## Writing Vectors

Posted in mathematics, The Universe and Stuff with tags , , , on October 11, 2021 by telescoper

Once again it’s time to introduce first-year Mathematical Physics students to the joy of vectors, or specifically Euclidean vectors. Some of my students have seen them before, but probably aren’t aware of how much we use them theoretical physics. Obviously we introduce the idea of a vector in the simplest way possible, as a directed line segment. It’s only later on, in the second year, that we explain how there’s much more to vectors than that and explain their relationship to matrices and tensors.

Although I enjoy teaching this subject I always have to grit my teeth when I write them in the form that seems obligatory these days.

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

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

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

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

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

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

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 a vector laboriously in terms of base vectors:

$\vec{r} = x\hat{\imath} + y \hat{\jmath} + z \hat{k}$

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. The only amusing thing about this is that you get to tell students not to put a dot on the “i” or the “j” – it always gets a laugh when you point out that these little dots are called “tittles“.

Worse still, for the purpose of teaching inexperienced students physics, it offers the possibility of horrible notational confusion. In particular, the unit vector $\hat{\imath}$ is too easily confused with $i$, the square root of minus one. Introduce a plane wave with a wavevector $\vec{k}$ and it gets even worse, especially when you want to write $\exp(i\vec{k}\cdot \vec{x})$, and if you want the answer to be the current density $\vec{j}$ then you’re in big trouble!

Call me old-fashioned, but I’ll take the row and column notation any day!

(Actually it’s better still just to use the index notation, $a_i$ which generalises easily to $a_{ij}$ and, for that matter, $a^{i}$.)

Or perhaps being here in Ireland we should, in honour of Hamilton, do everything in quaternions.

## Messers, Dreamers and Misfits

Posted in Art, Education, Music with tags , , , on September 5, 2021 by telescoper

After the death of Charlie Watts last week, Fintan O’Toole wrote a piece in the Irish Times (here, unfortunately behind a paywall) pointing out that, along with a large fraction of the English rock musicians that began their careers in the 1960s, Charlie Watts went to Art School; Harrow Art School in his case. O’Toole goes on to argue that society needs to find ways to nurture its creative talents and that modern education is far too utilitarian to allow space for “Messers, Dreamers and Misfits”.

I agree with the broad thrust of Fintan O’Toole’s argument. I think the School and University systems are far too focussed on examination and assessment at the expense of genuine education. What I disagree with is the idea that creativity is only to be found in the Arts. When I saw the phrase “Messers, Dreamers and Misfits” it struck me that this could very well describe many of my colleagues in physics, and in science generally – and I don’t mean that in any way as an insult!

There is an explicit assumption in much of the world that creativity is only to be found in the Arts. That’s just not true. Who can say that Einstein didn’t have a creative mind? It is true that if you want to be, say, a theoretical physicist you do have to do formal training in the methods used, especially mathematics. But that is no different from an art school really. To be a painter you have to learn the techniques needed to manage the media you are using. To be a musician you have to learn the basics of harmony and solve the technical problems involved with playing an instrument. Artists have to pay their dues just like scientists. I wrote about this here, in the context of the great Jazz pianist Bill Evans, where I said:

All subjects require technical skill, but there is more to being a great jazz musician than mastery of the instrument just as there’s more to being a research scientist than doing textbook problems. So here’s to creativity wherever it is found, and let’s have a bit more appreciation for the creative aspects of science and engineering!

Anyway, here in Ireland, the Leaving Certificate results came out on Friday and next week we’ll begin the process that determines how many students we’ll have doing Mathematical and Theoretical Physics at Maynooth. It always surprises me how many students choose study subjects other than Physics, but then I remember that I went from School to Cambridge in 1982 to read Natural Sciences, fully expecting to specialize in Chemistry but just found Physics more interesting and, yes, more fun.

I don’t know whether I count as a creative person at all, but I’m definitely a misfit, prone to dreaming and – especially at the moment in the middle of unpacking my belongings – my house is a mess!

Anyway, here is a message for students just about the start their Third Level education here in Ireland or elsewhere. The most important advice I can give is to choose the subject that you will enjoy most, but pursue your other interests too. Charlie Watts was interested in music while at art school. There’s no reason why a theoretical physicist can’t pursue an interest in music too. I can think of at least one prominent example of a person who managed to become a pop musician and a physicist.

Given my own background I read Fintan O’Toole’s article as clear encouragement to students to pick theoretical physics.

