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Universities Challenged

Posted in Education, Politics with tags , , , , , on June 10, 2010 by telescoper

The news headlines over the last couple of days have been dominated by remarks made by David Willetts, Minister for Universities and Science, who has called for a radical overhaul of the way UK universities are organized and funded. Predictably, his comments set alarm bells ringing about the savage cuts likely to be coming our way, but I hope it’s not just about slash-and-burn and that some imagination is applied to the problem of sorting out the mess the system has become. We’ll see.

According to a piece in the Guardian, for example, Willetts suggested that some students could study at smaller local colleges instead of going to a big university, but these colleges would teach courses designed and administered by the larger “elite” institutions, such as the University of London. This suggestion isn’t  exactly new because it’s actually how things used to work many years ago. In fact, Nottingham University, where I used to work used to be Nottingham University College and its degrees, along with those of a number of similar provincial universities, were University of London degrees. Nottingham University only got the power to award its own degrees in 1948. Of course, there wasn’t really such a thing as distance learning in those days, so there’s a possibility that a 21st Century revival of this basic idea could turn out very differently in terms of how things are actually taught.

On the up side of this suggestion is the fact that it would be a lot easier to maintain standards, if examinations were set by a common body. On the down side is the fact that the distinctive flavour of speciality courses taught in different colleges, which is a strength of research-led teaching, would be lost. In between these positives and negatives there is a huge grey area of questions, such as where the funding would go, precisely which universities should administer the changes and so on. A lot of thinking and planning will be  needed before anything like this could be implemented.

Let me add two more specific comments to this. First, I think Willetts’ suggestion would make a lot of sense here in Wales where it could be easily implemented by returning to the old University of Wales.  As I’ve mentioned before, as well as suffering from many of the problems besetting the English university system, the Principality has a few extra ones all its own. Among the most pressing is the proliferation of small colleges and the consequent duplication of administrative systems. I think a great deal of money could be saved and teaching quality improved by cutting out the unnecessary bureaucracy and having the smaller places administered by a larger central University (as Willetts imagined with the University of London).

My other comment is specific to my own subject, physics (and astronomy). The problem with this – and other laboratory based STEM subjects – is that it’s very difficult to imagine how they can actually be taught at all at degree level without access to research laboratories for, e.g., project work. This is why physics is only taught in 40-0dd of the 131 universities and colleges around the UK. You can call me old-fashioned, but I just don’t think it’s either possible or desirable to separate teaching from research in science subjects in the way this plan seems to suggest. I know some colleagues of mine disagree strongly with this, but there you go.

Behind this proposal is the issue of student funding, as it is at least partly motivated by the suggestion that students could stay at home and study at a local college instead of moving to a university further away, which would necessitate them taking out student loans which the Treasury has to pay out. 

There’s also the issue of fees. At the moment students in England are expected to pay a flat-rate annual fee of £3225. In addition to this the government pays to the University concerned an amount called the “Unit of Resource”. Last year, in England, the basic amount was around £4K but there is multiplier for more expensive courses. Clinical medicine, for example, attracts four times the basic rate. Subjects like physics and chemistry get a multiplier of 1.7 (so each student comes with around £6.7K of funding). Subjects with no laboratory component, i.e. most Arts and Humanities courses,  just get the bog-standard amount.

I think there’s an obvious problem with this system, namely that physics (and other science subjects) are  much more expensive to teach than the formula allows for. The total income per student for an arts subject would be about £7.2K, while that for physics is about £10K. Why bother with all that expensive laboratory space and shiny new kit when the funding differential is so small. That’s another reason why so many universities have scrapped their physics departments in favour of cheaper disciplines that generate a profit much more easily.

Coincidentally I attended a lunch yesterday with some of our soon-to-be-graduating students. I’ve been a member of a committee working on updating our Physics courses and we wanted to discuss the proposed changes with them. One of the group was a mature student who had already done an English degree (at another university). She said that a physics drgee was much harder work, but was impressed at how much more contact she had with staff. Like most physics department, virtually all our teaching is done by permanent academic staff. Students doing  Physics at Cardiff get about three times as many contact hours with staff as students doing English. It’s unfair to compare apples with oranges, but I’m convinced the funding model is stacked against STEM subjects.

The awful financial climate we’re in has led to a general sense of resignation that the government contribution to university education (the Unit of Resource) is going to decrease and the student contribution go up to compensate. However, there’s a Catch-22 here for the Treasury. If the tuition fee goes up students will have to borrow more, and the Treasury doesn’t want to take on more  subsidised student loans. It seems much more likely to me that the cuts will be achieved by simply reducing the number of funded places. However, in the light of what I argued above, I think this is a great opportunity to think about what is the correct Unit of Resource for different subjects. If we all agree the country needs more scientists and engineers, not less, I’d argue that funded places elsewhere should be cut, and that the difference between arts and science units of resource also be substantially increased.

I’d even go so far as to suggest that there should be zero-rated courses, i.e. those which students are welcome to take if they pay the full cost but to which the government will not contribute at all. That should put an end to the Mickey Mouse end of Higher Education provision once and for all.

PS. A review of the tuition fee system is currently taking place but isn’t due to report until the autumn. It is led by Lord Browne who was formerly the boss of BP. I wonder if there’ll be any leaks?

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Pecha Kucha

Posted in Education with tags , , , , , on June 8, 2010 by telescoper

A few months ago I was invited to take part in an evening of Pecha Kucha in a hotel in Geneva. I’ll try anything once, so I agreed. I have to admit, though, that I wasn’t actually very good at it. Neither were any of the other scientists present.

No idea what a Pecha Kucha is? Well then you’re probably not an architect or an artist or a designer. Then again, you’re reading this blog so that’s pretty much a given anyway. Pecha Kucha is a style of presentation at which arty types display their portfolios in a strictly disciplined format. The standard form is twenty slides with twenty seconds allowed for each one, i.e. a total time of 6 minutes and 40 seconds. The timing is ruthlessly regulated.

