In the Dark

Spinning Out

Posted in Cricket, The Universe and Stuff with tags angular momentum, cricket, friction, googly, Graham Swann, legspin, moment of inertia, offspin, Physics, Shane Warne, The Rubaiyat of Omar Khayyam on September 6, 2010 by telescoper

I don’t know why, but last week was my most popular week ever, at least in terms of blog hits! I was going to follow up with a foray into the role of spin in quantum mechanics, but decided instead to settle for a less ambitious project for this evening.

Yesterday I walked past the cricket ground at the SWALEC Stadium in Sophia Gardens, Cardiff, during the Twenty20 international between England and Pakistan. There is another match of this type tomorrow night which I’ll actually be going to, as long as it’s not rained off, but I have too many things to do to go to both games. Anyway, England’s excellent off-spinner Graham Swann was bowling when I watched through a gap in the stands at the river end of the stadium. He seemed to be getting an impressive amount of turn, and I got wondering about how fast a bowler like “Swannee” actual spins the ball.

For those of you not so familiar with cricket here’s a clip of another prodigious spinner of the ball, Australia’s legend of legspin Shane Warne:

For beginners, the game of cricket is a bit similar to baseball (insofar as it’s a game involving a bat and a ball), but the “strike zone” in cricket is a physical object ( a “wicket” made of wooden stumps with bails balanced on the top) unlike the baseball equivalent, which exists only in the mind of the umpire. The batsman must prevent the ball hitting the wicket and also try to score runs if he can. In contrast to baseball, however, he doesn’t have to score; he can elect to play a purely defensive shot or even not play any short at all if he judges the ball is going to miss, which is what happened to the hapless batsman in the clip.

You will see that Warne imparts considerable spin on the ball, which has the effect of making it change direction when it bounces. The fact that the ball hits the playing surface before the batsman has a chance to play it introduces extra variables that you don’t see in baseball, such as the state of the pitch (which generally deteriorates over the five days of a Test match, especially in the “rough” where bowlers have been running in). A spin bowler who causes the ball to deviate from right to left is called a legspin bowler, while one who makes it turn the other way is an offspin bowler. An orthodox legspinner generates most of the spin from a flick of the wrist while an offspinner mainly lets his fingers do the torquing.

Another difference that’s worth mentioning with respect to baseball is that the ball is bowled, i.e. the bowler’s arm is not supposed to bend during the delivery (although apparently that doesn’t apply if he’s from Sri Lanka). However, the bowler is allowed to take a run up, which will be quite short for a spin bowler, but long like a javelin thrower if it’s a fast bowler. Fast bowlers – who can bowl up to 95 mph (150 km/h) – don’t spin the ball to any degree but have other tricks up their sleeve I haven’t got time to go into here. A typical spin bowler delivers the ball at speeds ranging from 45 mph to 60 mph (70 km/hour to 100 km/hour).

The physical properties of a cricket ball are specified in the Laws of Cricket. It must be between 22.4 and 22.9 cm in circumference, i.e. 3.57 to 3.64 cm in radius and must weigh between 155.9g and 163g. It’s round, made of cork, and surrounded by a leather case with a stitched seam.

So now, after all that, I can give a back-of-the-envelope answer to the question I was wondering about on the way home. Looking at the video clip my initial impression was that the ball is deflected by an angle as large as a radian, but in fact the foreshortening effect of the camera is quite deceptive. In fact the ball deviates by less than a metre between pitching and hitting the stumps. There is a gap of about 1 metre between the popping crease (where the batsman stands) and the stumps – it looks much less from the camera angle shown – and the ball probably pitches at least 2 metres in front of the crease. I would guess therefore that it actually deflects by an angle less than twenty degrees or so.

What happens physically is that some of the rotational kinetic energy of the ball is converted into translational kinetic energy associated with a component of the velocity at right angles to the original direction of travel. In order for the deflection to be so large, the available rotational kinetic energy must be non-negligible compared to the original kinetic energy of the ball. Suppose the mass of the ball is $M$ , the translational kinetic energy is $T=\frac{1}{2} Mv^2$ where $v$ is the speed of the ball. If the angular velocity of rotation is $\omega$ then the rotational kinetic energy $\Omega =\frac{1}{2} I \omega^2$ , where $I$ is the moment of inertia of the ball.