## An Olympic Story

Posted in Sport with tags , , , , on July 26, 2021 by telescoper

Just a quick post to mention a wonderful Olympic story. Louise Shanahan (pictured above) from Cork is competing in the Olympic Games in Tokyo in the 800m for Ireland. She is also in the second year of a PhD in Physics in the University of Cambridge, working in the Atomic, Mesoscopic and Optical Physics (AMOP) group in the Cavendish Laboratory. I wish her all the best in the heats on Friday 30th July and hopefully beyond!

UPDATE: Louise came seventh in Heat 3 so is now eliminated. She kept pace with the leaders before falling away on the final 150m stretch, finishing in a time of 2:03.57.

## R. I. P. Steven Weinberg (1933-2021)

Posted in Biographical, The Universe and Stuff with tags , , , on July 24, 2021 by telescoper

I just heard this morning via Twitter of the death at the age of 88 of the physicist Steven Weinberg. The news media don’t seem to have caught on yet but I’ll add links to appropriate tributes when they do.

UPDATE: You will find an appreciation from UT Austin here and an Associated Press article here.

Steven Weinberg is probably most famous in physics circles for his work on electroweak unification, together with Seldon Glashow and Abdus Salam, for which he jointly won the Nobel Prize for Physics in 1979. He did many other things besides, of course, and his influence is felt across huge swathes of particle physics, quantum field theory and cosmology. As well as a researcher he was a prolific writer, both of technical books – his Gravitational and Cosmology is a classic text on the principles and applications of the general theory of relativity – but also of works for the general public. He was an author of rare elegance and lucidity with some wonderful turns of phrase and a beautifully articulated secular view of the human condition. For example

If there is no point in the universe that we discover by the methods of science, there is a point that we can give the universe by the way we live, by loving each other, by discovering things about nature, by creating works of art. And that—in a way, although we are not the stars in a cosmic drama, if the only drama we’re starring in is one that we are making up as we go along, it is not entirely ignoble that faced with this unloving, impersonal universe we make a little island of warmth and love and science and art for ourselves. That’s not an entirely despicable role for us to play.

I bought Weinberg’s popular book The First Three Minutes about 40 years ago, and I still have a copy today. It’s no exaggeration to say that this book played a major part in my decision to continue a career in theoretical physics. I know I’m not the only physicist of my generation (or others) for whom this is the case. Although I never met Steven Weinberg in person, he was definitely an inspiration and he will be greatly missed.

Rest in peace, Steven Weinberg (1933-2021).

## What should universities keep after Covid?

Posted in Education, Maynooth with tags , , , on May 7, 2021 by telescoper

On the last day of teaching for this academic year, with reasonably encouraging signs of some form of reopening of campus education being possible in September, it will shortly be time to think about how we proceed next academic year.

It seems obvious to me that although university staff have worked very hard over the last year the Covid-19 restrictions have meant that we have not been able to provide the level of education we would have liked and most of us are longing to get back to some form of face-to-face teaching. On the other hand, the restrictions imposed upon us have generated some new approaches and it would be silly just to abandon what we have learnt and go straight back to the way things were before the pandemic.

I have two main things in mind, one on teaching and one on assessment, both of them relating to my own discipline – theoretical physics – but hopefully of some wider interest.

Implementing the Lane-Emden equation as two coupled first-order ODEs.

First, on teaching. Over the past year we have mainly been delivering lectures and tutorials remotely, using a mixture of platforms (Zoom, Panopto, Microsoft Teams, etc). Most lecturers have done lectures as live webcasts as well as recording the sessions to be viewed later. I have used Panopto for most of mine, actually. I am actually looking forward to being able to dismantle the setup I have in my study for this, to reclaim a bit of space, but probably won’t do so until we know for sure what we’ll be doing next Semester!

(By the way does anyone know where I should send the bill to my employers for their use of my study over the last year?)

For the record, I have found about 50% of the registered students have watched the lectures as live broadcasts from my home; the rest watch the recordings offline.

Maynooth didn’t have any facilities for lecture capture on campus until September 2020, in contrast to my two previous employers – the University of Sussex and Cardiff University – who both had systems in place long before the pandemic. I blogged about this 8 years ago, in fact. In Cardiff they actually use Panopto; all lectures were recorded as standard. In my view the benefits of lecture capture far outweigh the disadvantages, and we should incorporate recordings of lectures as part of our standard teaching provision, as a supplement to learning rather than to replace face-to-face sessions.

It seems to me that much of the argument against providing lecture recordings is from older staff who thing the younger generations should learn exactly the same way they themselves did despite the reality that classroom teaching in schools is now utterly different from what my generation experienced.

My view is that every student learns in a different way and we should therefore be doing as much as we possibly can to provide a diverse range of teaching resources so that each can find the combination that suits them best. Technology allows us to do this far better now than in the past.