Those of us scientists used to taking at least a few  minutes per slide find this format very challenging, but then that’s because we tend to have text and equations on our slides and they take some explaining. Designers and the like tend to just show pictures, and these should – if they’re any good – be pretty self-explanatory. I guess this is why the Pecha Kucha format is de rigeur in such disciplines while it has yet to catch on in physics.

I only just survived my initiation into the strange world of Pecha Kucha. Before being told what it was I thought it was a mountain in the Andes. I was reminded about it this morning by a tweet from John Butterworth (a particle physicist who, incidentally, has a nice blog of his own) confessing similar trepidation to what I experienced before I lost my Pecha Kucha virginity. The first time can be disappointing, but I hope he survived his inauguration.

Looking back on it though I think this might be an interesting idea to try in a physics context. We’re trying increasingly hard these to teach our physics and astronomy students transferable skills, but when it comes to presentations we’re fixated by the traditional presentation format. Why not get undergraduate students to do a Pecha Kucha about their project, instead of a 20-minute lecture? Why not include a Pecha Kucha in the PhD viva?

The more I think about it, the more attractive the idea seems. Has anyone out there tried a physics Pecha Kucha?

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Experiments and Observations

Posted in Science Politics, The Universe and Stuff with tags , , , , , on May 8, 2010 by telescoper

It’s nice to be able to pass on some upbeat news for once.

The first thing is that, after a lot of delays and a bit of haggling, the School of Physics & Astronomy at Cardiff University has finally issued advertisements for a bunch of new Faculty positions in Experimental Physics. The positions, which are tenured,  involve both Chair and Lecturer/Reader levels and there are several positions available. The School and University  have  put together a handsome start-up package for a new group and there’s plenty of spanking new experimental laboratory space to set up shop. Coupled with the fact that Cardiff is a great city to live in, with low costs and great sporting and cultural infrastructure, this should prove a tempting opportunity for someone to set up their own group.

It’s also a welcome vote of confidence from Cardiff University which, despite cuts in its overall budget, has decided to invest heavily in the School’s strategic plan. I hope and believe we’ll attract a strong field for these appointments and look forward to seeing what develops. We need a shot in the arm and this might just deliver it.

What’s particularly interesting about this clutch of new appointments is that they are open to people working in any area of physics, with the exception of astrophysics. Given the massive cuts in STFC’s budget, this is no time to be expanding in areas covered by its remit. I say that as an astrophysicist, with considerable regret but pragmatism in the face of the changing landscape of British science funding. In times of risk you have to broaden your portfolio. However, that’s not to say that astrophysics at Cardiff is downbeat. Far from it, in fact.

ESA held an international press conference to present exciting new results from the Herschel Observatory at the European Space Research and Technology Centre, Noordwijk, The Netherlands, on Thursday 6 May. A webcast of the press conference with Cardiff’s Professors Matt Griffin and Steve Eales taking part, can be seen at from http://www.esa.int/SPECIALS/Herschel. At the conference Steve Eales talked about the latest results from the Herschel ATLAS survey: an ATLAS of the Universe. ATLAS will cover one eightieth of the sky, four times larger than all the other Herschel surveys combined and is led by Professor Eales and Dr Loretta Dunne at Nottingham University.

Herschel ATLAS has measured the infrared light from thousands of galaxies, spread across billions of light-years. Each galaxy appears as just a pinprick but its brightness allows astronomers to determine how quickly it is forming stars. Roughly speaking, the brighter the galaxy the more stars it is forming. The Herschel images show that in the past there were many more galaxies forming stars much faster than our own Galaxy. But what triggered this frantic activity is not completely understood. Steve Eales said

every time astronomers have observed the universe in a new waveband, they have discovered something new. So as well as our regular science programmes, I am hoping for the unexpected.

I am hoping to get involved with the ATLAS data myself at some point as I am formally a member of the consortium, but I’ve been too busy doing other things to get involved in these initial stages so am not on any of the preliminary science papers. I hope I can get properly involved in this project sooner rather than later…

The ATLAS survey, image courtesy of ESA and the ATLAS consortium

The full press release also includes surprises on how stars are formed including work carried out by Cardiff’s Professor Derek Ward-Thompson. Herschel’s star formation surveys are beginning to reveal the mysteries behind how massive stars are created.

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Lecture Notes

Posted in Education with tags , , , on April 25, 2010 by telescoper

One week to go before the end of teaching term, and it’s time for the dreaded questionnaires to be handed out for the purpose of gauging student feedback on our teaching. The responses from the students go off somewhere to be counted and I’ll get a summary back in due course and learn what the students made of the  series of chaotic and rambling performances I strung together to masquerade as lecture courses. At the end of the year we usually get to see a league table of who’s popular and who isn’t, but the scores aren’t very useful beyond that. More important than the tick boxes are the comments that students write about what’s good and what isn’t. I read through all those and they’re often very helpful in suggesting things to be done differently in subsequent years.

Lecturing has changed an enormous amount since I was at university almost thirty years ago. In those days we got very little in the way of printed notes and we were expected to write everything down in classes that were primarily delivered in the chalk-and-talk style, although some lecturers used overhead projectors. The disadvantage of the latter over the former was a tendency to go too quickly through the material.

As a student I just accepted this was the way things were and developed my own note-taking strategy. I trained myself to be able to write things down about as fast as the lecturer could speak. I did this by cutting out the biggest hindrance to taking notes quickly, which is the business of  making your eyes go backwards and forwards between the blackboard (or projection screen) and paper in front of you. I just wrote everything I could on the paper without looking at it. Although my handwriting was scrappy when I did this, I could keep track of just about everything that was said as well as what was written by the lecturer. Later on, I’d turn these notes into a neat copy and in the process of doing that I tried to iron out any bugs in the original notes as well as figure out things I couldn’t make sense of.

When I started lecturing I primarily used blackboards and chalk. I was teaching quite mathematical things and found this the best way to do it. For one thing the physical effort of writing made me go through the material at a reasonable pace. The other advantage is that I think mathematical proofs and derivations should not just be presented, but should happen as a process for the students to see. I always felt that a lecture would be more interesting if it appeared to be spontaneous rather than delivered from a pre-prepared script. Even if the students disagreed, I certainly enjoyed lecturing much more if there was an element of improvisation in the performance.