Approximating the ball as a uniform sphere of mass $M$ and radius $a$ , the moment of inertia is $I=\frac{2}{5}Ma^2$ . Putting $T=\Omega$ , cancelling $M$ on both sides and ignoring the factor of $\frac{2}{5}$ – because I’m lazy – we see that the rotational and translational kinetic energies are comparable if

$v^2 \simeq a^2\omega^2,$

or $\omega \simeq \frac{v}{a}$ , which makes sense because $a\omega$ is just the speed of a point on the equator of the ball owing to the ball’s rotational motion. This equation therefore says that the speed of sideways motion of a point on the ball’s surface must be roughly comparable to speed of the ball’s forward motion. Taking $v=80$ km/h gives $v\simeq \frac{80 \times 10^3}{60 \times 60} \simeq 20$ m/s and $a\simeq 0.036$ m gives $\omega \simeq 600$ radians per second, which is about 100 revolutions per second. This would cause a huge deviation (about 45 degrees), but the real effect is rather smaller as I discussed above (see comments below). If the deflection is actually around 15 degrees then the rotation speed needed would be around 30 rev/s.

This estimate is obviously very rough because it ignores the direction of spin and the efficiency with the ball grips on the pitch – friction is obviously involved in the change of direction – but it gives a reasonable ballpark (or at least cricketground) estimate.

Of course if the bowler does the same thing every time it’s relatively easy for the batsman to allow for the spin. The best bowlers therefore vary the amount and angle of spin they impart on each ball. Most, in fact, have at least two qualitatively different types of ball but they disguise the differences in the act of delivery. Offspinners typically have an “arm ball” which doesn’t really spin but holds its line without appearing to be any different to their spinning delivery. Legspinners usually have a variety of alternative balls, including a topspinner and/or a flipper and/or a googly. The latter is a ball that comes out of the back of the hand and actually spins the opposite way to a legspinner while being produced with apparently the same action. It’s very hard to bowl a googly accurately, but it’s a deadly thing when done right.

Another thing also worth mentioning is that the rotation of the cricket ball also causes a deviation of its flightpath through the air, by virtue of the Magnus effect. This causes the ball to curve in the air in the opposite direction to which it is going to deviate on bouncing, i.e. it would drift into a right-handed batsman before breaking away from him off the pitch. You can see a considerable amount of such movement in the video clip, away from the left-hander in the air and then back into him off the pitch. Nature clearly likes to make things tough for batsmen!

With a number of secret weapons in his armoury the spin bowler can be a formidable opponent, a fact that has apparently been known to poets, philosophers and astronomers for the best part of a thousand years:

The Ball no Question makes of Ayes and Noes,
But Right or Left, as strikes the Player goes;
And he that toss’d Thee down into the Field,
He knows about it all — He knows — HE knows!

The Rubaiyat of Omar Khayyam [50]

17 Comments »

The First Four Last Songs

Posted in Music with tags Four Last Songs, Karita Mattila, Kirsten Flagstad, Richard Strauss, September, Simon Rattle, Wilhelm Furtwangler on September 5, 2010 by telescoper

Just a quickie today, as I have a lot to do this afternoon. Last night I stayed in and listened to Prom 66, the penultimate Saturday evening concert of the 2010 season of BBC Promenade Concerts from the Royal Albert Hall in London. In fact it was broadcast live on BBC Radio 3, and then shown on BBC television a bit later, a strange arrangement but one that at least let me listen to some of the music twice.

I haven’t listened to all that many of the Saturday concerts this year – on a weekend the scheduling is often somewhat orthogonal to my tastes – but this one was one I’d been looking forward to for ages. It didn’t disappoint. The performance featured the Berlin Philharmoniker conducted by Sir Simon Rattle in a very varied programme of music, including the Prelude to Act I of Parsifal by Richard Wagner and three marvellous orchestral suites by Arnold Schoenberg (Five Pieces for Orchestra, Op. 16) and two of his students from the 2nd Vienna School Anton Webern (Six Pieces for Orchestra, Op. 6) and Alban Berg (Three Pieces for Orchestra, Op. 6). All of these were played quite beautifully by an Orchestra whose name is synonymous with the highest standards of musicianship.

Even better than these, however, was the centrepiece of the concert, Four Last Songs by Richard Strauss, sung by the wonderful Finnish soprano Karita Mattila. I particularly wanted to hear this because the very first recording I bought of the Four Last Songs was by her (conducted by Claudio Abbado). It got mixed reviews when it came out about 10 years ago, but it’s still one of my favourites. Anyway, I thought her performance last night was as moving as any I’ve heard. Ten out of ten.

I’ve always known that the Four Last Songs were published after his death, so Strauss never heard them performed. What I didn’t know before the discussion on TV during the interval immediately after the performance was that the very first time they were performed was in 1950 at the Royal Albert Hall, by the London Philharmonia, so this was an occasion especially redolent for those who love this exquisite work. One can only imagine what it must have been like for the orchestra making this music live for the very first time. Apparently the first time any of them had seen the score was when they turned up for the rehearsal. I’m sure they knew as soon as they started playing that it was a masterpiece.

