Gauge this!

This one is just about as fundamental as it gets. All of physics comes together because of "gauge fields".

The first to be discovered was gravity, (we studied the effects but didn't "know" what it was) and then electromagnetism. There are two more. But what are gauge fields?

To quote Sir Patrick Moore, "Well, we just don't know". We know what they DO, just not what they ARE.

Taking the first two forces to be discovered, gravity and electromagnetism, this is what happens.

Place something where there is a gravity gauge field (ie "gravity"), and it moves towards another thing according to rules we fully understand. The "field" is like a slope running downhill from the smaller to the bigger.

Place something in an electromagnetic gauge field, and nothing happens. That is, UNLESS that thing has something on it we call "electric charge". We also do not know what this is either, just that things either have it or not, and, what is more, it comes in two types. We call them "positive" and "negative". (We tried "Laurel" and "Hardy" but it didn't sound quite right). All we knew was that two of the same type were unfriendly and pushed each other away, and two which were different, became lovers and wanted to be together all the time. The "field" was like a slope running downhill from one type to another.

Once more, we do not know what these things are, just what they do.

We knew that these "charged particles" exchanged another, the photon, in order to be friendly or hostile. We called this an "exchange" particle.

Then we formulated a theory that to hold an atomic nucleus together, there had to be another gauge field that was stronger than the electromagnetic field, but only at very short distances. If you think abut it, the atomic nucleus has lots of particles in, some with no "charge", others with the SAME charge. Yes, I know I said stuff of the SAME type are hostile, that is why we figured out there must be a stronger "attractive" force only available inside this nucleus thing, otherwise it would fly apart (if you get a big enough nucleus, it actually does "fly" apart because it is simply too big, and we call this "radioactivity". Some guys (embarrassingly calling themselves "nuclear scientists" - my own field) even used this to make a very big bomb called "The Atom Bomb". Because of this force which acts at extremely short range, unlike the electromagnetic field (which has infinite range and therefore whose exchange particle, the photon, has zero mass), the exchange particle had to have real mass. (the mass of the exchange particle is inversely proportional to the range of whichever force we study)

Then, just before the atom smashing started, Hideki Yukawa predicted from analogy with the electromagnetic field in 1935, that the exchange particle for this "strong" force, would be 260-270 times heavier than the electron, and it would show up in our atom smashers.

What we found was startling. We found a new particle only 207 times the electron mass.

What was going on? It was actually the wrong particle, and we now know it as the mu-meson, or muon.

The one we were looking for duly turned up and we called it the pi-meson, or the pion. ("mu" and "pi" are Greek letters, "pi" being better known in mathematical circles)

The upshot was this:

All gauge fields "exchange" particles, rather like they play "catch", in order to do stuff.

Now, we had four forces, electromagnetism; the "strong" force (the one that prevents the nucleus from flying apart); the "weak" force (which doesn't do much except help change particles to others with help from the conversation of energy); and er, gravity (that thing that helps apples to fall from trees)

What are their exchange particles?

electromagnetism: the photon (has been known for years, it is "light")

strong: initially, pions, that stuck nuclear stuff together, but as wisdom blossomed, gluons, that stick the quarks together (theoretically, cannot be detected, only indirectly observable in "three-jet events* (see my post "Three jets are better than two")) ("quarks" are still more bits inside the bits inside the nuclei - a bit like those Russian Dolls - and yes, the buck stops there)

weak: Intermediate vector bosons, the W+ , W- and the Z-0. (predicted in the 1970's, observed in the 1980's)

gravity: ?

Gravity is, and always will seem to be, a complete headache. It's exchange particle is speculated to be the "graviton", but has never been seen, directly or indirectly.

WARNING - SUBJECTIVE BIT!

It is the opinion of this author that Einstein failed in his attempt to unify gravity with the other three forces because gravity is NOT a gauge field, but a geometric effect. I submit that there are only three forces, plus gravity, which Einstein himself proved to be geometric.

I find it hard to believe that if gravity IS a gauge field, spacetime would be, by definition, quantized, that is, there is a smallest bit of space beyond which we cannot measure because nothing smaller exists.

END OF SUBJECTIVE BIT

Spank Me

Here is a true story. Way back in the 60's, when atom smashing was just starting, it was thought that the basic building blocks of matter were not really that basic after all.

Protons and neutrons were bashed together so violently that they acted like they were made of smaller bits.

These bits were named "quarks", after the word from the book Finnegan's Wake. At first, two were enough to explain all of normal matter, and we called the "up" and "down" quarks. However, strange things started to happen soon after.

Literally.

Particles flew out which did not behave as they should have. Then to add insult to injury, us scientists added to the "up" ans "down" quarks another "flavor" of quark.

We called these strange particles "strange" particles.

This one was much heavier than the previous two, so it took much more energy to produce it.

The prediction was then made of a further "flavor" of quark, available at still higher energies.

Once again, our verbosity got the better of us and we named it "charm".

This quark first appeared in what we lovingly called a "charmonium" - a particle made up of one charm and it's antiparticle.