However, I soon noticed that many students didn’t really know how to take notes even at the modest speed I was going. They would generally only write down what I wrote on the board, not the little verbal explanations and embellishments I put in. My response to this observation was to make sure I wrote down more and consequently went through the material even more slowly. When I got to sit in as a peer reviewer of other staff lecturers, I looked at what the students around me were doing and realised that the vast majority simply didn’t know how to take notes efficiently or accurately. For many the act of writing things down took so much effort that they weren’t listening to the lecturer. I guess this stems from the changing style of teaching in schools, but even if that is true it is something that university teachers need to come to terms with.

Incidentally, I have from time to time given final-year undergraduate lectures at Italian universities (in English). When I used the same style there as at home – writing full notes on the board rather than just the equations – the students asked me why I was doing it. They all expected to have to write down what I was saying. If they could manage to do that with lectures in their second language, I don’t really see why our students can’t do it in their mother tongue!

Gradually the ubiquitous powerpoint has largely the old-fashioned style of lecturing to the extent that many lecture theatres don’t even have a blackboard. We’re generally expected to hand out complete sets of printed notes, with the result that the students don’t have to take notes of their own but also turning a lecture into an entirely passive experience.

I resisted the move to powerpoint for undergraduate lecturing for many years, but gave up and went with the flow when I moved to Cardiff.  However, what I do is a bit different from the others who teach this way. I generally use slides which have only a few bits of text, key equations and figures on them. I hand out copies of these slides at the start of each lecture and then go through them during the class, and also make the powerpoint files available on the web. This gives them all the important things, but I tell the students I expect students to annotate the handouts and make their own set of notes based on the skeleton I’ve handed out. However, it is clear that many students don’t write anything down at all during the lecture. We’ll see from the forthcoming exams how much they have actually learned.

Newer educational technology should enable us to improve the standards of teaching in universities, but I think there’s still a long way to go before we work out how to use it effectively.  In particular I think we need to question whether lectures in the old-fashioned sense should continue to provide the primary mode of teaching. My personal opinion is that we should be moving to more independent, problem-based, learning and much less of the passive spoon-feeding.  I think we should be aiming to cut the number of lectures we give by about 50% across the school and use the time and effort saved in more creative and effective ways.

We’re in the middle of a review of our course structure in the School of Physics & Astronomy at Cardiff University and I hope we take the opportunity to make radical changes not just to the curriculum but also to the way we present it. Not everyone in the School is keen on really radical changes. I think I understand why. I actually enjoy lecturing. I always have. It’s fun and it’s also a lot easier to give a lecture than to prepare large numbers of problems and write pages and pages of printed notes. Looking back at my time as a student, though, I am bound to admit that I learnt next to nothing from lectures. This was partly because many of the lecturers I had were poorly delivered but also partly because I’m not sure lectures are the best way to teach physics. We carry on doing it this way just because it’s what we’re used to.

Perhaps the biggest problem with the way we teach physics these days is that it encourages students to think of each module as a bite-sized piece that can be retained until the examinations, regurgitated, and then forgotten.  I’ve no doubt that memorizing notes  is how many students pass the examinations we set.  Little genuine understanding or problem-solving ability is needed. We promote physics as a subject that nurtures these skills, but I don’t think many physics graduates – even those with good degrees – actually possess them at the end. We should be making much more of an effort in teaching students how to use their brains in other ways than as memory devices.

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(Guest Post) The Emperor’s New Math

Posted in The Universe and Stuff with tags , on April 20, 2010 by telescoper

Time for another guest post from my old chum Anton, this time on the topic of mathematics. I’m not sure any mathematicians reading this piece will be too happy, but if that applies to you then blame him not me. As usual, comments are welcome through the proper channel at the bottom of the page..

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Nowadays a page of mathematics looks to a physicist or engineer like gobbledygook. This was not always so: a century ago a physicist might hope to understand everything in journals of mathematics, and even contribute to them. Fifty years ago a physicist might not be able to understand everything written there, but the mathematics would appear comprehensible in principle. A qualitative change has since taken place.

This change has coincided, roughly, with acceptance of the distinction between ‘pure’ and ‘applied’ (impure??) mathematics, and with the consequent, deliberate, emancipation of ‘pure’ mathematics. This is a new departure: for centuries mathematics evolved side by side with physics, and the mathematics that was studied was the mathematics used in tackling physical problems. Galileo had said (in his work Il Saggiatore, The Assayer) that

…the universe… cannot be understood unless one first learns to comprehend the language… in which it is written. It is written in the language of mathematics.

So the change is recent, and it is huge. I suggest that it is a change for the worse; that in divorcing themselves from physical science pure mathematicians have cut off their air supply; and that the suffocating style of modern pure mathematics is a result. Mathematics was not born in a vacuum, and it will not ultimately flourish in one.

A pure mathematician might respond that I would say that, since I am a physicist. But perhaps an outsider is needed to see the problem; insiders generally adopt the party line. The justification for my stance is this. Mathematicians acknowledge that their subject is the formal study of patterns. And mathematicians think in patterns, not formulae – which are really a highly efficient way to express their thoughts. Crucially, the patterns arising in the natural world are far richer and more diverse than the patterns that even the best pure mathematicians can pull out of their heads by introspection. Even number theory is not an exception, for the positive integers are abstractions – ideals – of the physical realisations of one, two, three etc sheep in a field, or boats on a lake.

The role of pattern explains the “unreasonable effectiveness” of mathematics in physical science (as Eugene Wigner put it), since physics is concerned with relations – correlations – between variables in space and time, and correlation is synonymous with pattern. The theoretical physicist Lev Landau vehemently believed that the best mathematics is the mathematics used in physics. An opposing point of view was taken by the pure mathematician Paul Halmos, in an essay titled Applied Mathematics is Bad Mathematics. Not all mathematicians share Halmos’ view, however. The mathematician Morris Kline was the author of many books about mathematics and its embedding in the cultures which nurtured it. In his book Mathematics: The Loss of Certainty, Kline demonstrated that the history of mathematics in the 20th century has not been the smooth progression that it appears to the outsider; and that arguments about the foundations have led not to resolution, but to schism into differing schools – based on different foundations – that do not talk to each other. Mathematics is not in fact a one-way road running from self-evident axioms to consequences, but is open at both top and bottom.