Anyway, I’ve posted a version of one of the Four Last Songs already – the last one, which happens to be my favourite. I thought I’d put up another one today and, given the historical connection, it seemed apt to pick a recording of the World Premiere of the work from 1950, by the London Philharmonia Orchestra conducted by Wilhelm Furtwängler and featuring the legendary Norwegian soprano Kirsten Flagstad. You have to make some allowance for the sound quality given that it’s such an old live recording, but it’s fascinating to listen to it. For one thing it’s a very different tempo to that of most modern recordings. Here they are performing the second song which, appropriately enough given the time of year, is called September.

11 Comments »

Get thee behind me, Plato

Posted in The Universe and Stuff with tags electrons, gluons, multiverse, ontology, Particle Physics, Physics, Platonic realism, Platonism, Quantum Mechanics, quarks, SU(3) on September 4, 2010 by telescoper

The blogosphere, even the tiny little bit of it that I know anything about, has a habit of summoning up strange coincidences between things so, following EM Forster’s maxim “only connect”, I thought I’d spend a lazy saturday lunchtime trying to draw a couple of them together.

A few days ago I posted what was intended to be a fun little item about the wave-particle duality in quantum mechanics. Basically, what I was trying to say is that there’s no real problem about thinking of an electron as behaving sometimes like a wave and sometimes like a particle because, in reality (whatever that is), it is neither. “Particle” and “wave” are useful abstractions but they are not in an exact one-to-one correspondence with natural phenomena.

Before going on I should point out that the vast majority of physicists are well away of the distinction between, say, the “theoretical” electron and whatever the “real thing” is. We physicists tend to live in theory space rather than in the real world, so we tend to teach physics by developing the formal mathematical properties of the “electron” (or “electric field”) or whatever, and working out what experimental consequences these entail in certain situations. Generally speaking, the theory works so well in practice that we often talk about the theoretical electron that exists in the realm of mathematics and the electron-in-itself as if they are one and the same thing. As long as this is just a pragmatic shorthand, it’s fine. However, I think we need to be careful to keep this sort of language under control. Pushing theoretical ideas out into the ontological domain is a dangerous game. Physics – especially quantum physics – is best understood as a branch of epistemology. What is known? is safer ground than what is there?

Anyway, my little piece sparked a number of interesting comments on Reddit, including a thread that went along the lines “of course an electron is neither a particle nor a wave, it’s actually a spin-1/2 projective representation of the Lorentz Group on a Hilbert space”. That description, involving more sophisticated mathematical concepts than those involved in bog-standard quantum mechanics, undoubtedly provides a more complete account of natural phenomena associated with the electrons and electrical fields, but I’ll stick to my guns and maintain that it still introduces a deep confusion to assert that the electron “is” something mathematical, whether that’s a “spin-1/2 projective representation” or a complex function or anything else. That’s saying something physical is a mathematical. Both entities have some sort of existence, of course, but not the same sort, and the one cannot “be” the other. “Certain aspects of an electron’s behaviour can be described by certain mathematical structures” is as I’m prepared to go.

Pushing deeper than quantum mechanics, into the realm of quantum field theory, there was the following contribution:

The electron field is a quantum field as described in quantum field theories. A quantum field covers all space time and in each point the quantum field is in some state, it could be the ground state or it could be an excitation above the ground state. The excitations of the electron field are the so-called electrons. The mathematical object that describes the electron field possesses, amongst others, certain properties that deal with transformations of the space-time coordinates. If, when performing a transformation of the space-time coordinates, the mathematical object changes in such a way that is compatible with the physics of the quantum field, then one says that the mathematical object of the field (also called field) is represented by a spin 1/2 (in the electron case) representation of a certain group of transformations (the Poincaré group, in this example).I understand your quibbling, it seems natural to think that “spin 1/2″ is a property of the mathematical tool to describe something, not the something itself. If you press on with that distinction however, you should be utterly puzzled of why physics should follow, step by step, the path led by mathematics.

For example, one speaks about the ¨invariance under the local action of the group SU(3)” as a fundamental property of the fields that feel the nuclear strong force. This has two implications, the mathematical object that represents quarks must have 3 ¨strong¨ degrees of freedom (the so-called color) and there must be 3²-1 = 8 carriers of the force (the gluons) because the group of transformations in a SU(N) group has N²-1 generators. And this is precisely what is observed.

So an extremely abstract mathematical principle correctly accounts for the dynamics of an inmensely large quantity of phenomena. Why does then physics follow the derivations of mathematics if its true nature is somewhat different?