To scientists, this was not good enough - we wanted pure charm, ie charm on it's own. So the search for "bare charm" was begun.

As the energy of the accelerators increased, sure enough, bare charm was found.

That brought the quarks to four (up, down, strange, charmed).

It gets even funnier. The theorists told us that the had figured out that there were no more than a total of six.

So we got our thinking caps on again and came up with two more wonderful names - "bottom", and "top" , each one still heavier than the last. At first, we found the "bottom" quark, then eventually, the "top"

Imagine the remarks that were made when we tried searching for "bare bottom".

That just about sums it up really

The Atom - Part Four

WARNING! THIS POST CAN BE BAD FOR YOUR HEALTH!

Now we had a real model of the atom to work with. All that was needed now was a mathematical formalism to describe it.

Independently Werner Heisenberg and Erwin Shrodinger worked it out, the former's was called Quantum Mechanics, the latter, Matrix Mechanics. They were exactly equivalent. Both worked the same way, one using differential operators, the other, matrices.

The guys "married" the two theories and came up with the single most important formula in physics, surpassing even Einstein's E = mc2.

The reason why the Shrodinger Equation is not widely well-known is because the pearl of wisdom in it is buried underneath the scary-looking math.

Here it is, plus the solution:




For those interested, it is an "EigenValue" equation. The left-hand side of the top line simply says, perform the calculation on Psi (the strange "Y" like letter), and you get a value for the energy of the particle of mass m on the right.

The V(x) is the potential energy, only applicable in electric fields (technically, gravitational fields also).

If we suppose the particle is clear of any such fields, V(x) is zero and the solution drops out on the bottom line.

That bottom expression is the simple equation of a travelling wave, ie the particle itself expressed mathematically.

The letter "i" is the square root of -1.

All of quantum mechanics can be described by this single equation.

Are you Positive?

Let's make a quantum jump ("quantum" jump these days means to most people "a specific" jump) to what Paul Dirac did.

In those days, the early thirties, quantum mechanics was still in it's infancy, as was special relativity.

What Paul Dirac did was to try to marry quantum mechanics with relativity for the electron, the simplest particle.

His results were astonishing.

He wrestled with the maths involved, turned the mathematical handle, and out came the washing.....But there were TWO results, not just the one.

The one he was after, the electron, was quite clear. What was NOT clear, however, was the other answer. The maths had produced something no experimentalist had ever done!

Paul Dirac simply shrugged off the shock and said "There must be another particle out there, with the same mass as the electron, but oppositely charged".

Outrageous!

In 1932, Carl Andersson found it in his cloud chamber. The path it took had the look of an electron, but curved the opposite way. It was henceforth named the "positron".

Paul Dirac and Carl Andersson had just revealed anti-matter!

Outrageous!

The Atom - Part Three

It was the insight of the century (OPINION!).

By the time the Bohr atom had come along, everyone knew that light behaved like particles AND waves, depending on the experiment.

Prince Louis De Broglie took this concept and turned it on its head, reasoning thus:

"So, the quanta of light, which we know as waves, sometimes behave like particles. WHY CAN'T particles, sometimes behave as waves?"

He even backed this insight up with a formula for the wavelength of a particle:

wavelength = (Planck's constant) divided by (particle momentum)

His PhD thesis was one side of A4 (mine was a mini-book of 10 chapters).

This got the theorists thunking. Then two guys name Davisson and Germer successfully DIFFRACTED a stream of electrons, PROVING that the stream behaved like light.

Then the A-HA came (as Martin Gardner would have said).

If electrons behaved as waves sometimes, if they were in an atom, unless their "orbits" were whole numbers of electron wavelengths, there was no possibility of any other configuration because of destructive interference. Have you ever tried to poing a ruler on the end of a desk any other way than the notes you get?

Yes, resonance was the key. The electron waves resonated around the nucleus exactly like notes in a piano. Only certain frequencies "poinged".

So the picture that emerged was this one:





Ripples in a pond. Rulers poinging. Piano strings vibrating. If you will pardon the pun, does this ring any bells?

WAVES.

The electrons behaved as waves in the atom.

The upshot of this realisation was that Quantum Mechanics was born. (sorry Max)

Enter Erwin Shrodinger......

The "Discovery" of Absolute Zero

This is a chart to show what happens when we chill out. More specifically, what happens to a gas.

Everyone knows what happens when it gets cold.....things SHRINK (men would know more about this).

It has to do with vibration (don't get excited, girls).

Heat = energy and the less heat an object has, the less it's vibration.

So the size decreases.

If you are talking gas (Ethel Merman go home) then for the same amount of gas, if you cool it, the container you need to keep it in will have to be smaller.

There was once a guy named Charles, who figured this out, but he did not stop there.
He said to himself, what if, the gas did not go to a liquid, bur instead stayed as a gas? He called it an "Ideal Gas".

The buckshot (whoops, Upshot) was that he continued the picture right back to zero volume ( as per the picture) and he found out that no matter WHAT gas was used, they all ended up at zero volume at EXACTLY the same "temperature".

It was from then on called "Absolute Zero". and the number is 273.16 degrees Celsius below the freezing point of water.