Already in the 19th century a formal style was developing in the mathematical study of logic, and such distinguished noses as Henri Poincaré (in Science et Methodes, part II) protested as early as 1909 that this tended to hide misleading or negligible content. To no avail: the dominance of the formalistic logical viewpoint led to the adoption of its house style across the whole of mathematics. Below university level, mathematics is still taught today as it used to be, with the emphasis on the understanding of ideas rather than their formal presentation. Freshmen are often shocked when they first meet the new way of doing things, in university lectures given by professional mathematicians. I doubt that the form of modern mathematical writing is governed by its content, for whenever my research has demanded I read some contemporary mathematics, and I have had to translate a piece of modern mathematical writing into something comprehensible to scientists, I have found it difficult to distinguish substantial points from trivia. When, for instance, four axioms are needed to establish a result, they will typically be presented as having equal weight, even if one is the crucial axiom that allows most of the proof to be constructed, and another is used only in closing loopholes. Acknowledging the quality of axioms, as well as the quantity, does not compromise rigour.

When I think of the work of Andrew Wiles and Grigori Perelman, I realise that magnificent work is done today by mathematicians far beyond my own competence. But might mathematicians question whether what they regard as the only way to write mathematics is actually a convention, and not necessarily a good one? If they wrote mathematics as they did fifty years ago, others might be able to see for themselves. More fundamentally, might they also realise what their predecessors understood, that by its abstraction mathematics is given an autonomy of its own, and that to look to the physical world for inspiration is not to make mathematics a slave of physics? The present divorce between mathematics and physics impoverishes everyone.

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Space without Physics…

Posted in Uncategorized with tags , , , , , on March 24, 2010 by telescoper

I’m indebted to a colleague (Annabel Cartwright) for sending me this (coincidentally topical) sample question, illustrating the quality of a modern British school science examination.

Since it’s now clear  that there is no room for science in the new era of the UK Space Agency, I suppose we should get used to the removal of science from other things too. Starting with science exams.

This question is taken from a GCSE Physics examination.

Some people think that governments spend too much money on space research.

Which ONE of the following statements is true?

  1. Science can tell us what the planets are made of, and whether they ought to be explored.
  2. Science can tell us what the planets are made of, but not whether they ought to be explored.
  3. Science cannot tell us what the planets are made of but can tell us whether they ought to be explored.
  4. Science cannot tell us what the planets are made of, nor whether they ought to be explored.

Apparently one (and only one) answer is correct. Any offers?

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Scientists in Residence

Posted in Biographical, The Universe and Stuff with tags , , , on February 23, 2010 by telescoper

I’ve managed to get through the hectic  first couple of days of what promises to be a very hectic week without feeling too much of the strain, which is quite a pleasant surprise given my advancing senility.

This week a whole bunch of Cardiff astronomers are taking part in a Scientists in Residence scheme at Monkton Combe School which nestles in among the lovely hills in the picturesque countryside near Bath. The idea was to try to give the pupils some sort of idea what it’s like being a scientist – specifically an astronomer – by having an intensive series of teaching sessions run by scientists who visit the school for several days running.  A whole range of different types have taken part, from graduate students and postdoctoral researchers all the way down to Professors. Some, in fact, have been staying overnight there too.; it’s a boarding school, in fact.

As with most things these days, I’ve been a bit of a freeloader in this thing – the course materials were prepared by others, principally Chris North, so all I had to do was turn up and lend a hand on the day. Members of the department with duties at Cardiff have only been able to go for part of the time and even that has meant, for me at least, a bit of dashing backwards and forwards on the train. On Monday I had a full complement of meetings, lectures and exercise classes in Cardiff before heading off to Bath to give an evening lecture on The Big Bang to what turned out to be quite a large and attentive audience of sixth-form students. When I finished I had to get the train back to Cardiff – about 70 minute journey – in order to be able to give Columbo his evening insulin fix in good time.

This morning I was up at six to get the train again to Bath – after doing the necessary with Columbo again – in order to take part in a classroom session where we took the students through activities centred around the idea of using the orbital motions of astronomical objects to work out masses. I found this very interesting. On the one hand the students were keen and very easy to interact with, but on the other this experience reinforced the impression that today’s A-level physics students are given a syllabus that is diluted beyond all recognition compared with what older generations of physicists learned. Even in a private school, with excellent laboratoty facilities and highly motivated teachers, it is difficult for todays 16-18 year olds to learn anything meaningful about what physics is really like.

Not having kids of my own, I’ve only observed the changes in educational standards over the last decade indirectly, so this couple of days was a bit of a reality check for me. Unless someone can be persuaded to force schools to teach science properly again, university lecturers will have to carry on doing what is essentially remedial teaching.

Anyway, I’ve found the last couple of days very interesting and I hope the others taking part in the week will enjoy it as much as I did.

You might reasonably ask why a bunch of University academics – mainly funded by the taxpayer – should be running backwards and forwards organizing activities for a posh private school? The mercenary answer is, of course, that some of the kids we’ve been talking to might actually turn into Cardiff undergraduates one day and even if only one does so, the income that generates for the School of Physics & Astronomy more than pays for the number of person-hours we have put in. But even if that doesn’t happen it’s still worth it. Our plan is to offer this type of activity to all kinds of schools in  local areas, not only for our own recruitment, but also for the general purpose of “outreach”, communicating an interest in science in the society beyond academia. This week is the first time we’ve done it. Undoubtedly some things will work and others won’t. This week we will iron out some of the problems before we take it on the road to more challenging audiences.

It will need to be a good show if it is to go down well in the Valley Comprehensives, and what better way to improve it than to practice on the rich kids?

