Monday 20 February 2012

Mathematical physics

Ernst Stückelberg
One of the things that both amazes and disappoints me when it comes to physics is mathematics. Using mathematical tools theoretical physicists have been able to come up with some of the most fantastic theories to describe the universe we live in.

Some of these equations can only be described as breath taking. The most famous is probably E=mc2. The equation linking energy and mass. Another is Newton's F=ma. Planck's E=hf is a third.

Simple equations that give us insight into the way the universe behaves. Yet there are other parts of physics where the mathematics is exceptionally difficult.

Take the Einstein field equations, still can't figure most of this out. Well hard. Most of quantum mechanics, or take the strong nuclear force, before you know it your staring down at something like this bad boy







it is at this point that most of us reach for our coats and head off home. This is just the way of it sometimes. There are lots  of areas in physics were the best models are still mind numbing for the majority of people. These are the best solutions we have found so far. For me this raises two questions. Are there actually better theories out there that are more simplistic that we haven't found yet? Or is mathematics alone not the best method of describing everything in Physics?

The greatest step forward in "alternative" methods of describing the physical universe are probably the Feynman diagrams (first thought up by Ernst Stückelberg apparently, who was also the first to propose that a positron may be an electron travelling backwards in time).

Richard Feynman used the diagrams back in 1948 as a pictorial representation of events.

Take something "simple", an electron and a positron meeting each other, annhiliating each other to create a photon that then converts back to a positron and an electron. The maths for this is based on something called a Scattering-matrix that looks something like this


Let's be honest it takes you half an hour just to try and say it out loud. Then half a day trying to figure out what each of the terms in the equation actually represents.

The Feynman diagram on the other hand looks like the diagram below left. You can see the positron and electron meet, create a photon that then turns back into a positron and an electron.


Electron - Positron interaction
This representation is far easier for many to understand and the Feynman diagrams have become very popular over the years for describing  many subatomic interactions.

They are so handy that there will be a future post completely dedicated to the use  of Feynman diagrams and the rules governing their use. There you will find out how to describe some of the most complex physics ideas without the use of a single equation. Which brings me back to the point of this post.

Physics will always rely on mathematics to a large degree. The 20th century advancement in physics as been driven by mathematics.  I wonder though, if we will reach a point where maths can take us no further and in order to open up the next great epoch in physics advancement will require a new set of tools based on something we can't even imagine just yet. Best get my thinking cap on.

I can feel a great idea coming on, where are those crayons?



Wednesday 15 February 2012

Sometimes even the greats get it wrong

A really clever idea!
There is no doubt that Aristotle was one of the greatest thinkers there as every been. In the history of human intelligence he is definitely up there amongst the greats and in a way that is the problem.

This problem is not Aristotle's fault, after all he was well dead and buried for the majority of the time his legacy was causing problems. It is part of the human condition I suppose. To go up against a great man is to some how imply that you are equal or greater than the man and is ideas. This is wrong, but is often the way.

The upshot of this is that it can become more difficult to dispute the ideas of a great man after his death than during his life. As time goes on the ideas mature and solidify in the mind of those left behind, even if the idea is complete nonsense. One of these is Aristotle's idea on weight and its behaviour.

Aristotle believed that the falling speed of an object was proportional to its weight. The heavier the object the higher is "falling speed". This it turns out is complete rubbish. But it took almost 2000 years before it was proven to be well wide of the mark. Why did it take so long? two reasons I think. One is that it was actually difficult, without the use of an accurate clock, to come up with any experiment that could prove things one way or another. The other reason is that I think many people accepted Aristotle's thinking and did not both to question it.

Eventually both of these reasons were addressed and solved by Galileo. He solved it with the rather elegant experiment, he also raised doubt with an equally elegant thought experiment. It goes like this...

you have 2 stones, let's call them A and B. One, A, is twice the size and weight of the other, B.