Nothing has ever been cooled further since.

This whole issue is couched in quantum mechanics, although at the time, it wasn't even thought of. The reason why quantum mechanics comes into it is down to Heisenberg again - the Uncertainly Principle. If something stops moving, then you can say where it is (Trivial, I know, but hang on). This is completely forbidden by quantum mechanics. Moving targets are a must!

So stuff need to keep moving. and if you are at absolute zero, you stop. Therefore, you cannot get there (thermodynamics again).

Physics is wonderful because basically all the stuff around us is related to everything else. It's only when you study really hard do you get to know (and wonder at) how it all fits together.

This is the be-all and end-all of why the subject is so dear to me. You can call me a "bore" but at least I know the answers to the questions we have all asked our parents as kids.

Water, or di-hydrogen oxide, is the strangest of all, and blatantly refuses to conform....!

Why does ice float?

Just how DOES the sun shine?

Sequel to the post "THIS is Quantum mechanics".

For at least twenty years, we did not know how the sun worked, ie shone.

Yes, we had the equations, the PP1 and PP2 chains, and the C-N-O Bi-cycle, all nuclear physics equations, which predicted so many neutrinos.

So Dr Paul Davies in 1967 made the most interesting telescope ever. A huge tank of carbon tetrachloride buried in an old mine in Dakota.

Just to detect the neutrinos.

The problem was about those darn neutrinos, the smallest particles in the multi-cosmiverse. No mass, no charge, only "spin". They pass through lead without hitting anything in it for at least 18 million million miles.

And for years, Dr Davies detected about a third of the quantity predicted by our equations.

We could not figure it out.

Then a breakthrough.

It was put forward that maybe we detected only "electron" related neutrinos.

In order to understand this, it had to be realised that we now know there are three different types of neutrinos. What they are is no important.

What IS important is that the neutrinos coming from the sun were actually a "quantum superposition" of all three.

Dr Davies' telescope was only detecting one of them.

So, other experiments were set up to detect at least one of the others ( the third one is so rare, to detect them was almost impossible).

And we found it. This bumped up the number coming from sun to about two-thirds of what the equations predicted.

The next surprise was that there came results from experiments that hinted at an actual MASS for the neutrino. This changed things, as if the neutrino had an actual mass, the neutrino could "oscillate" between all three varieties, just like the K-0 meson could.

That would explain why we could only detect a third of the neutrinos. We would only detect the electron neutrinos.

Bearing in mind that the "oscillation" period depended on the reciprocal of the mass difference ( as measured by the K-0 effect), this explained an awful lot.

Finally, we had worked out just "how" the sun shone!!!!

THIS is Quantum Mechanics

The Quantum world is really strange. Some of the effects defy explanation completely.

This is the one that threw my imagination off it's axis.

It is called "regeneration".

To put it normal terms, a particle dies and is reborn.

It's astonishing what happens....

The particle was one of the first of the "strange" particles, and it is called the K-0 meson.

And this thing was really strange.

First, here is a description of what was seen.

The beam of particles was produced. We shone it onto a detector (basically, a wall) which stopped them.

Quantum machanics predicted some would come out of the other side of the "wall". Its part of the rules these things obey.

After that, with another "wall", we detected the ANTIparticle of the K-0 meson.

This told us that the beam was a mixture of the K-0 meson and its antiparticle.

It gets really weird now.

The first "wall" stopped the K-0. The second stopped the antiparticle. So basically, nothing should have got through.

Yet, because of that same rule of quantum mechanics, the second "wall" did NOT stop all the particles.

So we set up a third "wall".

You'd expect to get just K-0 antiparticles.

We didn't. We got the K-0 again!

They were dead, and yet, somehow they were "reborn".

The explantion is really simple if you know QM.

In simple terms it's not one particle, but two.

It's very hard to explain.

It's called "Quantum Superposition".

In other words, it IS both particles at the same time. The reason why it is "reborn" is because it is NOT just the K-0 we detect. We detect both, but the effigies that "get through" both "walls" are neither K-0 nor it's antiparticle. The things we detect are only what the "walls" "feel". The stuff that comes out of any "wall" is a mixture of both particle and antiparticle.

The really weird thing is that if we shift the "wall" back 3 meters or so, we detect the OTHER one!

In other words, the "thing" we call the K-0 meson actually shifts between being itself and it own antiparticle. Over a distance of about 3 meters. We actually have calculated how far. It depends on the difference between their masses! (actually it depends on the reciprocal but there you go!)

This topic is closely related to how the sun shines, but that is another post!

The Road Goes Ever On

A bit more math. Has anyone ever heard of Clebsch-Gordan and Racah Coefficients? No-one outside quantum mechanics, I expect, as only quantum mechanics uses them. We certainly need them and they are very complex indeed.

There are no numbers, just symbols. Forgive me if I cut and paste from the 'net as there simply is not any other way to show them here.

These are the Clebsch-Gordan Coefficients.



These are the Racah Coefficients.

Before I lose the reader completely, let me tell you a story.