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Killing Vectors

Posted in The Universe and Stuff with tags , , , on February 16, 2010 by telescoper

I’ve been feeling a rant coming for some time now. Since I started teaching again three weeks ago, actually. The target of my vitriol this time is the teaching of Euclidean vectors. Not vectors themselves, of course. I like vectors. They’re great. The trouble is the way we’re forced to write them these days when we use them in introductory level physics classes.

You see, when I was a lad, I was taught to write a geometric vector in the folowing fashion:

\underline{r} =\left(\begin{array}{c} x \\ y \\ z \end{array} \right).

This is a simple column vector, where x,y,z are the components in a three-dimensional cartesian coordinate system. Other kinds of vector, such as those representing states in quantum mechanics, or anywhere else where linear algebra is used, can easily be represented in a similar fashion.

This notation is great because it’s very easy to calculate the scalar (dot) and vector (cross) products of two such objects by writing them in column form next to each other and performing a simple bit of manipulation. For example, the scalar product of the two vectors

\underline{u}=\left(\begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right) and \underline{v}=\left(\begin{array}{c} 1\\ 1 \\ -2 \end{array} \right)

can easily be found by multiplying the corresponding elements of each together and totting them up:

\underline{u}\cdot \underline{v} = (1 \times 1) + (1\times 1) + (1\times -2) =0,

showing immediately that these two vectors are orthogonal. In normalised form, these two particular vectors  appear in other contexts in physics, where they have a more abstract interpretation than simple geometry, such as in the representation of the gluon in particle physics.

Moreover, writing vectors like this makes it a lot easier to transform them via the action of a matrix, by multipying rows in the usual fashion, e.g.

\left(\begin{array}{ccc} \cos \theta & \sin\theta & 0 \\ -\sin\theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array} \right) \left(\begin{array}{c} x \\ y \\ z \end{array} \right) = \left(\begin{array}{c} x\cos \theta + y\sin\theta \\ -x \sin \theta + y\cos \theta \\ z \end{array} \right)

which corresponds to a rotation of the vector in the x-y plane. Transposing a column vector into a row vector is easy too.

Well, that’s how I was taught to do it.

However, somebody, sometime, decided that, in Britain at least, this concise and computationally helpful notation had to be jettisoned and students instead must be forced to write

\underline{r} = x \underline{\hat{i}} + y \underline{\hat{j}} + z \underline{\hat{k}}

Some of you may even be used to doing it that way yourself. Why is this awful? For a start, it’s incredibly clumsy. It is less intuitive, doesn’t lend itself to easy operations on the vectors like I described above, doesn’t translate easily into the more general case of a matrix, and is generally just …well… awful.

Worse still, for the purpose of teaching inexperienced students physics, it offers the possibility of horrible notational confusion. In particular, the unit vector \underline{\hat{i}} is too easily confused with i, the square root of minus one. Introduce a plane wave with a wavevector \underline{k} and it gets even worse, especially when you want to write \exp(i\underline{k}\cdot\underline{x})!

No, give me the row and column notation any day.

I would really like to know is who decided that our schools had to teach the horrible notation, rather than the nice one, and why? I think everyone who teaches physics knows that a clear and user-friendly notation is an enormous help and a bad one is an enormous hindrance.  It doesn’t surprise me that some student struggle with even simple mathematics when its presented in such a silly way. On those grounds, I refuse to play ball, and always use the better notation.

Call me old-fashioned.

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A Little Bit of Quantum

Posted in The Universe and Stuff with tags , , , , , , , , , , , on January 16, 2010 by telescoper

I’m trying to avoid getting too depressed by writing about the ongoing funding crisis for physics in the United Kingdom, so by way of a distraction I thought I’d post something about physics itself rather than the way it is being torn apart by short-sighted bureaucrats. A number of Cardiff physics students are currently looking forward (?) to their Quantum Mechanics examinations next week, so I thought I’d try to remind them of what fascinating subject it really is…

The development of the kinetic theory of gases in the latter part of the 19th Century represented the culmination of a mechanistic approach to Natural Philosophy that had begun with Isaac Newton two centuries earlier. So successful had this programme been by the turn of the 20th century that it was a fairly common view among scientists of the time that there was virtually nothing important left to be “discovered” in the realm of natural philosophy. All that remained were a few bits and pieces to be tidied up, but nothing could possibly shake the foundations of Newtonian mechanics.

But shake they certainly did. In 1905 the young Albert Einstein – surely the greatest physicist of the 20th century, if not of all time – single-handedly overthrew the underlying basis of Newton’s world with the introduction of his special theory of relativity. Although it took some time before this theory was tested experimentally and gained widespread acceptance, it blew an enormous hole in the mechanistic conception of the Universe by drastically changing the conceptual underpinning of Newtonian physics. Out were the “commonsense” notions of absolute space and absolute time, and in was a more complex “space-time” whose measurable aspects depended on the frame of reference of the observer.

Relativity, however, was only half the story. Another, perhaps even more radical shake-up was also in train at the same time. Although Einstein played an important role in this advance too, it led to a theory he was never comfortable with: quantum mechanics. A hundred years on, the full implications of this view of nature are still far from understood, so maybe Einstein was correct to be uneasy.

The birth of quantum mechanics partly arose from the developments of kinetic theory and statistical mechanics that I discussed briefly in a previous post. Inspired by such luminaries as James Clerk Maxwell and Ludwig Boltzmann, physicists had inexorably increased the range of phenomena that could be brought within the descriptive framework furnished by Newtonian mechanics and the new modes of statistical analysis that they had founded. Maxwell had also been responsible for another major development in theoretical physics: the unification of electricity and magnetism into a single system known as electromagnetism. Out of this mathematical tour de force came the realisation that light was a form of electromagnetic wave, an oscillation of electric and magnetic fields through apparently empty space.  Optical light forms just part of the possible spectrum of electromagnetic radiation, which ranges from very long wavelength radio waves at one end to extremely short wave gamma rays at the other.

