to give us the math and explain it, right there, instead of explaining something which we haven't even seen. My mind can't make the connection as easily without the numbers. But like I said, if you JUST have the number then I probably won't know what anything means. I don't have a clue what Schrodinger's equation means, for instance, because all of those symbols mean nothing to me, besides the ones you discussed.

I'm confused about what "L" is, in terms of a real world counterpart. Am I wrong in thinking that Psi is a piecewise function

0, (-inf, 0)
sin(x), [0, L]
0, (L, +inf)

Is it less simple than that? How do we determine what L is? I mean, I know the one-dimensional box is simplified in that it only includes the x without taking into account the rest of the information we'd need to determine the position, but there must be some real world analog to L.

Is L determined by seeing where Psi(x) stops being 0 and starts being sin(x)? (doesn't sin(x)=0 at numerous points between 0 and L? Does that mean the particle can't be there, either?)

To be completely honest, I did not really get anything out of the latest post, other than that "there are certain places where the particle can't be". Sans any real use of Schrodinger's equation, I didn't get a mathematical reason for that, and even if you'd included one, the connection between that equation and the way stuff actually works is still shadowy and mysterious. I'm not even sure what it means to say that a particle can't be somewhere. It can't be there given what information?


... despite all that, I find your posts accessible and they seem like a pretty good way to gain conceptual knowledge about QM. I'm not sure whether conceptual knowledge translates into anything useful, though.

My last math course was calculus in 12th grade (which I just finished a couple months ago, and will be taking calculus 2 in the fall). What level of math is recommended/required to understand the nitty gritty stuff? Are there any books that someone at my level could use to develop math skills to the point where they could read technical treatments of QM?

[Dragar]

Is it less simple than that? How do we determine what L is?

It's as simple as that. The actual function is Asin(kx), and you can work out what k is by throwing the solution into the equation and running it through - it's just a constant.

The constant A at the beginning is set by a process called normalisation - which I'll explain in a bit.

One of the other things I hope you've got out of it is that k can only take certain values dependant upon the size of the well (and other constants within the equation). Certain frequencies are only allowed. It's very similar to waves on a string fastened at both ends, where the string can only vibrate at set frequencies.

You'd work out what L is simply by actually measuring the size of the well. Presumably if you'd actually built the quantum well, you'd have an idea of what size it would be (and we're on the order of nano metres).

(doesn't sin(x)=0 at numerous points between 0 and L? Does that mean the particle can't be there, either?)

One of the curious things is we can't actually measure psi, though we can work it out through measurements. And you're quite correct; in some of the solutions there are various other points where psi falls to 0, and that means the particle cannot be at that 'point'.

A lot of the other questions I will be answering in the next part, and then you'll be able to start seeing the relevence of all this maths. Once both are finished, I'm going to work through them again taking into account all this feedback, and make it a bit more readable and understandable. I'm well aware there are improvements that can be made.

I'm not even sure what it means to say that a particle can't be somewhere. It can't be there given what information?

If you prefer, try thinking of it as 'quantum mechanics predicts that we will never make a successful measurement of finding the particle there'.

My last math course was calculus in 12th grade (which I just finished a couple months ago, and will be taking calculus 2 in the fall).

I'm afraid that means nothing to me. However, if you've covered partial differential equations, complex numbers, operators and non-commutative algebra (and a bit of Fourier analysis) you should be all set to pick up a book on quantum mechanics and have a crack. The best one I've found so far is Quantum Mechanics (http://www.amazon.com/exec/obidos/tg/detail/-/0750308397/104-6484106-2646325?v=glance) by Alistair I. M. Rae.

Thank you for this feedback, by the way. I prefer a lot of questions and criticisms to 'that's great Dragar!' because it means I know where to improve and can help you understanding what I'm trying to say.

[Joshua Adams]

It's as simple as that. The actual function is Asin(kx), and you can work out what k is by throwing the solution into the equation and running it through - it's just a constant. Hmm? The solution to what, and which equation? Sorry.