It begins with Newton's Laws, the first one to be precise. Harking back to my post on these laws, it states that "an object moving will keep on moving, or if at rest, will stay put unless moved"

We say a moving object has "momentum" and a stopped one, "inertia". The above math is all about momentum, and we measure momentum (p) most easily by saying it is " mass (m) x velocity (v) " . However the above is not about "linear" momentum, which is "mv", but "angular" momentum, which is defined similarly, thus: Angular momentum L is (angular inertia) I x (angular velocity) w

These are roughly the same as their linear counterparts. I is the equivalent of mass, and w is the equivalent of velocity.

I is a bit difficult to understand, it is the "moment of Inertia" of a body, and different bodies have different I's, some of them even have more than one. Here is a table to illustrate:


These objects have 2 distinct moments of inertia depending on which way you spin it.

Notice earlier how the words "velocity" and "angular velocity" were used. "Speed" and "spin" were not used because velocity is what we call a "vector" - it depends which way it points. Example, my car's speed is 30 kph no matter which way I turn it. However, if I drive north at 30 kph for an hour, then turn round and go south at the same speed for an hour, I will end up where I started. The speed was the same, but I used two different but equal (and opposing) vectors that cancelled each other out.

This, in essence is what these coefficients are all about. Example, if I take a vector east and travel for 12 km, and then south for 5 km, I won't finish 17 km away from the start, but 13km. (how do I know this? Try Pythagoras' Theorem). So when you add vectors, you have to notice which way they point and do the geometry.

Quantum mechanics does this, but at this level, momentum and angular momentum are quantised (only come in little bits at once), so special math are needed to do the geometry. These are the Clebsch-Gordan and Racah Coefficients.

Well, that is the background, angular momentum and vectors. Now for the rub.

The key to the idea is this. It is best to go back to my East-South car journey. How many ways are there to get from the start to the finish? I already gave one, but there are others, 5 km west, then 17 km east, and 12 km south is another. But the best way is the original one, 12 then 5 km.

Quantum mechanics, though, has to deal with all possibilities of the combinations of these vectors, and though none are equally likely, we need math to tell us how likely the car would would go West5, East17 (Brian, sit down!), South 12.

The Clebsch-Gordans tell us how many ways we can do it with two vectors (E - S , hence the j1 and j2), and the Racahs tell us how to do it with three (W - E - S, hence the j1, j2 and j3). The twiddly bits j12 and j23 just ask us which is first.

The "| >" , "< |" and "< | >" bits are actually indefinite integrals (yes, calculus) and the big mouths (the big bold symbols after the equals), are simply the adding up of all the bits we know about that follow after it.

That is the interpretation of the above. "Coefficients" is the name of a number we put in front of something we don't know to tell us how much to multiply it by.

The best way to put it is this: How do we get from A to B quantum mechanically? Answer: Add up all the probabilities of all the different ways we can do it to get "1" ( it is certain we will arrive at B). In this case, we have several quantum mechanical angular momenta, and we want to add them up. These formulae tell us how to do it and get an answer as to how probable it is that we will end up with a certain answer.

We don't know what it will be, but essentially we plug in the numbers for the J's and m's and M and we will get a big J. The j's are the individual angular momenta, and m's are the spin angular momenta, and the J and M are the probability of getting a certain total at the end from all the others.

Plug in the numbers, turn the handle, and out come the answers.

This is once more just an introduction, and the Road Goes Ever On.

The "Unfriendly" elements

There are elements call "Noble" - the name is historical. They are in group 8, that is, they already have enough electrons to cope with.

They can't take any more.

So they do not combine chemically with anything. They are known by the names Helium, Neon, Argon, Xenon, Krypton( beware superman) Xenon and the radioactive Radon.

Only Xenon is known to be remotely friendly. Due to the structure of the electrons, it can sometimes form compounds. But the rest remain safely uncooperative.

The most interesting is Helium. This element was first identified in the sun, from the light it gives out when heated, and then on earth. Because it is so unfriendly, it replaced Hydrogen as the gas used to fill airships, after the R101 etc.

But the real interesting stuff comes when you cool Helium down, because try as you may you cannot freeze it. It is a physical impossibility. Water you get as steam, water(of course), and ice, as the temperature goes down. You can solidify any element you like as long as you cool it enough.

But Helium point blank refuses to go into a solid. Why?

We don't actually know why Helium, but we do know what it does....

It pours itself out of its own container, it is has zero viscosity ( easier to move through than empty space), has zero resistance to electricity.

We have explanations for this behavior, but we still have little idea as to WHY it is Helium only that it was possible theoretically before.

Trumpet Voluntary - the Triple Bond

Ok, it now behests me, after coming full circle with this blog, to demonstrate the power of what can be learned. This post is just about the most difficult to understand of them all.

The story starts thus: Chemistry talks about "bonding" between atoms. That is, for example, when you get wet, you are not spattered with either oxygen or hydrogen, but water, and water is simply H2O - two hydrogen atoms and an oxygen atom, "bonding" together.

What is this bonding, then? It's when atoms come together rather like Jigsaw pieces do, and they can fit perfectly.