No doubt this line of reasoning is why so many theoretical physicists seem to adopt a view of the world that regards mathematical theories as being, as it were, “built into” nature rather than being things we humans invented to describe nature. This is a form of Platonic realism.

I’m no expert on matters philosophical, but I’d say that I find this stance very difficult to understand, although I am prepared to go part of the way. I used to work in a Mathematics department many years ago and one of the questions that came up at coffee time occasionally was “Is mathematics invented or discovered?”. In my experience, pure mathematicians always answered “discovered” while others (especially astronomers, said “invented”). For what it’s worth, I think mathematics is a bit of both. Of course we can invent mathematical objects, endow them with certain attributes and proscribe rules for manipulating them and combining them with other entities. However, once invented anything that is worked out from them is “discovered”. In fact, one could argue that all mathematical theorems etc arising within such a system are simply tautological expressions of the rules you started with.

Of course physicists use mathematics to construct models that describe natural phenomena. Here the process is different from mathematical discovery as what we’re trying to do is work out which, if any, of the possible theories is actually the one that accounts best for whatever empirical data we have. While it’s true that this programme requires us to accept that there are natural phenomena that can be described in mathematical terms, I do not accept that it requires us to accept that nature “is” mathematical. It requires that there be some sort of law governing some of aspects of nature’s behaviour but not that such laws account for everything.

Of course, mathematical ideas have been extremely successful in helping physicists build new physical descriptions of reality. On the other hand, however, there is a great deal of mathematical formalism that is is not useful in this way. Physicists have had to select those mathematical object that we can use to represent natural phenomena, like selecting words from a dictionary. The fact that we can assemble a sentence using words from the Oxford English Dictionary that conveys some information about something we see doesn’t not mean that what we see “is” English. A whole load of grammatically correct sentences can be constructed that don’t make any sense in terms of observable reality, just as there is a great deal of mathematics that is internally self-consistent but makes no contact with physics.

Moreover, to the person whose quote I commented on above, I’d agree that the properties of the SU(3) gauge group have indeed accounted for many phenomena associated with the strong interaction, which is why the standard model of particle physics contains 8 gluons and quarks carrying a three-fold colour charge as described by quantum chromodynamics. Leaving aside the fact that QCD is such a terribly difficult theory to work with – in practice it involves nightmarish lattice calculations on a scale to make even the most diehard enthusiast cringe – what I would ask is whether this description in any case sufficient for us to assert that it describes “true nature”? Many physicists will no doubt disagree with me, but I don’t think so. It’s a map, not the territory.

So why am I boring you all with this rambling dissertation? Well, it brings me to my other post – about Stephen Hawking’s comments about God. I don’t want to go over that issue again – frankly, I was bored with it before I’d finished writing my own blog post – but it does relate to the bee that I often find in my bonnet about the tendency of many modern theoretical physicists to assign the wrong category of existence to their mathematical ideas. The prime example that springs to my mind is the multiverse. I can tolerate certain versions of the multiverse idea, in fact. What I can’t swallow, however is the identification of the possible landscape of string theory vacua – essentially a huge set of possible solutions of a complicated set of mathematical equations – with a realised set of “parallel universes”. That particular ontological step just seems absurd to me.

I’m just about done, but one more thing I’d say to finish with is concerns the (admittedly overused) metaphor of maps and territories. Maps are undoubtedly useful in helping us find our way around, but we have to remember that there are always things that aren’t on the map at all. If we rely too heavily on one, we might miss something of great interest that the cartographer didn’t think important. Likewise if we fool ourselves into thinking our descriptions of nature are so complete that they “are” all that nature is, then we might miss the road to a better understanding.

15 Comments »

Alone

Posted in Poetry with tags Edgar Allan Poe, Poetry on September 3, 2010 by telescoper

From childhood’s hour I have not been
As others were; I have not seen
As others saw; I could not bring
My passions from a common spring.
From the same source I have not taken
My sorrow; I could not awaken
My heart to joy at the same tone;
And all I loved, I loved alone.
Then – in my childhood, in the dawn
Of a most stormy life – was drawn
From every depth of good and ill
The mystery which binds me still:
From the torrent, or the fountain,
From the red cliff of the mountain,
From the sun that round me rolled
In its autumn tint of gold,
From the lightning in the sky
As it passed me flying by,
From the thunder and the storm,
And the cloud that took the form
(When the rest of Heaven was blue)
Of a demon in my view.

by Edgar Allan Poe.

2 Comments »

Hawking and the Mind of God

Posted in Books, Talks and Reviews, Science Politics, The Universe and Stuff with tags Physics, Religion, Science, Stephen Hawking, Universe, William Hague on September 2, 2010 by telescoper

I woke up this morning to the news that, according to Stephen Hawking, God did not create the Universe but it was instead an “inevitable consequence of the Law of Physics”. By sheer coincidence this daft pronouncement has come out at the same time as the publication of Professor Hawking’s new book, an extract of which appears in todays Times.