Some really enjoy live lecture sessions, but others don’t. Others have reasons (such as disability) for not being able to attend in-person lectures, so providing recordings can help them. But why not in that case provide recordings for everyone? That seems to me to be a more inclusive approach.

The problem with continuing lecture capture beyond September 2021 in Maynooth is that we will need to improve the cameras and recording equipment in the large lecture rooms to make this possible for lectures with a significant mathematical content, as the existing setups in teaching rooms do not easily allow the lecturer to record material on a whiteboard or blackboard. In Cardiff the larger rooms have more than one camera, usually one on the lectern and one on the screen or whiteboard (which has to be placed further away and therefore needs to be of higher resolution). In Maynooth we only have small podium cameras in the teaching rooms.

The next topic is assessment. Since we were forced to switch to online timed assessments last May we have been doing most of our assessments that way. The student is given an exam paper at the appointed time, which they do on their own, then scan and upload their answers online (in our case via Moodle).

This mode of assessment has its problem. One is the possibility that students can collude (as there are no invigilators). Another is that not all students have a home environment conducive to taking an examination, nor a decent internet connection.

We decided to implement these as truly “open book” exams in which students are free to consult their notes, textbooks and internet resources. That format means it is pointless to ask the students to regurgitate definitions or learn derivations by rote so we concentrate on problem-solving, testing the understanding and application of concepts. Although it makes it a little harder to construct the examination papers, I think this a good way of assessing ability and knowledge of physics. If we can go do exams back on campus I think we should retain this approach at least for advanced topics, providing supervised spaces on campus to prevent collusion.

There are doubtless many other innovations we have brought in over the last year that people feel strongly about (one way or the other). Feel free to share them through the comments!

## Hawking and the Mind of God

Posted in Books, Talks and Reviews, Science Politics, The Universe and Stuff with tags , , , , , on March 3, 2021 by telescoper

There’s a new book out about Stephen Hawking which has triggered a certain amount of reaction (see, e.g., here) so I thought I’d mention a book I wrote, largely in response to the pseudo-religious nature of some of Hawking’s later writings.

I have in the past gone on record, both on television and in print, as being not entirely positive about the “cult” that surrounds Stephen Hawking. I think a number of my colleagues have found some of my comments disrespectful and/or churlish. I do nevertheless stand by everything I’ve said. I have enormous respect for Hawking the physicist, as well as deep admiration for his tenacity and fortitude, and have never said otherwise. I don’t, however, agree that Hawking is in the same category of revolutionary thinkers as Newton or Einstein, which is how he is often portrayed.

In fact a poll of 100 theoretical physicists in 1999 came to exactly the same conclusion. The top ten in that list were:

1. Albert Einstein
2. Isaac Newton
3. James Clerk Maxwell
4. Niels Bohr
5. Werner Heisenberg
6. Galileo Galilei
7. Richard Feynman
8. Paul Dirac
9. Erwin Schrödinger
10. Ernest Rutherford

The idea of a league table like this is of course a bit silly, but it does at least give some insight into the way physicists regard prominent figures in their subject. Hawking came way down the list, in fact, in 300th (equal) place. I don’t think it is disrespectful to Hawking to point this out. I’m not saying he isn’t a brilliant physicist. I’m just saying that there are a great many other brilliant physicists that no one outside physics has ever heard of.

It is interesting to speculate what would have happened if the list had been restricted to living physicists. I’d guess Hawking would be in the top ten, but I’m not at all sure where…

And before I get accused of jealousy about Stephen Hawking’s fame, let me make it absolutely clear that if Hawking was like a top Premiership footballer (which I think is an appropriate analogy), then I am definitely like someone kicking a ball around for a pub team on a Sunday morning (with a hangover). This gulf does not make me envious; it just makes me admire his ability all the more, just as trying to play football makes one realise exactly how good the top players really are.

I am not myself 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.

Anyway, this is the book I wrote:.

And here is the jacket blurb:

Stephen Hawking has achieved a unique position in contemporary culture, combining eminence in the rarefied world of theoretical physics with the popular fame usually reserved for film stars and rock musicians. Yet Hawking’s technical work is so challenging, both in its conceptual scope and in its mathematical detail, that proper understanding of its significance lies beyond the grasp of all but a few specialists. How, then, did Hawking-the-scientist become Hawking-the-icon? Hawking’s theories often take him into the intellectual territory that has traditionally been the province of religion rather than science. He acknowledges this explicitly in the closing sentence of his bestseller, A Brief History of Time , where he says that his ultimate aim is to know the Mind of God . Hawking and the Mind of God examines the pseudo-religious connotations of some of the key themes in Hawking’s work, and how these shed light not only on the Hawking cult itself, but also on the wider issue of how scientists represent themselves in the media.