With Maxwell’s theory in hand, it became possible to think about how atoms and molecules might exchange energy and reach equilibrium states not just with each other, but with light. Everyday experience that hot things tend to give off radiation and a number of experiments – by Wilhelm Wien and others – had shown that there were well-defined rules that determined what type of radiation (i.e. what wavelength) and how much of it were given off by a body held at a certain temperature. In a nutshell, hotter bodies give off more radiation (proportional to the fourth power of their temperature), and the peak wavelength is shorter for hotter bodies. At room temperature, bodies give off infra-red radiation, stars have surface temperatures measured in thousands of degrees so they give off predominantly optical and ultraviolet light. Our Universe is suffused with microwave radiation corresponding to just a few degrees above absolute zero.

The name given to a body in thermal equilibrium with a bath of radiation is a “black body”, not because it is black – the Sun is quite a good example of a black body and it is not black at all – but because it is simultaneously a perfect absorber and perfect emitter of radiation. In other words, it is a body which is in perfect thermal contact with the light it emits. Surely it would be straightforward to apply classical Maxwell-style statistical reasoning to a black body at some temperature?

It did indeed turn out to be straightforward, but the result was a catastrophe. One can see the nature of the disaster very straightforwardly by taking a simple idea from classical kinetic theory. In many circumstances there is a “rule of thumb” that applies to systems in thermal equilibrium. Roughly speaking, the idea is that energy becomes divided equally between every possible “degree of freedom” the system possesses. For example, if a box of gas consists of particles that can move in three dimensions then, on average, each component of the velocity of a particle will carry the same amount of kinetic energy. Molecules are able to rotate and vibrate as well as move about inside the box, and the equipartition rule can apply to these modes too.

Maxwell had shown that light was essentially a kind of vibration, so it appeared obvious that what one had to do was to assign the same amount of energy to each possible vibrational degree of freedom of the ambient electromagnetic field. Lord Rayleigh and Sir James Jeans did this calculation and found that the amount of energy radiated by a black body as a function of wavelength should vary proportionally to the temperature T and to inversely as the fourth power of the wavelength λ, as shown in the diagram for an example temperature of 5000K:

Even without doing any detailed experiments it is clear that this result just has to be nonsense. The Rayleigh-Jeans law predicts that even very cold bodies should produce infinite amounts of radiation at infinitely short wavelengths, i.e. in the ultraviolet. It also predicts that the total amount of radiation – the area under the curve in the above figure – is infinite. Even a very cold body should emit infinitely intense electromagnetic radiation. Infinity is bad.

Experiments show that the Rayleigh-Jeans law does work at very long wavelengths but in reality the radiation reaches a maximum (at a wavelength that depends on the temperature) and then declines at short wavelengths, as shown also in the above Figure. Clearly something is very badly wrong with the reasoning here, although it works so well for atoms and molecules.

It wouldn’t be accurate to say that physicists all stopped in their tracks because of this difficulty. It is amazing the extent to which people are able to carry on despite the presence of obvious flaws in their theory. It takes a great mind to realise when everyone else is on the wrong track, and a considerable time for revolutionary changes to become accepted. In the meantime, the run-of-the-mill scientist tends to carry on regardless.

The resolution of this particular fundamental conundrum is accredited to Karl Ernst Ludwig “Max” Planck (right), who was born in 1858. He was the son of a law professor, and himself went to university at Berlin and Munich, receiving his doctorate in 1880. He became professor at Kiel in 1885, and moved to Berlin in 1888. In 1930 he became president of the Kaiser Wilhelm Institute, but resigned in 1937 in protest at the behaviour of the Nazis towards Jewish scientists. His life was blighted by family tragedies: his second son died in the First World War; both daughters died in childbirth; and his first son was executed in 1944 for his part in a plot to assassinate Adolf Hitler. After the Second World War the institute was named the Max Planck Institute, and Planck was reappointed director. He died in 1947; by then such a famous scientist that his likeness appeared on the two Deutschmark coin issued in 1958.

Planck had taken some ideas from Boltzmann’s work but applied them in a radically new way. The essence of his reasoning was that the ultraviolet catastrophe basically arises because Maxwell’s electromagnetic field is a continuous thing and, as such, appears to have an infinite variety of ways in which it can absorb energy. When you are allowed to store energy in whatever way you like in all these modes, and add them all together you get an infinite power output. But what if there was some fundamental limitation in the way that an atom could exchange energy with the radiation field? If such a transfer can only occur in discrete lumps or quanta – rather like “atoms” of radiation – then one could eliminate the ultraviolet catastrophe at a stroke. Planck’s genius was to realize this, and the formula he proposed contains a constant that still bears his name. The energy of a light quantum E is related to its frequency ν via E=hν, where h is Planck’s constant, one of the fundamental constants that occur throughout theoretical physics.

Boltzmann had shown that if a system possesses a  discrete energy state labelled by j separated by energy Ej then at a given temperature the likely relative occupation of the two states is determined by a “Boltzmann factor” of the form:

n_{j} \propto \exp\left(-\frac{E_{j}}{k_BT}\right),

so that the higher energy state is exponentially less probable than the lower energy state if the energy difference is much larger than the typical thermal energy kB T ; the quantity kB is Boltzmann’s constant, another fundamental constant. On the other hand, if the states are very close in energy compared to the thermal level then they will be roughly equally populated in accordance with the “equipartition” idea I mentioned above.

The trouble with the classical treatment of an electromagnetic field is that it makes it too easy for the field to store infinite energy in short wavelength oscillations: it can put  a little bit of energy in each of a lot of modes in an unlimited way. Planck realised that his idea would mean ultra-violet radiation could only be emitted in very energetic quanta, rather than in lots of little bits. Building on Boltzmann’s reasoning, he deduced the probability of exciting a quantum with very high energy is exponentially suppressed. This in turn leads to an exponential cut-off in the black-body curve at short wavelengths. Triumphantly, he was able to calculate the exact form of the black-body curve expected in his theory: it matches the Rayleigh-Jeans form at long wavelengths, but turns over and decreases at short wavelengths just as the measurements require. The theoretical Planck curve matches measurements perfectly over the entire range of wavelengths that experiments have been able to probe.