The constant A at the beginning is set by a process called normalisation - which I'll explain in a bit.I've heard of normalisation; I'll be looking forward to finding out what it is.

One of the other things I hope you've got out of it is that k can only take certain values dependant upon the size of the well (and other constants within the equation). Certain frequencies are only allowed.I worked with sine a few years ago and can recall, hazily, the various things we can do to the function to change the graph. I think the general form was a*sin(b-kx), or something. These numbers affected things like the amplitude, period, and 'offset' (i.e., which x values the mins and maxes are at). Is A the amplitude of the sine part of Psi?

It's very similar to waves on a string fastened at both ends, where the string can only vibrate at set frequencies.I have no real background in classical mechanics, so I don't relate to the vibrating string example. On the other hand, I can imagine we'd run out of string if the frequency got too high. But it's not clear to me whether/how that applies to something that is not a string.

Speaking of which, what exactly is this sine wave depicting? Are points on it the possible positions of the particle? This may seem like a dumb question, but how'd you graph sin(x), which incorporates the y-axis, in a one dimensional box? Or did we abandon that device long before I noticed?

You'd work out what L is simply by actually measuring the size of the well. Presumably if you'd actually built the quantum well, you'd have an idea of what size it would be (and we're on the order of nano metres).Well... that makes sense. Is this something that we can do, in practice?

A lot of the other questions I will be answering in the next part, and then you'll be able to start seeing the relevence of all this maths. Once both are finished, I'm going to work through them again taking into account all this feedback, and make it a bit more readable and understandable. I'm well aware there are improvements that can be made.Well, I figured that the next one would elaborate more, but the latest one feels very incomplete without it, maybe to the point that they should've been released together. In my opinion.

If you prefer, try thinking of it as 'quantum mechanics predicts that we will never make a successful measurement of finding the particle there'.That doesn't alleviate the problem. Basically, I'm not sure why a particle can't be in a given place. That fundamentally makes no sense to me at all. There must be some information that leads to that prediction, such as (for example) already-established information on the particle's position, etc. Since that's the kind of stuff you pop into Schrodinger's equation, I would think that my thinking is somewhat on track, there. But that would obtain even if I were wrong, so...

I'm afraid that means nothing to me. However, if you've covered partial differential equations, complex numbers, operators and non-commutative algebra (and a bit of Fourier analysis) you should be all set to pick up a book on quantum mechanics and have a crack. The best one I've found so far is Quantum Mechanics (http://www.amazon.com/exec/obidos/tg/detail/-/0750308397/104-6484106-2646325?v=glance) by Alistair I. M. Rae.I am not there yet. I can handle PDE and operators, if I'm right about what you mean by operators; but I haven't worked with complex numbers in any depth, or even touched the rest. Still, thanks for the book recommendation. You see, it's exceedingly unlikely that I'll pursue any advanced physics courses, so I'm going to have to learn on my own, and I'd prefer to be at least somewhat thorough.

[Dragar]

Hmm? The solution to what, and which equation? Sorry.

The Schrödinger Equation, in both cases.

Speaking of which, what exactly is this sine wave depicting? Are points on it the possible positions of the particle? This may seem like a dumb question, but how'd you graph sin(x), which incorporates the y-axis, in a one dimensional box? Or did we abandon that device long before I noticed?

The y-axis is the magnitude of psi.

Well... that makes sense. Is this something that we can do, in practice?

Yup.

Well, I figured that the next one would elaborate more, but the latest one feels very incomplete without it, maybe to the point that they should've been released together. In my opinion.

I came to this conclusion myself, also. Most of the answers to your questions will be in the next part.

[ceptimus]

How about a few examples of what QM would be like if it applied on a human scale. What would it be like trying to drive your car through some toll booths? How would pool balls behave on a table when making a shot?

What have you done to that cat Schrodinger? It looks half-dead!