Now according to Aristotle, the larger, A, of the two will fall fastest. Ok. But if we now tie the two together, with a long piece of string, what happens?

According to Aristotle, A will fall faster than B and at some point will be far enough ahead of B that the string will go taught.  B, the slower of the two will then actually slow down A, so the combination of A + B will be slower than A alone. Or would the two weights attached by the taught string suddenly realise the have a combined weight of A+B and start to fall even faster?

This is the paradox that confronts us if we choose to believe Aristotle.

The reality is that neither of the outcomes given above actually happens. Provided air resistance can be ignored, which it pretty much can for heavy weights, such as marble balls. All objects, irrespective of their individual weights fall at the same speed. (This experiment was performed on the moon using a feather and a hammer. There is no atmosphere on the moon to produce air resistance. The result was that the hammer and the feather fell at the same rate and touched the ground at the same time.)

What I think is really clever is the way the that Galileo proved this. He did a number of different experiments including dropping object from a large tower, what he found was that irrespective of the weight, they all fall at the same rate. He also discovered that this rate was not constant but changed, being dependent on the height of the fall.

He also did some clever work to determine the rate of the acceleration. He did not have a clock accurate enough to perform detailed measurements. So what he needed to do was slow things down, so that he could get away using a less accurate clock. He achieved this by rolling balls down slopes of different inclines. This is a far clever idea than it may first appear and is not at all as obvious as it seems in hindsight. Galileo discovered that

v = at + u (1)

v - velocity when you reach bottom of slope (or tower)
t - time it takes to reach the bottom
u - the initial velocity
a - acceleration

now equation (1) looks very similar to y = mx + c (equation of a straight line), so we would expect Galileo's equation to generate a straight line, which is what it does. What Galileo had discovered was a value for "a", which turns out to be acceleration due to gravity, these days taken to be about 10 m/s2.

This is probably the first known calculation of the parameter.

What is interesting is that "a" is not a constant. It gets higher the further North or South of the equator we go. It also gets less the higher we get. So acceleration due to gravity at the top of Everest (28oN) is lower than acceleration due to gravity at see level at 28oN. This would have been unknown to Galileo though because the degree of accuracy to determine these differences was probably smaller than he could detect with experiments.


So it just goes to show that even the greats get it wrong sometimes. Newton, Einstein, they have all backed the odd donkey in their day. The point is that it does not matter that they have come up with some really shockers. The fact that they came up with anything at all is the important thing.

More important maybe that although they have had some truly brilliant insights it does not mean that they are infallible. Even their most amazing ideas must be held up for scrutiny. People should not feel intimidated when they do question the greats be them alive or dead.

Sunday 12 February 2012

Inertial Frame Of Reference

In Physics especially were motion is concerned you will often hear the term "Inertial Frame of Reference" or "Inertial Reference Frame", but what is it? Well, technically it can be described as follows (this one is taken from Wikipedia);


In physics, an inertial frame of reference (also inertial reference frame or inertial frame or Galilean reference frame) is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.

All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not accelerating in the sense that an accelerometer at rest in one would detect zero acceleration.

It goes on to say

Measurements in one inertial frame can be converted to measurements in another by a simple transformation (the Galilean transformation in Newtonian physics and the Lorentz transformation in special relativity). In general relativity, in any region small enough for the curvature of spacetime to be negligible one can find a set of inertial frames that approximately describe that region.

Glad we cleared that one up! See this can sometimes be the problem with Physics. In order to describe something in such a manner that it is not open to misunderstanding, misinterpretation or ambiguity you can end up sounding like a lawyer.  Worse, you write something that most people cannot understand! Let's be honest the definition above is a bit of a shocker. So, is there a way we can state it in plain English without losing fidelity? Can we describe the same thing in 2 words?  Let's have another look at what has just been said

an inertial frame of reference (also inertial reference frame or inertial frame or Galilean reference frame) - this section is just telling us what it is called. I'm going to call it "my garage".


describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner. - tells us that time is uniform (a second is a second) and that space it is same in each direction and doesn't change with time. In the case of "my garage" the dimensions of the garage do not change over time, and an hour continues to be an hour, hour after hour.