Curiously perhaps, Planck stopped short of the modern interpretation of this: that light (and other electromagnetic radiation) is composed of particles which we now call photons. He was still wedded to Maxwell’s description of light as a wave phenomenon, so he preferred to think of the exchange of energy as being quantised rather than the radiation itself. Einstein’s work on the photoelectric effect in 1905 further vindicated Planck, but also demonstrated that light travelled in packets. After Planck’s work, and the development of the quantum theory of the atom pioneered by Niels Bohr, quantum theory really began to take hold of the physics community and eventually it became acceptable to conceive of not just photons but all matter as being part particle and part wave. Photons are examples of a kind of particle known as a boson, and the atomic constituents such as electrons and protons are fermions. (This classification arises from their spin: bosons have spin which is an integer multiple of Planck’s constant, whereas fermions have half-integral spin.)

You might have expected that the radical step made by Planck would immediately have led to a drastic overhaul of the system of thermodynamics put in place in the preceding half-a-century, but you would be wrong. In many ways the realization that discrete energy levels were involved in the microscopic description of matter if anything made thermodynamics easier to understand and apply. Statistical reasoning is usually most difficult when the space of possibilities is complicated. In quantum theory one always deals fundamentally with a discrete space of possible outcomes. Counting discrete things is not always easy, but it’s usually easier than counting continuous things. Even when they’re infinite.

Much of modern physics research lies in the arena of condensed matter physics, which deals with the properties of solids and gases, often at the very low temperatures where quantum effects become important. The statistical thermodynamics of these systems is based on a very slight modification of Boltzmann’s result:

n_{j} \propto \left[\exp\left(\frac{E_{j}}{k_BT}\right)\pm 1\right]^{-1},

which gives the equilibrium occupation of states at an energy level Ej; the difference between bosons and fermions manifests itself as the sign in the denominator. Fermions take the upper “plus” sign, and the resulting statistical framework is based on the so-called Fermi-Dirac distribution; bosons have the minus sign and obey Bose-Einstein statistics. This modification of the classical theory of Maxwell and Boltzmann is simple, but leads to a range of fascinating phenomena, from neutron stars to superconductivity.

Moreover, the nature the ultraviolet catastrophe for black-body radiation at the start of the 20th Century perhaps also holds lessons for modern physics. One of the fundamental problems we have in theoretical cosmology is how to calculate the energy density of the vacuum using quantum field theory. This is a more complicated thing to do than working out the energy in an electromagnetic field, but the net result is a catastrophe of the same sort. All straightforward ways of computing this quantity produce a divergent answer unless a high-energy cut off is introduced. Although cosmological observations of the accelerating universe suggest that vacuum energy is there, its actual energy density is way too small for any plausible cutoff.

So there we are. A hundred years on, we have another nasty infinity. It’s a fundamental problem, but its answer will probably open up a new way of understanding the Universe.


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A Little Bit of Chaos

Posted in The Universe and Stuff with tags , , , , , , , , on November 21, 2009 by telescoper

The era of modern physics could be said to have begun in 1687 with the publication by Sir Isaac Newton of his great Philosophiae Naturalis Principia Mathematica, (Principia for short). In this magnificent volume, Newton presented a mathematical theory of all known forms of motion and, for the first time, gave clear definitions of the concepts of force and momentum. Within this general framework he derived a new theory of Universal Gravitation and used it to explain the properties of planetary orbits previously discovered but unexplained by Johannes Kepler. The classical laws of motion and his famous “inverse square law” of gravity have been superseded by more complete theories when dealing with very high speeds or very strong gravity, but they nevertheless continue supply a very accurate description of our everyday physical world.

Newton’s laws have a rigidly deterministic structure. What I mean by this is that, given precise information about the state of a system at some time then one can use Newtonian mechanics to calculate the precise state of the system at any later time. The orbits of the planets, the positions of stars in the sky, and the occurrence of eclipses can all be predicted to very high accuracy using this theory.

At this point it is useful to mention that most physicists do not use Newton’s laws in the form presented in the Principia, but in a more elegant language named after Sir William Rowan Hamilton. The point about Newton’s laws of motion is that they are expressed mathematically as differential equations: they are expressed in terms of rates of changes of things. For instance, the force on a body gives the rate of change of the momentum of the body. Generally speaking, differential equations are very nasty things to solve which is a shame because most a great deal of theoretical physics involves them. Hamilton realised that it was possible to express Newton’s laws in a way that did not involve clumsy mathematics of this type. His formalism was equivalent, in the sense that one could obtain the basic differential equations from it, but easier to use in general situations. The key concept he introduced – now called the Hamiltonian – is a single mathematical function that depends on both the positions q and momenta p of the particles in a system, say H(q,p). This function is constructed from the different forms of energy (kinetic and potential) in the system, and how they depend on the p’s and q’s, but the details of how this works out don’t matter. Suffice to say that knowing the Hamiltonian for a system is tantamount to a full classical description of its behaviour.

Hamilton was a very interesting character. He was born in Dublin in 1805 and showed an astonishing early flair for languages, speaking 13 of them by the time he was 13. He graduated from Trinity College aged 22, at which point he was clearly a whiz-kid at mathematics as well as languages. He was immediately made professor of astronomy at Dublin and Astronomer Royal for Ireland. However, he turned out to be hopeless at the practicalities of observational work. Despite employing three of his sisters to help him in the observatory he never produced much of astronomical interest. Mathematics and alcohol seem to have been the two real loves of his life.

It is a fascinating historical fact that the development of probability theory during the late 17th and early 18th century coincided almost exactly with the rise of Newtonian Mechanics. It may seem strange in retrospect that there was no great philosophical conflict between these two great intellectual achievements since they have mutually incompatible views of prediction. Probability applies in unpredictable situations; Newtonian Mechanics says that everything is predictable. The resolution of this conundrum may owe a great deal to Laplace, who contributed greatly to both fields. Laplace, more than any other individual, was responsible to elevated the deterministic world-view of Newton to a scientific principle in its own right. To quote:

We ought then to regard the present state of the Universe as the effect of its preceding state and as the cause of its succeeding state.