It’s interesting that such a fatuous statement managed to become a lead item on the radio news and a headline in all the national newspapers despite being so obviously devoid of any meaning whatsoever. How can the Universe be “a consequence” of the theories that we invented to describe it? To me that’s just like saying that the Lake District is a consequence of an Ordnance Survey map. And where did the Laws of Physics come from, if not from God?

Stephen Hawking is undoubtedly a very brilliant theoretical physicist. However, something I’ve noticed about theoretical physicists over the years is that if you get them talking on subjects outside physics they are generally likely to say things just as daft as some drunk bloke down the pub. I’m afraid this is a case in point.

Part of me just wants to laugh this story off, but another part is alarmed at what must appear to many to be an example of an arrogant scientist presuming to pass judgement on subjects that are really none of his business. When scientists complain about the lack of enthusiasm shown by sections of the public towards their subject, perhaps they should take seriously the alienating effect that such statements can have. This kind of thing isn’t what I’d call public engagement. Quite the opposite, in fact.

In case anyone is interested, I am not religious but I do think that there are many things that science does not – and probably will never – explain, such as why there is something rather than nothing. I also believe that science and religious belief are not in principle incompatible – although whether there is a conflict in practice does depend of course on the form of religious belief and how it is observed. God and physics are in my view pretty much orthogonal. To put it another way, if I were religious, there’s nothing in theoretical physics that would change make me want to change my mind. However, I’ll leave it to those many physicists who are learned in matters of theology to take up the (metaphorical) cudgels with Professor Hawking.

No doubt this bit of publicity will increase the sales of the new book, so I’ve decided to point out that I have written a book myself on precisely this question, which is available from all good airports bookshops. I’m sure you’ll understand that there isn’t a hint of opportunism in the way I’m drawing this to your attention. If you think this is a cynical attempt to cash in then all I can say is

BUY MY BOOK!

I also noticed that today’s Grauniad is offering a poll on the existence or non-existence of God. I noticed some time ago that there’s a poll facility on WordPress, so this gives me an excuse to try repeating it here. Anything dumb the Guardian can do, I can do dumber. However, owing to funding cuts I’ve decided to do a single poll encompassing several topical news stories at the same time.

45 Comments »

September Song

Posted in Jazz with tags Django Reinhardt, Kurt Weill, September Song on September 1, 2010 by telescoper

Well, the summer’s over and soon we’ll be welcoming the new academic year. I thought I’d mark the occasion of the First of September with a piece of music that celebrates the genius of Django Reinhardt, the great Belgian-born gypsy guitarist who overcame the terrible injuries he suffered as a child (in a fire in his caravan) to become one of the greatest jazz musicians of all time. He had an unparalleled gift for melodic improvisation and a unique style of playing the guitar he invented himself to get around the fact that the third and fourth digits on his left hand were so badly burned he could effectively only use two fingers. Add him to your list of famous Belgians right away!

Here he’s playing the beautifully poignant September Song, by Kurt Weill:

Oh, it’s a long, long while from May to December
But the days grow short when you reach September

1 Comment »

STFC Grants Consultation

Posted in Science Politics with tags astronomy, Particle Physics, research grants, STFC, Universities on August 31, 2010 by telescoper

I thought I’d put my community service badge on today and draw the attention of any astronomers or particle physicists reading this blog that the Science and Technology Facilities Council (STFC) is consulting on proposed changes to the ways it funds research grants. I can hardly over-emphasize the importance of this issue, especially for those of us working in University departments who rely on grant funding in order to carry out our research.

There is a consultation form on which you can post comments on the alternatives outlined in the accompanying document.

Regrettably, only three options are offered. In brief, they are

All grants to be 3-year “standard” grants (i.e. no more “rolling” grants at all)
Some (a small number?) of 6-year “core” grants introduced, mainly to cover the cost of technical support staff.
The status quo (i.e. mixture of 3-year “standard” and 5-year “rolling” grants).

I’m not going to comment on these here, as my intention is just to draw your attention to the fact that this consultation is open and that the deadline is very soon: Monday 6th September 2010, at 4pm. I would have thought it’s probably a good idea for groups to submit collective responses where possible, but I’m sure all feedback would be welcomed.

We don’t know how much of a grant programme will remain after the forthcoming Comprehensive Spending Review, but it’s even more important to make the system as efficient and fair as possible when we know money is going to be tight.