How? Well, broadly, it is this....atoms like electrons flowing around them (yes, FLOWING! after all, moving charges are a current, are they not?) and the number they like to have is about 8 or 16, but they are only happy with certain numbers. Once that "shell" is "full" they become uncooperative (they will not do any more chemistry).

Once the atoms discovered this, they became VERY friendly.

A short course in chemistry is in "order" here.

Electrons, as they whizz, or "flow" around the nucleus (my field of research) like to group together but not always in the same way.

The first group has 2, the second eight, the third eight also. And so on.

Hygrogen was lonely. It has only one electron in a group that wanted two. Oxygen was also lonely, it had the full first group (2) but only six of the second group, which wanted eight.

Then Oxygen and Hydrogen met and their problems were solved in one swell floop!. They could "share" electrons!

Two hydrogens could "share" with oxygen to give oxygen the illusion of eight electrons, whilst hydrogen could "pretend" that both of it's atoms had two!

End of lesson. Well done. That is how water works, said the crying nun.

Now to the point of this post. The above is called a chemical bond. A "single" chemical bond. Oxygen-hydrogen - Easy!

BUT ( there is always one of these is physics, even without the goat) it gets worse. Here's what happens.....

The behaviour of the electrons in atoms can be illustrated by "seeing" where they are. This is where the math comes in, because no-one can actually "see" them.

First off, how do I tell you where I am? Well, right now, I am about 1 meter away from the window and 2 meters away from the the TV in a 90 degress direction. Whoops! I am on the 33rd floor, so I am about 130 meters UP.

That's 3 numbers. One for each dimension. Those 3 numbers tell anybody where I am. It's very easy because I live in a rectangular apartment, and am 130 meters directly up from level ground.

To do the same with electrons is not as easy. They live in a circular apartment and are going round it. Round.... circular.......

Why not use different numbers to tell where the electron is? How about "angle from somewhere" and "how far it is from the somewhere"?

My apartment and I are referred to as "rectangular" coordinates (the three numbers). The electron in the atom can be found by using a more convenient way, using "Spherical" coordinates
(also 3 numbers).

It is now time to show how "easier" it is this way.. copy this link.....

http://en.wikipedia.org/wiki/Spherical_harmonics
(This is the math)

and then this one..

http://oak.ucc.nau.edu/jws8/dpgraph/Yellm.html
(these are the pics that come out of the math)

There are two points to note...1. This is the EASIER way....2. I see past the waves of unintelligible letters in link 1 and all I see are the salient points, which in this case, is the shape of the thing when you draw the picture of it in link 2.

That picture is essential to the understanding of the 021, sorry, "triple bond".

This is another thing I failed to understand. I got the single bit, but double and even triple bonds? Try as I might, my teachers were unable to tell me what they were.

Here is why....

In the chemistry of Carbon, this element has 6 electrons, 2 (full shell) plus 4 (half the next shell). It would like four more. It gets this with hydrogen , but whereas oxygen gets 2 hydrogen atoms, carbon gets 4, to give it the eight electrons it "needs" The molecule formed is "methane", the simplest hydrocarbon ( see my previous post on oil). CH4.

But sometimes, there are only 2 hydrogens available, and what we see is the "double bond", that is, the carbon has a "double bond"with itself and with only two hydrogen atoms, double "2" making four. C2H4 - Ethene. What happens when there is only one hydrogen? It does complex chemical gymnastics and does a "triple bond" with itself and the one hydrogen.

It is best to think about this with "hooks and fish". Hydrogen has one hook, carbon has four hooks or fishes.

Carbon wants a full shell, either by losing four or gaining four, electrons, it doesn't care which.

For 2 carbon atoms, it is easy, just "share" the outer four and both can pretend that either have 8 ( a full shell). In this case, carbon is definitely heterosexual. It doesn't happen.

But HOW can we visulise this, after all, pictures often help. This is what none of my teachers were able to do.

Take the simplest Spherical Harmonic, the ball-shaped "cloud". Hydrogen atoms each have one of these, and when they overlap - cool! we have H2, or chemical hydrogen, two electrons sharing one "orbit", but how can four electrons share one orbit? This is the clever bit. This is the point where the maths take over. Have a look at the oak pictures (second link). Only the first one is like a ball, all the others have different "bits" sticking out of them, like the balloon sculptures that clowns make. Well, the key is understanding what the bits actually "mean".

The reason they are that shape is that the further out the electron gets from the nucleus, the more angular momentum it has, and using Spherical Harmonics is the only way to illustrate this.

For example, carbon looks like the Y(2,1) picture, basically a 4-pointed star. You can't see it, but there is an extra one in the middle, the "ball" shape. Imagine two of these face to face. Red bits opposite the red bits, blue opposite blue. Push them together and they overlap just like hydrogen's "balls". THIS is the triple bond. Red-red, blue-blue and the "ball" shape which is not shown . 1, 2, 3 electrons being "shared" .

The way to really understand this is to think of the shapes as "probability densities" that tell you just how likely it is to find an electron in "that" shape.