All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not accelerating in the sense that an accelerometer at rest in one would detect zero acceleration. - everything is moving in a straight line at a constant speed (think of Newton's first law - ...an object in motion stays in motion with the same speed and in the same direction...), or does not appear to be moving at all, a bit like my garage.

So, a reasonable representation of an Inertial frame of reference is .... "Your garage".

Feel free to experiment in this garage and when publishing your results remember to start with ....

The experiments where carried out in an inertial frame of reference (also inertial reference frame or inertial frame or Galilean reference frame), ie a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.

This inertial frame was assumed to be in a state of constant, rectilinear motion with respect to any other inertial reference frame; it was not accelerating in the sense that an accelerometer at rest in the inertial frame would detect zero acceleration.


 or you could say you did them in your garage.

Saturday 11 February 2012

Quasars

PKS 1127-145 - 10 Billion light years from earth
This post is about the Quasar. When I was younger it was the quasar that made the papers much in the way that Black Holes tend to these days. Over the years though the light of Quasars began to dim has the light of the Black Hole intensified and these days they seem to be hardly ever mentioned.

The Quasar was, and still is, one of the greats of the heavens. The stats for these fellows is truly awe inspiring. In fact the numbers are so large that they really are way beyond comprehension.

A Quasar, or quasi-stellar radio source to give it its full title, are really energetic light sources often billions of light years away. To give you some idea of how bright these objects are, consider this...

The sun is about 8 light minutes away. It takes light about 8 minutes to travel the 150 million kilometers from Sun to earth. The nearest star is alpha century, which is about 4 light years away. It is so far that it is seen a star, one amongst many, granted it is a little brighter than most, but it is what we would call a regular star all the same.

If a Quasar was 33 light years away, 8 times the distance of Alpha Century, it would burn so bright as to be almost indistinguishable from the sun on a sunny day. It is thought to be as bright as two trillion (2x1012) suns. This is one of those numbers that is too large to grasp, it is genuinely massive.

The Sun itself radiates an incredible amount of energy every second of every day and yet a quasar takes this to an entirely different level. To try and give you some idea think of this,

A year is about 31 million seconds, 2x1012 /31,000,000 = 65,000. A quasar gives off the same amount of light in 1 second as our sun does in about 65,000 years!

It is estimated that the Sun emits 3.3×1031 Joules of energy every second. Given E=mc2, this amount of energy is the same as 3.6×1014 Kg of mass begin converted into pure energy every second. In 65,000 years this amounts to 7.4x1025 Kg of mass. The mass of the earth is about 6 x 1024 Kg.

In other words, if we could convert the entire earth into pure energy, then it would produce enough energy to keep a Quasar going for about 0.1 seconds!!!! That's 600 earths a minute. Talk about greedy!

Apart from the 0.1 seconds, which is actually over very quickly, the rest of the numbers discussed above are beyond out comprehension, suffice to say they really are huge.

So, we know that a Quasar is running on high octane, we also know that they are not that big. Once again, this is a relative thing. They are thought to be about the size of our solar system, which, lets be honest is pretty big. But considering how bright they burn, it is a small volume.


There are about 200,000 known Quasars and most appear to be upwards of 3 billion light years away. The furthest upwards of 10 billion years. If the Big Bang theory is correct then these older Quasars formed about 3 Billion years after the start of the universe.

They were first detected in the 1950s and measurements using the Doppler effect have shown that some are moving away from us at a tremendous rate, around 37% of the speed of light!

For several years after their discovery many theories tried to explain their brightness, antimatter or white holes, though it is now thought that it is due to the accretion disc energy mechanism. It is thought that at the center of each Quasar is a super massive black hole.