3 Comments »

Dragons and Unicorns

Posted in Education, The Universe and Stuff with tags de Broglie, Double-Slit Experiment, Heisenberg, Niels Bohr, Physics, Quantum Mechanics, Schrodinger, Young's fringes on August 30, 2010 by telescoper

When I was an undergraduate I was often told by lecturers that I should find quantum mechanics very difficult, because it is unlike the classical physics I had learned about up to that point. The difference – or so I was informed – was that classical systems were predictable, but quantum systems were not. For that reason the microscopic world could only be described in terms of probabilities. I was a bit confused by this, because I already knew that many classical systems were predictable in principle, but not really in practice. I blogged about this some time ago, in fact. It was only when I had studied theory for a long time – almost three years – that I realised what was the correct way to be confused about it. In short, quantum probability is a very strange kind of probability that displays many peculiarities and subtleties that one doesn’t see in the kind of systems we normally think of as “random”, such as coin-tossing or roulette wheels.

To illustrate how curious the quantum universe is we have to look no further than the very basic level of quantum theory, as formulated by the founder of wave mechanics, Erwin Schrödinger. Schrödinger was born in 1887 into an affluent Austrian family made rich by a successful oilcloth business run by his father. He was educated at home by a private tutor before going to the University of Vienna where he obtained his doctorate in 1910. During the First World War he served in the artillery, but was posted to an isolated fort where he found lots of time to read about physics. After the end of hostilities he travelled around Europe and started a series of inspired papers on the subject now known as wave mechanics; his first work on this topic appeared in 1926. He succeeded Planck as Professor of Theoretical Physics in Berlin, but left for Oxford when Hitler took control of Germany in 1933. He left Oxford in 1936 to return to Austria but fled when the Nazis seized the country and he ended up in Dublin, at the Institute for Advanced Studies which was created especially for him by the Irish Taoiseach, Eamon de Valera. He remained there happily for 17 years before returning to his native land at the University of Vienna. Sadly, he became ill shortly after arriving there and died in 1961.

Schrödinger was a friendly and informal man who got on extremely well with colleagues and students alike. He was also a bit scruffy even to the extent that he sometimes had trouble getting into major scientific conferences, such as the Solvay conferences which are exclusively arranged for winners of the Nobel Prize. Physicists have never been noted for their sartorial elegance, but Schrödinger must have been an extreme case.

The theory of wave mechanics arose from work published in 1924 by de Broglie who had suggested that every particle has a wave somehow associated with it, and the overall behaviour of a system resulted from some combination of its particle-like and wave-like properties. What Schrödinger did was to write down an equation, involving a Hamiltonian describing particle motion of the form I have discussed before, but written in such a way as to resemble the equation used to describe wave phenomena throughout physics. The resulting mathematical form for a single particle is

$i\hbar\frac{\partial \Psi}{\partial t} = \hat{H}\Psi = -\frac{\hbar^2}{2m}\nabla^2 \Psi + V\Psi$ ,

in which the term $\Psi$ is called the wave-function of the particle. As usual, the Hamiltonian $H$ consists of two parts: one describes the kinetic energy (the first term on the right hand side) and the second its potential energy represented by $V$ . This equation – the Schrödinger equation – is one of the most important in all physics.

At the time Schrödinger was developing his theory of wave mechanics it had a rival, called matrix mechanics, developed by Werner Heisenberg and others. Paul Dirac later proved that wave mechanics and matrix mechanics were mathematically equivalent; these days physicists generally use whichever of these two approaches is most convenient for particular problems.

Schrödinger’s equation is important historically because it brought together lots of bits and pieces of ideas connected with quantum theory into a single coherent descriptive framework. For example, in 1911 Niels Bohr had begun looking at a simple theory for the hydrogen atom which involved a nucleus consisting of a positively charged proton with a negatively charged electron moving around it in a circular orbit. According to standard electromagnetic theory this picture has a flaw in it: the electron is accelerating and consequently should radiate energy. The orbit of the electron should therefore decay rather quickly.

Bohr hypothesized that special states of this system were actually stable; these states were ones in which the orbital angular momentum of the electron was an integer multiple of Planck’s constant. This simple idea endows the hydrogen atom with a discrete set of energy levels which, as Bohr showed in 1913, were consistent with the appearance of sharp lines in the spectrum of light emitted by hydrogen gas when it is excited by, for example, an electrical discharge. The calculated positions of these lines were in good agreement with measurements made by Rydberg so the Bohr theory was in good shape. But where did the quantised angular momentum come from?