I’m sure you’ll understand that there isn’t a hint of opportunism in the way I’m drawing this to your attention because my book is long out of print so you can’t buy it unless you get a copy second-hand…

## The Start of Spring Semester

Posted in Biographical, Education, Maynooth with tags , , , , , , , on February 1, 2021 by telescoper

It’s February 1st 2021, which means that today is Imbolc, a Gaelic festival marking the point halfway between the winter solstice and vernal equinox, i.e. it’s a Cross-Quarter Day. To be pedantic, Imbolc is actually the period between this evening and tomorrow evening as in the Celtic calendar days were counted from sunset to sunset.

The first Day of February is also the Feast day of St Brigid of Kildare (c. 451-525), one of Ireland’s patron saints along with Saints Patrick and Colm Cille. One of her miraculous powers was the ability to change water into ale, which perhaps explains her enduring popularity among the Irish.

In Ireland this day is sometimes regarded as the first day of spring, as it is roughly the time when the first spring lambs are born. It corresponds to the Welsh Gŵyl Fair y Canhwyllau and is also known as the Cross Quarter Day’ or (my favourite) The Quickening of the Year’. According to legend it is also the day on which jackdaws mate. Given how many of them there are around Maynooth there should be a lot of action today.

Today is, appropriately enough in the light of all this, the start of the Spring Semester of teaching at Maynooth University, the fourth Spring Semester I will have experienced here although this is obviously not like the others in that we’ll be teaching online at least for the first half and probably for the entirety. I was planning to stay at home today but I realised I’d left some things I need in the office on campus so will have to go to collect them. That’s why I’m up early. That and the need to shake myself out of the lockdown torpor that has afflicted me since New Year. It’s time to get my act together, pull my finger out, put my best foot forward, etc.

This Semester I am teaching Engineering Mathematics II, Computational Physics I and Advanced Electromagnetism. The former, what you would probably call a `service course’, covers a mixture of things, mainly Linear Algebra but with some other bits thrown in for fun, such as Laplace transforms. Interestingly I find the Mathematical Physics students do not encounter Laplace Transforms in the first year, but perhaps engineers use them more often than physicists do? I think I’ve written only one paper that made use of a Laplace transform. Anyway, I have to start with this topic as the students need some knowledge of it for some other module they’re taking this semester. I reckon six lectures will be enough to give them what they need. That’s two weeks of lectures, there being three lectures a week for this module.

Once again my teaching timetable for this module is quite nice. I have lectures on Monday, Tuesday and Wednesday and then the students have a choice of tutorial (on either Thursday or Friday). That means I can get through a decent amount of material each week before each tutorial. I don’t do the tutorials, by the way: that’s left to one of our PhD students, who gets paid for doing that and correcting the weekly coursework. There are about 50 students on this module, divided into two courses: Electronic Engineering and Robotics and Intelligent Devices. We don’t have Civil or Mechanical or Chemical Engineering, etc, at Maynooth, in case you were wondering. Lectures will be done as webcasts using Panopto but also recorded for later viewing.

My first Computational Physics lecture, which I will do from home, is on Thursday, after which there is a lab session which we will do via Microsoft Teams. That’s the way we did it after lockdown last year and it worked OK. Students attend one two-hour lab session in addition to the lecture, on either Thursday or Tuesday. The first lecture being on Thursday the first lab session will be Thursday afternoon, with the same material being covered the following Tuesday. Fortunately, Python is free to download and easy to install so it’s quite straightforward to run the labs remotely. Teams has a screen sharing facility so it’s quite easy for myself or my demonstrator to see what is wrong in the same way we would do in a laboratory class.

The Advanced Electromagnetism module is a new one for me but I’m quite looking forward to it. Being a final-year module its content is less prescriptive than others and I’ll be adding a few things that I find interesting. Both lectures for that one are on Wednesdays and again will be given as webcasts with recordings available later.

Today is a particularly busy day because in addition to my first lecture (at 2pm) I have a meeting of Academic Council (3pm via Teams), a Euclid telecon (via Zoom) and a meeting with my PhD student via Teams. I have also been trying to sort out tutors and tutorials for the forthcoming Semester: these don’t start until next week so there’s time, but it has been quite a challenge to get everyone sorted. Fortunately I think that’s now done.

Oh, and another thing. I signed up for Irish language lessons (Beginners Level) and will be having classes once a week from now on.