It is also thought by some that most galaxies, including our own Milky Way, have gone through a Quasar stage and are now quiet because they have run out of energy to feed into their central black holes to generate radiation.

I am not entirely convinced of the existence of black holes so I don't agree with some of this. That said, I do like the idea that our galaxy was once a Quasar. It's nice to think that our galaxy really was a bright shining object in its day.

I was brought up on Quasars, what is so cool about them is that there is no doubt they exist, you can see them with a telescope. They are beautiful, fantastic and utterly magnificient. I think that they are great.

Quasar, I salute you.

Thursday 9 February 2012

Fine Structure Constant

1 / 137.03597
These days the word genius is applied to just about everything, football, cooking, acting, you name it someone will say "oh yeah that guy, absolute genius, he's amazing". This is a shame because for me it actually takes something away from people who really are geniuses.

A Genius (plural geniuses) is something or someone embodying exceptional intellectual ability, creativity or originality typically to a degree that is associated with the achievement of unprecedented insight.

In other words, something or someone truly, truly amazing. Someone who will be remembered long after they have gone on their way. Newton, Mozart Einstein, The Beatles and for me and many others... Richard Feynman.

The reason why I love Feynman is not because he was absolutely fantastic at Physics, or that he put together some of the greatest Physics lecture notes ever. It's not that he could explain the most complicated ideas really clearly, though he could. It was that fact that he was not afraid to say that he didn't know things in Physics. In fact he often drew attention to them.

He won the Nobel Prize for his work on Quantum Electro Dynamics, a brilliant theory that has so far stood the test of time. It has given Physics some of the most accurate predictions ever and these have been backed up time and again by experiment. It really is brilliant. Now, you would think that having come up with something this clever, you would spend your time giving yourself a little pat on the back and saying "yep, hit that mark with that one", instead, Feynman says this...

There is a most profound and beautiful question associated with the observed coupling constant, e - the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." 

Oh cool is that?! Wins a Nobel prize and then points out that there is a number used in a part of his theory, a number that he as not got the first clue about where it comes from or what it is means?!

How can you not love him?

So, what is this magic number? It is called the fine structure constant and has an approximate value of 1/137. It was first introduced to the world in 1916 by Arnold Sommerfeld.  It is defined by the equation below
\alpha = \frac{e^2}{\hbar c}
where:
  • e is the elementary charge;
  • ħ = h/2π, known as the reduced Planck constant
  • c is the speed of light in a vacuum
e, is the charge that an electron has. We don't know why it has this charge, it just does. c,  we don't know why light travels at this speed, it just does. h, Plancks constant, or the proprotionality constant, relates the frequency of light to energy, again we have no idea why it has the value it does.

So to summarise, we have 3 values that we don't really know why they have the values they do. But more interestingly when we multiple the charge on an electron by itself and divide by the speed of light and then divide by Plancks constant you end up with a number 1/137. A unitless constant which props up in various place in Physics. Can't help thinking that once we crack this fella we will be making a real big break through.



Wednesday 8 February 2012

Black holes - Part 1

Stars and planets warp space time
Black holes, now you could easily be mistaken for thinking that black holes are real and exist.

A day doesn't pass without something about another Super Massive Black Hole that has been spotted out in this galaxy or that galaxy.

Every suprise that the universe throws at us are something to do with black holes. But here is a thing, I don't think they exist.

What is the evidence for black holes? what are the theories? do we even need them to exist? Well that is what I am going to write about in this blog and the next probably, but first a quick summary of the state of play.

Newton  - published "Principia" a work in three book on 5 July 1687. Rightly regarded as one of the most important works in the history of science. It contains a law of Gravity. summary...


The force of Gravity is related to the mass of an object, the more mass, the greater gravitational force. 

Why is this so? He didn't say. Who knows, well… no one actually, still don’t. Moving on.

1783, geologist John Michell writes to the royal society to point out that a really big sun would have a really big gravity, so big in fact that not even light could escape, should light fall under the rules of gravity.