The Schrödinger equation describes some form of wave; its solutions $\Psi(\vec{x},t)$ are generally oscillating functions of position and time. If we want it to describe a stable state then we need to have something which does not vary with time, so we proceed by setting the left-hand-side of the equation to zero. The hydrogen atom is a bit like a solar system with only one planet going around a star so we have circular symmetry which simplifies things a lot. The solutions we get are waves, and the mathematical task is to find waves that fit along a circular orbit just like standing waves on a circular string. Immediately we see why the solution must be quantized. To exist on a circle the wave can’t just have any wavelength; it has to fit into the circumference of the circle in such a way that it winds up at the same value after a round trip. In Schrödinger’s theory the quantisation of orbits is not just an ad hoc assumption, it emerges naturally from the wave-like nature of the solutions to his equation.

The Schrödinger equation can be applied successfully to systems which are much more complicated than the hydrogen atom, such as complex atoms with many electrons orbiting the nucleus and interacting with each other. In this context, this description is the basis of most work in theoretical chemistry. But it also poses very deep conceptual challenges, chiefly about how the notion of a “particle” relates to the “wave” that somehow accompanies it.

To illustrate the riddle, consider a very simple experiment where particles of some type (say electrons, but it doesn’t really matter; similar experiments can be done with photons or other particles) emerge from the source on the left, pass through the slits in the middle and are detected in the screen at the right.

In a purely “particle” description we would think of the electrons as little billiard balls being fired from the source. Each one then travels along a well-defined path, somehow interacts with the screen and ends up in some position on the detector. On the other hand, in a “wave” description we would imagine a wave front emerging from the source, being diffracted by the screen and ending up as some kind of interference pattern at the detector. This is what we see with light, for example, in the phenomenon known as Young’s fringes.

In quantum theory we have to think of the system as being in some sense both a wave and a particle. This is forced on us by the fact that we actually observe a pattern of “fringes” at the detector, indicating wave-like interference, but we also can detect the arrival of individual electrons as little dots. Somehow the propensity of electrons to arrive in positions on the screen is controlled by an element of waviness, but they manage to retain some aspect of their particleness. Moreover, one can turn the source intensity down to a level where there is only every one electron in the experiment at any time. One sees the dots arrive one by one on the detector, but adding them up over a long time still yields a pattern of fringes.

Curiouser and curiouser, said Alice.

Eventually the community of physicists settled on a party line that most still stick to: that the wave-function controls the probability of finding an electron at some position when a measurement is made. In fact the mathematical description of wave phenomena favoured by physicists involves complex numbers, so at each point in space at time $\Psi$ is a complex number of the form $\Psi= a+ib$ , where $i =\sqrt{-1}$ ; the corresponding probability is given by $|\Psi^2|=a^2+b^2$ . This protocol, however, forbids one to say anything about the state of the particle before it measured. It is delocalized, not being definitely located anywhere, but only possessing a probability to be any particular place within the apparatus. One can’t even say which of the two slits it passes through. Somehow, it manages to pass through both slits. Or at least some of its wave-function does.

I’m not going to into the various philosophical arguments about the interpretation of quantum probabilities here, but I will pass on an analogy that helped me come to grips with the idea that an electron can behave in some respects like a wave and in others like a particle. At first thought, this seems a troubling paradox but it only appears so if you insist that our theoretical ideas are literal representations of what happens in reality. I think it’s much more sensible to treat the mathematics as a kind of map or sketch that is useful for us to do find our way around nature rather than confusing it with nature itself. Neither particles nor waves really exist in the quantum world – they’re just abstractions we use to try to describe as much as we can of what is going on. The fact that it doesn’t work perfectly shouldn’t surprise us, as there are are undoubtedly more things in Heaven and Earth than are dreamt of in our philosophy.

Imagine a mediaeval traveller, the first from your town to go to Africa. On his journeys he sees a rhinoceros, a bizarre creature that is unlike anything he’s ever seen before. Later on, when he gets back, he tries to describe the animal to those at home who haven’t seen it. He thinks very hard. Well, he says, it’s got a long horn on its head, like a unicorn, and it’s got thick leathery skin, like a dragon. Neither dragons nor unicorns exist in nature, but they’re abstractions that are quite useful in conveying something about what a rhinoceros is like.

It’s the same with electrons. Except they don’t have horns and leathery skin. Obviously.

19 Comments »

The Character of a Happy Life

Posted in Poetry with tags Henry Wotton, Poetry on August 29, 2010 by telescoper

How happy is he born or taught
That serveth not another’s will,
Whose armour is his honest thought,
And simple truth his highest skill;

Whose passions not his masters are;
Whose soul is still prepared for death,
Untied unto the world with care
Of princes’ grace or vulgar breath;

Who envies none whom chance doth raise,
Or vice; who never understood
The deepest wounds are given by praise,
By rule of state but not of good;

Who hath his life from rumours freed,
Whose conscience is his strong retreat,
Whose state can neither flatterers feed
Nor ruins make accusers great;

Who God doth late and early pray
More of his grace than goods to send,
And entertains the harmless day
With a well-chosen book or friend.