According to Laplace’s view, knowledge of the initial conditions pertaining at the instant of creation would be sufficient in order to predict everything that subsequently happened. For him, a probabilistic treatment of phenomena did not conflict with classical theory, but was simply a convenient approach to be taken when the equations of motion were too difficult to be solved exactly. The required probabilities could be derived from the underlying theory, perhaps using some kind of symmetry argument.

The s-called “randomizing” devices used in all traditional gambling games – roulette wheels, dice, coins, bingo machines, and so on – are in fact well described by Newtonian mechanics. We call them “random” because the motions involved are just too complicated to make accurate prediction possible. Nevertheless it is clear that they are just straightforward mechanical devices which are essentially deterministic. On the other hand, we like to think the weather is predictable, at least in principle, but with much less evidence that it is so!

But it is not only systems with large numbers of interacting particles (like the Earth’s atmosphere) that pose problems for predictability. Some deceptively simple systems display extremely erratic behaviour. The theory of these systems is less than fifty years old or so, and it goes under the general title of nonlinear dynamics. One of the most important landmarks in this field was a study by two astronomers, Michel Hénon and Carl Heiles in 1964. They were interested in what would happens if you take a system with a known analytical solutions and modify it.

In the language of Hamiltonians, let us assume that H0 describes a system whose evolution we know exactly and H1 is some perturbation to it. The Hamiltonian of the modified system is thus

 H(q_i,p_i)=H_0(q_i, p_i) + H_1 (q_i, p_i)

What Hénon and Heiles did was to study a system whose unmodified form is very familiar to physicists: the simple harmonic oscillator. This is a system which, when displaced from its equilibrium, experiences a restoring force proportional to the displacement. The Hamiltonian description for a single simple harmonic oscillator system involves a function that is quadratic in both p and q:

H=\frac{1}{2} \left( q_1^2+p_1^2\right)

The solution of this system is well known: the general form is a sinusoidal motion and it is used in the description of all kinds of wave phenomena, swinging pendulums and so on.

The case Henon and Heiles looked at had two degrees of freedom, so that the Hamiltonian depends on q1, q2, p1 and p2:

H=\frac{1}{2} \left( q_1^2+p_1^2 + q_2^2+p_2^2\right)

 However, in this example, the two degrees of freedom are independent, meaning that there is uncoupled motion in the two directions. The amplitude of the oscillations is governed by the total energy of the system, which is a constant of the motion. Other than this, the type of behaviour displayed by this system is very rich, as exemplified by the various Lissajous figures shown in the diagram below. Note that all these figures are produced by the same type of dynamical system of equations: the different shapes are consequences of different initial conditions and different coefficients (which I set to unity in the form above).

 

 If the oscillations in each direction have the same frequency then one can get an orbit which is a line or an ellipse. If the frequencies differ then the orbits can be much more complicated, but still pretty. Note that in all these cases the orbit is just a line, i.e. a one-dimensional part of the two-dimensional space drawn on the paper.

More generally, one can think of this system as a point moving in a four-dimensional phase space defined by the coordinates q1, q2, p1 and p2; taking slices through this space reveals qualitatively similar types of orbit for, say, p2 and q2 as for p1 and p2. The motion of the system is confined to a lower-dimensional part of the phase space rather than filling up all the available phase space. In this particular case, because each degree of freedom moves in only one of its two available dimensions, the system as a whole moves in a two-dimensional part of the four-dimensional space.

This all applies to the original, unperturbed system. Hénon and Heiles took this simple model and modified by adding a term to the Hamiltonian that was cubic rather than quadratic and which coupled the two degrees of freedom together. For those of you interested in the details their Hamiltonian was of the form

 H=\frac{1}{2} \left( q_1^2+p_1^2 + q_2^2+p_2^2\right) +q_1^2q_2+ \frac{1}{3}q_2^3

 

The first set of terms in the brackets is the unmodified form, describing a simple harmonic oscillator; the other two terms are new. The result of this simple alteration is really quite surprising. They found that, for low energies, the system continued to behave like two uncoupled oscillators; the orbits were smooth and well-behaved. This is not surprising because the cubic modifications are smaller than the original quadratic terms if the amplitude is small.  For higher energies the motion becomes a bit more complicated, but the phase space behaviour is still characterized by continuous lines, as shown in the left hand part of the following figure.

 

However, at higher values of the energy (right), the cubic terms become more important, and something very striking happens. A two-dimensional slice through the phase space no longer shows the continuous curves that typify the original system, but a seemingly disorganized scattering of dots. It is not possible to discern any pattern in the phase space structure of this system: it appear to be random.

 

Nowadays we describe the transition from these two types of behaviour as being accompanied by the onset of chaos. It is important to note that this system is entirely deterministic, but it generates a phase space pattern that is quite different from what one would naively expect from the behaviour usually associated with classical Hamiltonian systems. To understand how this comes about it is perhaps helpful to think about predictability in classical systems. It is true that precise knowledge of the state of a system allows one to predict its state at some future time.  For a single particle this means that precise knowledge of its position and momentum, and knowledge of the relevant H, will allow one to calculate the position and momentum at all future times.

But think a moment about what this means. What do we mean by precise knowledge of the particle’s position? How precise? How many decimal places? If one has to give the position exactly then that could require an infinite amount of information. Clearly we never have that much information. Everything we know about the physical world has to be coarse-grained to some extent, even if it is only limited by measurement error. Strict determinism in the form advocated by Laplace is clearly a fantasy. Determinism is not the same as predictability.

In “simple” Hamiltonian systems what happens is that two neighbouring phase-space paths separate from each other in a very controlled way as the system evolves. In fact the separation between paths usually grows proportionally to time. The coarse-graining with which the input conditions are specified thus leads to a similar level of coarse-graining in the output state. Effectively the system is predictable, since the uncertainty in the output is not much larger than in the input.

In the chaotic system things are very different. What happens here is that the non-linear interactions represented in the Hamiltonian play havoc with the initial coarse-graining. Phase-space orbits that start out close to each other separate extremely violently (typically exponentially) and in a way that varies from one part of the phase space to another.  What happens then is that particle paths become hopelessly scrambled and the mapping between initial and final states becomes too complex to handle. What comes out  the end is practically impossible to predict.

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