1796 Laplace promotes the same idea in the first couple of editions of one of his books and then chickened out and had it removed from later editions. If you’re going to put forward a daft idea, have the bottle to see it through eh?

Not much happens in the 18th and 19th century and then comes along Albert Einstein.

Einstein and General Relativity

1915 Albert Einstein develops his theory of General Relativity, a Geometric theory of gravitation.

Einstein realised that it may be possible to describe the universe in a rather odd way using equally odd mathematics.These turn out to be mindnumbingly complex. Fortunately the ideas themselves are not really that complex and so we can understand the physics without being able to do the maths. Which is as it should be.

Space time (an idea also dreamed up by Albert Einstein) can actually be considered to behave a little like a giant rubber sheet pulled tight. There’s not much in the universe, it is mostly nothing, so the sheet is flat. When you get near a star the sheet gets a little warped, imagine placing a weight on a trampoline, it distorts the surface, same idea.  Why does it behave like this? I'm not sure anyone really knows.

Years earlier a really clever mathematician called Riemann has spent some time working out the maths of geometric surfaces, Einstein applied this work to his idea and bingo, you have General Relativity. In other words, General Relativity is a mathematical model of how surfaces change their shape when you distort them. This can also be applied to the universe.

Einstein was able to develop a series of 10 equations, now known as the Einstein’s field equations (EFE) which describe how gravity can actually come about from the distortion of spacetime by matter and energy (energy being equivalent to mass thanks to E=mc2, who came up with that? Einstein, he was definitely on a roll!)

It wasn’t long before people began to realise that these equations had the potential to do some strange things and they began to wonder what this may actually mean in the physical world.

This turned out to be difficult, partly because the maths was difficult, but also because there was no chance of testing any of this in a lab, at least not in those days.

Take gravity waves, Einstein argued in 1916 that it should exist. These are waves that travel at the speed of light and would be produced in a binary star system for example. A binary star system being two stars very close together that orbit each other (such things do actually exist in nature). For more than 40 years people where not sure if Einstein was even right about his interpretation. It was only in 1957 when Feynman put forward a thought experiment that people figured that it may be right. This also had something to do with the fact that Feynman was one of the best theoretical physicists of his day.

But here is the thing, do gravity waves exist? As of 2011 we have NOT detected a single gravity wave. No experiment has so far proved their existence.  So it could still be wrong!
Back to black holes…

The next thing we need to think about is escape velocity,

Back on earth, we realise that we want to send someone to the moon, the problem is that gravity is determined to stop us from leaving. We throw up a stone and it comes back down, what goes up must come down right? But what happens if you throw it really fast, it goes higher and takes longer to come back down. So throw it really really fast, so fast that it gets high enough that gravity can no longer pull it back! The speed we need to throw it so that gravity doesn’t pull it back is known as the escape velocity.

We find that escape velocity is related to gravity, the greater the gravity the faster we have to go to escape. It’s easier to escape from the moons gravitational pull than the earths. Although an atmosphere complicates escape velocity, to escape the earth required a massive Saturn 5 rocket. The rocket that blasted off from the moon was tiny in comparison! That’s because the moon as far less mass, far less gravity! The sun on the other hand had a large escape velocity, about 620 km/s (over 200,000 km/hr).

Now Newton told us that has we get farther away from something the gravity gets smaller, double the distance the gravity is only a quarter, make it five times the distance and the gravity is only 1/25th. Keep that in mind.

Imagine taking something like the sun and pushing other things, like large planets and things into it, more and more and more. So its starts to get more massive and this according to Einstein, warps space time which increase the gravity which crushes the sun. But the sun is full of hot gases which push back against the gravity.