If you put 2 carbon atoms together like this, they form a "triple bond", and remembering that Carbon likes to have 4 hooks, with three hooks already taken with the triple bond, stick a hydrogen on and each of those left, you have the molecule H-C-C-H, which is called "Ethyne".

What has just been "explained" here is not chemistry, but "quantum chemistry", and along with powerful math, gives a beautiful explanation of how chemical bonding works. If you didn't understand, don't worry, it took me a good 4 years to work it out.

Are you Still Uncertain?

This post is aimed at someone on the Chess site www.chess.com who told me I don't know the Uncertainty Principle.

In short it is this:

Dp.Dx >= h/(2*pi)

where

Dx is the uncertainty in the position of a particle

Dp is the uncertainty in the momentum of it

h is Planck's constant which is:

6.626068 × 10^-34 m² kg / s

pi = 3.1415926.......

and ">=" means "greater than or equal to"

The Uncertainty here is best explained with reference to my car example in the Calculus post. If the car is travelling at 30 kph, it is not travelling at EXACTLY 30 kph because of bumps in the road, uneven tire pressures, wind, etc. A more accurate statement would be "approximately 30kph", or better still, "somewhere between" 29kph and 31 kph. The speed would then be 30kph "give or take" 1 kph. The "1kph" is called the "uncertainty".

Thus Heisenberg's Uncertainty Principle says that the uncertainty in position, multiplied by the corresponding uncertainty in momentum, can be no less, even in principle, than a certain number (h/(2*pi).

The astounding thing to come out of this is that there is even a number.

Ok, so the number is very small, and has often been quoted as the "measurement" of how small quantum mechanics has to be in order to effective.

This is deeply related to the double slit video posted earlier. If you see the electron then the photon's momentum in order to see it, and it's uncertainty, is enough to raise the uncertainty in the position of the electron so as to destroy the diffraction pattern .

The uncertainty relation has its roots in wave theory, where a wave "packet", caused by many waves of similar wavelengths superimposed, results in a bunch of waves, travelling at a particle's speed (the group velocity), but the waves themselves (the phase velocity) travel at a speed much greater than that of light. It's dealt with here:

http://gnabgib.blogspot.com/2005/06/groupie-or-just-phase.html

The uncertainty in the wavelength is equivalent to the uncertainty in momentum, according to the De Broglie formula (wavelength of "particle" equals "h" divided by momentum), and the uncertainty in the position is equivalent to how "big" the "bunch of waves" is.

What does it teach us? It teaches us that we cannot measure everything.

This is something that the books don't teach, and is a piece of knowledge only available through complete understanding. That means, you can find it nowhere on the web either, unless you ask Mr Feyman, who would probably say the same, but unfortunately he is rather indisposed at this time.

The fundamental obstacle that faces us all in quantum mechanics is trying to envisage "particles" as "waves". It did in 1924 and still does now.

That Darn Quantum Mechanics again....

Here is a cool introduction to the bizarre world of Quantum Mechanics. It is a paraphrase of the opening chapter to "The Feynman Lectures in Physics" Book 3. It is a very plain and very simple demonstration of what we were up against in the 1930's, and it is EXCELLENT.



However, it lacks in two areas:

1. It does not mention that any wave effect (waves or electrons) is only possible if the wavelength of the "waving" is about the same size as the aperture. (simple wave theory)

2. It does not say why looking at the electron affects the pattern. The reason why is that until you look, there is an interference pattern, but if you look, you have to bounce a photon off the electron back into your eye to see it. This photon has momentum according to p = E/c, and although tiny, is just enough to push the electron off course from the interference pattern. If you lower the "wiggling" (the frequency) of the light so that this effect becomes less, the amount of photon momentum needed to tell which slit the photon has gone through is enough to not destroy the pattern, but the light is so dim, it is too dark to see which slit it has gone through. This reciprocal effect has a name , and it is the Heisenberg Uncertainty Principle.

Just what IS Quantum mechanics?

This is a question that has been been asked many, many times.

Now, let's get one thing perfectly straight - the "mechanics" bit up there bears no resemblance to the kind of mechanics we usually talk about, such things as engines, nuts, bolts, screwdrivers, wheel- balancing, etc.

True, it is the same word, but in physics it means something very different and quite specific: "mechanics" in physics is nothing more than the study of the relative movement of things, like apples dropping from trees, billiard balls moving on a snooker table, and the occasional hammer hitting someone's head.

Nothing to do with cars or anything Richard Anderson might say in "Forbidden Planet".

A lot of the time, mechanics just deals with things hitting each other. When I say "deals with" I mean we use equations to find out where they will go next.

For example, what happens when two billiard balls collide, head on? Well, if one is standing still, the moving one hits it and transfers the motion, and the second one moves off.

In an ideal world, yes, but this is not what happens. It so happens that both balls end up moving. "How can this be?" I hear you ask. The answer lies in the way the balls are put together and what they are made of, BUT that is the realm of materials science, not physics.

So mechanics is when things hit other things. Now we're getting someplace, as Ollie would say.