This man is free from servile bands
Of hope to rise or fear to fall,
Lord of himself, though not of lands,
And having nothing, yet hath all.

by Sir Henry Wotton (1568-1639).

Political Correlation

Posted in Bad Statistics, Politics with tags Paul Whiteley, statistics, STEM, Times Higher, Universities on August 28, 2010 by telescoper

I was just thinking that it’s been a while since I posted anything in my bad statistics category when a particularly egregious example jumped up out of this week’s Times Higher and slapped me in the face. This one goes wrong before it even gets to the statistical analysis, so I’ll only give it short shrift here, but it serves to remind us all how feeble is many academic’s grasp of the scientific method, and particularly the role of statistics within it. The perpetrator in this case is Paul Whiteley, who is Professor of Politics at the University of Essex. I’m tempted to suggest he should go and stand in the corner wearing a dunce’s cap.

Professor Whiteley argues that he has found evidence that refutes the case that increased provision of science, technology, engineering and maths (STEM) graduates are -in the words of Lord Mandelson – “crucial to in securing future prosperity”. His evidence is based on data relating to 30 OECD countries: on the one hand, their average economic growth for the period 2000-8 and, on the other, the percentage of graduates in STEM subjects for each country over the same period. He finds no statistically significant correlation between these variates. The data are plotted here:

This lack of correlation is asserted to be evidence that STEM graduates are not necessary for economic growth, but in an additional comment (for which no supporting numbers are given), it is stated that growth correlates with the total number of graduates in all subjects in each country. Hence the conclusion that higher education is good, whether or not it’s in STEM areas.

So what’s wrong with this analysis? A number of things, in fact, but I’ll start with what seems to me the most important conceptual one. In order to test a hypothesis, you have to look for a measurable effect that would be expected if the hypothesis were true, measure the effect, and then decide whether the effect is there or not. If it isn’t, you have falsified the hypothesis.

Now, would anyone really expect the % of students graduating in STEM subjects to correlate with the growth rate in the economy over the same period? Does anyone really think that newly qualified STEM graduates have an immediate impact on economic growth? I’m sure even the most dedicated pro-science lobbyist would answer “no” to that question. Even the quote from Lord Mandelson included the crucial word “future”! Investment in these areas is expected to have a long-term benefit that would probably only show after many years. I would have been amazed had there been a correlation between measures relating to such a short period, so absence of one says nothing whatsoever about the economic benefits of education in STEM areas.

And another thing. Why is the “percentage of graduates” chosen as a variate for this study? Surely a large % of STEM graduates is irrelevant if the total number is very small? I would have thought the fraction of the population with a STEM degree might be a better choice. Better still, since it is claimed that the overall number of graduates correlates with economic growth, why not show how this correlation with the total number of graduates breaks down by subject area?

I’m a bit suspicious about the reliability of the data too. Which country is it that produces less than 3% of its graduates in science subjects (the point at the bottom left of the plot). Surely different countries also have different types of economy wherein the role of science and technology varies considerably. It’s tempting, in fact, to see two parallel lines in the above graph – I’m not the only one to have noticed this – which may either be an artefact of small numbers chosen or might indicate that some other parameter is playing a role.

This poorly framed hypothesis test, dubious choice of variables, and highly questionable conclusions strongly suggest that Professor Whiteley had made his mind up what result he wanted and simply dressed it up in a bit of flimsy statistics. Unfortunately, such pseudoscientific flummery is all that’s needed to convince a great many out there in the big wide world, especially journalists. It’s a pity that this shoddy piece of statistical gibberish was given such prominence in the Times Higher, supported by a predictably vacuous editorial, especially when the same issue features an article about the declining standards of science journalism. Perhaps we need more STEM graduates to teach the others how to do statistical tests properly.

However, before everyone accuses me of being blind to the benefits of anything other than STEM subjects, I’ll just make it clear that, while I do think that science is very important for a large number of reasons, I do accept that higher education generally is a good thing in itself , regardless of whether it’s in physics or mediaeval latin, though I’m not sure about certain other subjects. Universities should not be judged solely by the effect they may or may not have on short-term economic growth.

Which brings me to a final point about the difference between correlation and causation. People with more disposal income probably spend more money on, e.g., books than people with less money. Buying books doesn’t make you rich, at least not in the short-term, but it’s a good thing to do for its own sake. We shouldn’t think of higher education exclusively on the cost side of the economic equation, as politicians and bureaucrats seem increasingly to be doing, it’s also one of the benefits.

9 Comments »