We keep going more and more, so the space time gets more and more warped and gravity gets greater and greater crushing the sun even more. The escape velocity starts to clime 1000km/s, 10000 km/s, 100000 km/s. This goes on until you've got a Mr Creosote of a sun with an escape velocity of  300000 km/s. The gases keep pushing back but eventually they haven't got enough fuel to continue the struggle.

At 300000 km/s something interesting happens. We have managed to get a gravity so big that the escape velocity would have to be faster than the speed of light! So even light can't escape because it can't go fast enough!

What would happen then? What what would happen to a star with that much mass?  Can such a thing exist? These questions will be discussed in the next post on Black Holes.

Tuesday 7 February 2012

Superconductivity - an introduction

Meissner Effect
Superconductivity was discovered in 1911 by Heike Kamerlingh Onnes, a Dutch bloke who was very good at making fridges. He was awarded the 1913 Nobel Prize for Physics for this discovery.

While testing the "current best" theory on electricity he made a truly amazing discovery. He was cooling mercury and measuring it's resistance as a function of temperature. As the temperature of the mercury got less and less so did the electrical resistance of the mercury and it should have continued on this way until you got right down to absolute zero, which is the lowest temperature we can reach. Instead of this, when the temperature reached about 4.25 Kelvin, which is 4.25 degrees above absolute zero, a seriously cold temperature (outer space is 3 Kelvin) the resistance of mercury suddenly disappeared totally. 

It no longer had any measurable resistance whatsoever. Further research showed that this was not the only material to undergo this transition at low temperatures. Many metals have superconducting properties when cooled far enough. In recent times some ceramics have been found that demonstrate "high temperature" superconductivity. This is a relative term and often refers to materials that are superconducting just above the boiling point of liquid nitrogen, which is 77K (-196 Celcius).

For me there is nothing in physics that is more interesting, more puzzling and potentially more revealing about the universe than a superconductor. At the time of writing there is no accepted theory for high temperature superconductors.

Note: Low temperature superconductors can be described by the BCS theory (Bardeen Cooper Schrieffer - 1956), but it cannot account for high temperature materials.

One of the things that really amazes me is the flow of electric current in a superconducting material. Normally the world can be seperated into conductors at one end of the spectrum and insulators at the other, with every material fitting in somewhere between these two extremes. Metals are good conductors, most plastics are good insulators, Silicon is neither, it is a semiconductor. When a voltage is applied across a material a current will flow which is proportional to the voltage. The most common case of this are materials that are said to obey Ohm's law,

V = I R , V - voltage, I is current and R is resistance of the material. So if you have 240 V and the resistance is 120 Ohms, then

240 = I x 120, I =2 Amps.

But with a superconductor R = 0. This means that you can have a current flowing without a voltage. If the superconductor is like a doughnut shape and the current is circulating round the doughnut, it will continue to do so forever. It will never stop, it will just go on and on. Just like the Terminator!

So what do we know about them, besides them having no resistance? Well, when superconductors are placed in a magnetic field they create their own magnetic field opposite in magnitude to the field. So you get a South - South, or a North-North. This means that the magnetic field does not penetrate into the body of the superconductor. This can be seen most dramatically in the Meissner effect (see above).

The electrons in a superconductor pair up into what are known as Cooper pairs and it is these that move through the superconductor.

They can be used to produce extremely sensitive detectors of magnetic fields called SQUIDs or Superconducting Quantum Interference Devices.

They can often carry massive currents, way beyond the capacities of normal metals. 10 of thousands of Amps of current can be carried in superconducting wires no more than a few millimeters in diameter.
Should a room temperature superconductor be discovered the world will change as dramatically as it did in the last century because of oil and electricity. The uses that superconducting materials will be put to really is beyond the imagination. Already high speed maglev trains use the Meissner effect to float above the rail tracks. Speeds in excess of 500 km/h have been achieved by this train. There are also other "specialist" applications using superconductors, but this is small compared to what would be available if and when room temperature superconductors become available.

I truly believe that a theory of superconductivity will not only explain how high temperature superconductors work, but more fundamental properties of the universe will also be revealed. This subject will be covered in future posts.