There are different types of mechanics, all of which use the basic "mechanics" laws set down by Isaac Newton. There is Celestial Mechanics (how large things like planets behave), Terrestrial Mechanics (hammers hitting heads), Car Mechanics (nothing to do with this discussion), and Quantum Mechanics (how very small things behave, things like atoms, etc). Oh and there is Relatavistic Mechanics, and Relativistic Quantum Mechanics, too, but both these only come into play when you are travelling just a tad faster than Richard Hammond did last year.

So, WHAT is Quantum Mechanics? We, like I said, it is "how small things behave" but there is more to it than that. Scientists thought terrestrial mechanics ruled ok, ino questions asked, including the very small. The fact is, we had to invent a completely new branch of Mechanics, so as to predict where things ended up after the charabanc. The reason was, the very small particles acted in very different ways to what we expected. They actually acted like "waves". Hard to believe , ain't it? Moreover this "quantum Mechanics" gave right answers for all the other mechanics branches as well.

Depending on the experiment you were doing , the very small particles act both like "waves" and "particles". It is very hard to imagine, and even to this day 99% of scientists don't really know what is going on, BUT, what they DO know is how to predict what will happen when two of these "quantum" billiard balls collide, and what happens afterwards. Very clever.

So where does that leave us? Well, to the ordinary layman, you or I, it means nothing, because we as people are never going to deal with things THAT small ( I am saying nothing about Marvin's brain here), but some scientists have to deal with this every day.

They are called "particle physicists" and live in very tall ivory towers in Geneva. I, the layman (or laywoman) deal with particles just as small, but I am only called a lowly "nuclear physicist".

However, I still find fun mucking about with the equations in our computer programs, investigating "Quantum Mechanics". It is almost all a little girl can do.......

So, when it comes down to it, we have a perfectly good representation of the world as we know it with "Terrestrial Mechanics", so why do we need "Quantum Mechanics"?

A good question, and one which deserves an answer. With no Quantum Mechanics, we would not have discovered Light Amplification by the Stimulated Emission of Radiation, or the LASER. There would be no nuclear or particle physics, ie neither the Atom bomb nor the Hydrogen bomb would have been invented. Is that a good or a bad thing (bad for me, anyway!) - one for the philosophers, I am sure.

Another thing is that wihout Quantum Mechanics, we would have primitive computer controls still, the most important of all would be - no DVD's CD's etc, Guided missiles would be out too, as would be supermarket checkouts. TV remotes. The road goes ever on.

Maybe the most important thing we would be missing would be the Josephson Junction in electonics, but I wll leave the reader to find out what is THAT by himself.

Shrodinger will never know what a Pandora's Box he opened!

Three Jets are Better than Two!

For those of you who couldn't manage my sequence (see previous post on 1,1,1,2,3,3,4,5,5,6,6,6,8,8....etc), here is some more thought-provoking science stuff.

As intimated in my "Angels and Demons" post, there are certain things far more fascinating than electromagnetism.

There is that which holds the atomic nucleus together. It is called the "strong" or "color" force and overcomes the electrostatic repulsion of the protons by a factor of something like 1,000,000,000,000. It is called "color" force because unlike electromagnetism with two kinds of charge, this one has three. For want of a better name, we named it "color" charge, and called the charges "red", "blue" and "green", as 'opposed" to "positive" and "negative" for electromagnetism.

However, this "color force" is only for a very short range.

This means that as the nucleus approaches the size of a Uranium nucleus, physically, the electromagnetism takes over and the nucleus splits, due to the proton-proton repulsion, a process we call "fission", the essence of the atom bomb.

Carriers of this "color force" are called "gluons", analogous to the photons we all know and love.

However, there is a difference. Whereas photons are chargeless (electrically neutral), the gluons are not.

Gluons carry the "color" force as well, which the quarks inside the protons also have.

Thus: whereas two electrons interact with each other by exchanging a photon (they move away from each other because of electrostatic repulsion), quarks do a whole lot more gymnastics by exchanging gluons, which themselves exchange more gluons, etc etc.

Hurray for Feynman diagrams!

The whole thing is bound up in a theory we call "Quantum ChromoDynamics", or QCD. This is partial homage to the most successful theory that was ever thunked, "Quantum ElectroDynamics", or QED.

But even though QCD is insuperably more difficult mathematically, it does share stuff with QED.

QED explains that an electron accelerated in an electromagnetic field will "give off" radiation, in the form of electromagnetic waves. (This is the one thing that we scientists had trouble with envisioning the atom - how can an electron encircle the nucleus, thus having a circular acceleration, be stable? Surely it would radiate and spiral into the nucleus! - the problem was solved with "quanta" of radiation and thus was born quantum mechanics). This radiation was given the name "Bremstrahlung".

We derived much of QCD from the tenets of QED.

When enough energy was available at CERN (yes, it does exist, readers of Dan Brown!), we were in a position to identify the plethora of particles that exited the collisions as being directed as "sprays" of particles in two general directions. There were always two to begin with, corresponding with the sprays of particles "created" by two quarks - a quark and an anti-quark. These we called "two-jet events".