It’s going to be a very busy term but I reckon being busy is probably going to be a good way to get through the next few months.

## Physics versus Astrophysics

Posted in The Universe and Stuff with tags , on January 21, 2021 by telescoper

I don’t often post memes on here, but I couldn’t resist this one. Credit here.

## “And” Time Draws Nigh

Posted in History, Poetry, The Universe and Stuff with tags , , , , , , , on November 30, 2020 by telescoper

It’s November 30th 2020, which means we have just three teaching weeks to go until the end of term. I am currently teaching two modules: Mechanics 1 and Special Relativity for first-year students and Vector Calculus and Fourier Series for second years. We’re now getting to the “and” bit in both modules.

I didn’t want to present the two topics mentioned in the title of the second year module as completely disconnected, so I decided to link them with a lecture in which I use the divergence theorem of vector calculus to derive the heat equation, the solution of which led Joseph Fourier to devise his series in Mémoire sur la propagation de la chaleur dans les corps solides (1807), a truly remarkable work for its time that inspired so many subsequent developments.

That gives me an excuse to repost the following “remarkable” poem about Fourier by William Rowan Hamilton:

In the first-year module I will be spending most of this week talking about potentials and forces before starting special relativity next week, at the proper time.

This day and age we’re living in
Gives cause for apprehension
With speed and new invention
And things like fourth dimension
Yet we get a trifle weary
With Mr. Einstein’s theory
So we must get down to earth at times
Relax relieve the tension
And no matter what the progress
Or what may yet be proved
The simple facts of life are such
They cannot be removed

As time goes by, the other thing drawing nigh is the loosening of Ireland’s current Level 5 Covid-19 restrictions which were imposed about six weeks ago though, judging by the crowds drinking in Courthouse Square on Saturday night, a lot of folks have thrown the rules out the window already.

I think it’s a dangerous time. The daily cases are still hovering around the 250-300 mark and will undoubtedly start climbing even before Christmas itself:

The chances of us getting back to anything resembling normality during the early part of next year are exceedingly slim.

## Lecture Streaming

Posted in Education, Maynooth with tags , , , on October 7, 2020 by telescoper

When we were told, at the start of this term, to move all our teaching online my initial intention was to record most of the lectures in my office for the students to watch at their leisure rather than streaming them live.
The system we’re using, Panopto, allows for both webcast (i.e. live streaming) and pre-recorded (offline) videos. I thought that only very few students would want to watch the broadcast version. I have however changed my mind about this and am now streaming all my lectures (as well as recording them for later viewing). It also meant that I could record the lectures in advance at my without being constrained by the timetable.

Last week my office wasn’t usable for recording videos because of noise from building work so I had to find somewhere else to record the videos so I decided to go to the lecture theatre at the scheduled time primarily because I knew the room would be available at that time. When I started the first lecture I thought I might as well webcast it as well, thinking only a few students would tune in. In fact, out of my class of 45 or so second-year students, about 39 were online while I did the lecture. Since then I’ve done all the lectures live and plan to do so until further notice.

A handful of students even turn up in person to the lectures. I see no problem with this. The restrictions are designed to minimize as far as possible the number of students coming to campus, but if they are here anyway because of labs (which can’t be done virtually) then why shouldn’t they come to the lectures? (Provided, of course, that they follow the public health guidance, wear masks, wash their hands, practice social distancing, etc). I find their presence very helpful, actually. Talking to an audience is far easier than talking only to a camera. You do have to remember to look at the camera though!

It is possible to edit the webcast recording before sharing it with the students. That way you can get rid of all the mistakes, hesitations and other defective bits. Starting with a 50 minute lecture that usually means you end up with about 10 minutes of good material.

Having settled on this approach I was dismayed on Monday to find the Panopto system wasn’t working in either webcast or offline recording mode. I assumed at first that I was doing something wrong but it turns out it was a major outage affecting all of Europe that went on all day. Twitter was full of comments from academics complaining about! Panopto uses cloud storage with very little being held locally so when the connectivity fails the user is helpless. I did the lecture by Teams instead, but had lost some time faffing around trying to get Panopto to work.

Yesterday morning Panopto was back working and my office was quiet so I reran the lecture using the blackboard in my office and recorded it as an offline video. That way the students now have the lecture in the right space on the Moodle page. After that I did two more lectures as webcasts using Moodle – Tuesday is a busy teaching day this term – and everything worked fine.

I think there are two morals to be drawn from this. The first is not to assume that you know what students will find useful. The second is wherever possible to have a backup plan. Putting all your eggs in the basket marked Panopto is risky.