Monday 6 February 2012

Kepler's laws

Kepler's 2nd law
Johannes Kepler (1571-1630) was an astronomer who continued the work of Tycho Brahe (1546-1601).

Brahe, himself a great astronomer, had measured the heavens collecting a mass of data that fell to Kepler to unravel.

What Kepler discovered is one of the great moments in science and shows just how fantastic and simplistic the universe can be.

In order to derive these laws Kepler had to make a massive intellectual step one that contradicted his former bosses view. One of the difficulties was that even for the orbits of earth and Mars seen from Earth appear really odd and would be difficult to try and model mathemtaically. If however, it is accepted that the sun is the center of the orbit, the one can start to see that the orbits may actually be almost circular. By doing this Kepler was then able to use the data to develop his laws.

Keplers 3 laws can be described as follows 

Kepler's elliptical orbit law
The planets orbit the sun in elliptical orbits with the sun at one focus.

Kepler's equal-area law
The line connecting a planet to the sun sweeps out equal areas in equal amounts of time.

Kepler's law of periods
The time required for a planet to orbit the sun, called its period, is proportional to the long axis of the ellipse raised to the 3/2 power. The constant of proportionality is the same for all the planets.

The Ellipse. An ellipse is like a circle that has been squashed slightly. (In fact it makes more sense to me to say that a circle is a special case of an ellipse where both foci (plural of focus)are the same.

For the Earth this means that the closest it gets to the sun is 91 million miles and the furthest distance is 95 million.

So the first law tells us something that these days everyone takes for granted, the planets orbit the sun in almost circular orbits.

The second and third laws are no so well known. Both of these I find to be amazing, If you draw a line from the center of the sun to the center of a planet, it sweeps out equal AREAs in EQUAL amounts of TIME! (see the diagram at the beginning of this post)

For a circular orbit this would be expected in a way. But in an elliptical orbit it means that when the planet is nearer the sun it is moving faster relative to the sun than when the planet is at its extreme distance from the sun.

The last law says that the time needed to complete and orbit squared is related to the long axis of the elliptical orbit cubed! Lets try a simple calculation to see what this means. The Earth is 150 million km from the sun (just to make the maths a little bit easier at this point). If we take the cube of 150,000,000 we get

150,000,000 x 150,000,000 x 150,000,000 = 3,375,000,000,000,000,000,000,000

now we know that the earth takes about 365 days to orbit the sun, each day has 24 hours, made up of 60 minutes, which are in turn made up of 60 seconds. This means that in 1 years we have

1 year = 365 x 24 (hours) x 60 (minutes) x 60 (seconds) = 31,536,000,

so Kepler tells us that we need to square this

so for earth this is

31,536,000 x 31,536,000 ~ 995,000,000,000,000

now according to Kepler we have

995,000,000,000,000 = k x 3,375,000,000,000,000,000,000,000

cancelling a few zeros from each side gives

9 = k x 33,750,000,000

so k = 9.95 / 33,750,000,000 with units of seconds squared / km cubed in this case.

k ~ 2.9 x 10-10s2/km3, or 2.9 10x10-19 s2/m3

What this also implies, given the second law that the time to orbit is proprtional to r 3/2, means that the planets further out must be moving more slowly than the inner planets.

The fact that the outer planets moved more slowly can be explained if the sun's force was some how dependent on distance, getting weaker as the distance increased and it was Newton who showed this in his law of gravity.

So, Kepler was able to show that by taking the Sun as the center of the solar system he could  demonstrate that the motion of the planets could be explained by very simple school boy level mathematics. I find this absolutely amazing. Here in the universe in which we exist these massive planets move in paths described by remarkable simple maths.

For me this raises a question. Is it the case that all of physics can be explained by equally simplistic methods? and the fact that we currently use some exceptionally difficult maths to explain certain phenomena is just misunderstanding and a lack of understanding on our part and we have not truly realised the real nature of the universe?