The reason why quarks were created in the first place was dealt with in my earlier post "Angels and Demons - Reality Check!".

Now comes the crux.

As our beam energy grew larger, we occasionally got not two "jets', but three.

These are the "three-jet events" I hinted at in my earlier post.

It turns out that "three-jet events" can be explained not by QED bremstrahlung, but QCD bremstrahlung. The third jet is no more nor less than the product of a gluon itself.

The energy available in the "crash" is too much to create just a quark-antiquark pair, so physics "decides" somehow to create something else. Not enough energy to create another pair, but one of the quarks accelerates through so much "color" field, that it "radiates" a gluon just like the electron radiated a photon when accelerated through an electromagnetic field.

THIS is what a "three-jet event" is.

Funnily enough, it is this same reasoning that has prompted a stepping-up of the search for gravity waves, being a product of an accelerated mass in a gravitational field.

Groupie or just a Phase?

Time and again, it seems I am fascinated by quantum mechanics.

Here's the bit the relativists don't understand:

For a wave packet of a material particle, the phase velocity exceeds that of light.

Before anyone throws up their hands in disbelief and renders "Oh my! She has gone mad!", a little explanation may be in order.

In quantum mechanics, all stuff is represented by a "wavefunction". That may seem a little abstruse, but all it means is that whatever happens can be described by equations that look a lot like wave equations (yes, wave equations were already worked out years before).

If you know a little about waves, when you put two together, they can interfere, crest plus crest equals bigger crest, trough plus trough equals deeper trough, and trough plus crest equals zero if the amplitude is right.

Given that, if you superimpose two waves of slightly different wavelength, you will get no waves at all except for a small bump, called a "wave packet".

Quantum Mechanicists often call this a "particle", be it a neutrino, a proton or a car.

If you could watch the "waves" in this "wave packet", you would quickly see that although the "bump" is moving slowly, the waves that generate it inside are moving faster than light.

The velocity of the "bump" is called "group velocity" (the speed we actually see), and the velocity of the waves inside that generate it, is called the phase velocity.

For actual particles (ie ones with actual mass), it is a requirement of quantum mechanics that this "phase velocity" always exceeds that of light.

Oh, you can't detect it,THAT would violate relativity, but it is there.

For those of you who would say "Then how can you know?", the simple answer is "It is in the mathematics, so try and disprove it"!

I wonder what Kurt Godel would say?

Wanted, Dead or Alive - SHRODINGER'S CAT!

Being a nuclear structure physicist has its advantages. One of them being a familiarity with something that is totally baffling to most people.

I am talking about quantum mechanics, that realm of physics spurned by Einstein and shunned by everyone but those in the know. I am honored to say I am in the final category.

So what is this cat thing anyway? Well, it is only a "gedanken" experiment, that is, a "thought" experiment, possible in principle only.

A cat is in a closed box. With it, is a device that releases a poison gas only when a certain atom of radium decays. The fate of the cat is entirely dependent upon the random nature of the quantum decay of the atom. The atom could decay at any time.

No-one knows when.

The question is this: "Is the cat dead or alive?".

The answer can only be found if the box is opened.

But if you do not open it, can you say if the cat is dead or alive?

Funnily enough, the question is actually meaningless.

Quantum mechanics tells us that the cat is actually in a superposition of both states of being "dead" AND "alive" until you look at it. By disturbing the equilibrium, you force the quantum state into one the or the other by merely looking.

This is what is strange about quantum mechanics. The cat is merely an everyday representation of many of the effects we actually see on the subatomic level.

One of these recently solved a long-standing problem. The way the sun shines. Honest.

The tiniest particles detectable are called neutrinos. They were "invented" by Enrico Fermi to explain the anomalous electron momentum spectrum in Beta decay. Years later, they were actually detected. So small are they, that their mean free path in lead is about a parsec.

Mr Ray Davis, in 1967 went digging in Dakota. In fact he went deep down underground to create one of the most unusual telescopes ever. A huge swimming pool of cleaning fluid. Cleaning fluid (Carbon tetrachloride) sucks up neutrinos rather more than most stuff.

So, Mr Davis decided to detect them. From the sun. Trouble was, he only detected about 1/3 of the SNU's predicted (SNU = Solar Neutrino Units, pronounced "snoos"). Either there was a problem with our understanding of the way neutrinos interacted, or there was a problem with our understanding with the way the sun shone. Neither was acceptable.

Until recently.

Until recently, we thought neutrinos were massless, but sensitive experiments indicated that they DO have mass. The implication of this is that in their journey from the sun they can "oscillate" between different quantum states, which means we detect less of the expected ones on earth.

The same sort of effect the poor cat demonstrated.

A similar, but far more fascinating effect happens to K-0 mesons (strange particles they are!) when they decay. It is called "regeneration", and is something that happens that is quite beyond any common sense at all.

If THAT ain't strange enough, try doing a search on "alpha particle tunnelling", where particles pass through inpenetrable barriers.

Quantum mechanics challenges most people's common sense.

God not only plays dice, but he plays with a sense of humor bordering on cynicism.