Note: Kepler's laws (and Newton's law of gravity) do not appear to hold in galaxies. This has led to the emergence of a number of competing theories. It is unclear at this time which, if any, is correct.

Wednesday 1 February 2012

Heisenberg uncertainty principle

location or momentum
The Heisenbery uncertainty principle (HUP) is one of those things from Physics that, to me at least, is really odd.

In this post I'll try to explain what it is, and also why we might not really need it!

This theorem was an attempt to try and figure out what quantum mechanics actually means and has become part of what is called the  Copenhagen interpretation of quantum mechanics.

This is how odd HUP is, take the title of the paper itself

 "Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik".

which roughly translates as


"On the anschaulich content of quantum theoretical kinematics and mechanics"

See the word anschaulich, as no direct translation into English and by choosing different translations actually introduces a certain uncertainty into the title of the paper!

The HUP was discovered by Heisenberg in 1927, the idea being that on an atomic scale we cannot know the momentum of an object and its location simultaneously. To understand HUP it is necessary to understand what was just said in the previous sentence, well the bolded bit.

So, thought experiment, imagine an electron and its momentum is known, in other words, we know its mass, we have applied a known force to it for a certain amount of time, so we know its momentum. Let's say that it starts of stationary. We apply a force, we know

F = m a,  which is the same as F/m = a

so if we know F and m we can calculate a. We also know that

v = a t ,

where v is velocity, a is acceleration and t is time. But hold on one moment, we can replace "a"

v = (F t)/m , which becomes

mv = F t , but mv is just the definition of momentum. So if the particle starts out stationary we apply a force for t seconds then its momentum is just F multiplied by t (ignoring relativistic effects, this is being done a low velocities).

A little while later someone says, were has that electron got to? So we decide to take a look, using something like light or by firing other particles at it. This is were we run into trouble according to Heisenberg. By trying to measure were the particle is, using the technique described, the interaction will alter the momentum by an unknown and indeterminable amount. So while we may now have some idea of where it is, we no longer know its momentum accurately.

What Heisenberg is saying here is that this inability to know position and momentum is a fundamental property of quantum mechanics. It is not the case that we are poor experimenters.

That is it, that is the Heisenberg Uncertainty Principle.

A year later a guy called Kennard proved the theorem to give us

Δψp Δψq ≥ ℏ/2

the equation that is now associated with the HUP. Over the years it seems to me that this idea has come to be accepted without people actually stopping to think what it means.

By taking a second measurement it is possible to work out the change in momentum from the first measurement. What this implies is that the HUP is not true of the past. Heisenberg also says that before taking the second measurement we don't know the momentum. Furthermore, after the measurement we no longer know the momentum.

As with many things in physics, HUP, while being one of the most famous aspects of quantum mechanics, is not the only player. In 1932 Dirac, while pondering the positron was also dreaming up  an alternative idea to Schrödingers and Heisenberg's work. This lay mostly forgotten until a visiting physicist called Jehle told Richard Feynman about it in 1941.

The Dirac-Feynman idea is based on path integrals and will be covered in another post. It turns out to be equivalent to Schrödingers and Heisenbergs ideas. So it will contain the HUP.

Is it the best of the three? No one really knows. Each has its place and each can be used to solve problems that are difficult using the other theories.

Interestingly Dirac-Feynmans theory was mostly ignored until the 1970s because it was generally hard to work with. Schrödingers and Heisenberg's work offered more productive alternatives especially to undergrads. Then a Russian paper used the idea to advance Quantum Field Theory and since then it has made a serious come back.

This raises the question of whether we will still need HUP if Dirac-Feynman's approach gains more acceptance over time? Will we still be discussing it in 30 years? Who knows?

I can't help thinking that there is probably another solution out there that is even more elegant than those proposed so far. Once this is discovered maybe all three will be retired!

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