Cambridge NST Maths year 1 2019 Paper 2 solutions

 Here is the second in a sequence of my solutions to the Cambridge NST maths papers, This one is for the second of the papers set for the first year students in 2019 the last year before COVID. You can access it here 

https://drive.google.com/file/d/1SSePjGdcGjWatVFNHh5kVzYORGgaBUrD/view?usp=sharing

I have previously published my solutions to the first of the papers and for convenience I post the link again here for those who missed it first time around

https://drive.google.com/file/d/1SSePjGdcGjWatVFNHh5kVzYORGgaBUrD/view?usp=sharing

So there you have it a complete set of answers to the first year papers for NST students at Cambridge for 2019. Most of them will have sat there finals last summer and I hope they did well. 

I apologise in advance for any typos spelling errors etc. A review of the paper follows 

Again just like the first paper this was quite challenging and unfortunately I was unable to answer a question on Parseval's theorem properly. On the whole though I think I got the questions out, but I doubt if I could do well under exam conditions. 

Anyway just like the first paper there were 10 short questions which were relatively straightforward once you decoded what the examiner was getting at. The questions included solving a first order differential equation by the integrating factor method. Deriving a recurrence relation for an Integral (something they love testing people on). Calculating the stationary values of a function f(x,y). A slightly confusing question on pronability I really need more practice at questions involviing conditional probability. A volume integral and a surface integral. All this should take no longer than 30 mins but I suspect I would take a lot longer. Again there is no time to think and in the rush you would probably end up making silly mistakes,

The core of the paper is 10 questions of which answers to five must be submitted. The last two questions are reserved for those students deemed clever enough to do some advanced topics although I didn't think they were particularly difficult. 

Anyway here are the questons 

Question 11 was a geometric one involving the equation of a plane and finding the volume of the parallipped enclosed by 3 vectors. This was relatively straightforward once I had reminded myself of the vector equation of a plane. But there were quite a few parts. Although the question asked for a few diagrams I had the luxury of using MATLAB to draw the relevant pictures/ Not very exciting I must admit

Question 12 . Involved fiinding the stationary points of a function in f(x,y) and drawing a contour plot showing the function and the gradient. This was tedious but relatively straightforward and again I was able to use MATLAB to draw some pretty pictures. 

Question 13 Involved calculating the line integral of a vector function F for various paths and also finding a function satisfying curl F = 0. (a conservative function). I hadn't done a question like this for ages so I had to remind myself of how you go about calculating such things but it was relatively straightforward although finding the conservative function involved a little guesswork so not very satisfactory.

Question 14  Involved some questions on probability density functions. and evaluating their products and change of  variables. It was ok but not a very exciting topic. 

Question 15  This was a set of questions on solving second order differential equations with constant coefficients my favourite topic at this level. The first question was a homogeneous equation with boundary conditions and relatively straightforward to solve. The second part was an inhomogenous equation and whilst finding the complementary function was relatively straightforward. In order to find the particular integral you had use a function of the form  x^n f(x) and increase n until you found one that worked. This took a couple of goes and so would have been quite time consuming under exam conditions how nice of them 😊. The last part involved solving two differential equations simultaneously some what surprisingly they don't seem to teach how to solve such systems using matrices and their eigenvectors unlike the open university courses MST210 or MST224 so this is one occasion wihere the open university is better than Cambridge. Anyway compared to the tedium of the last few questions this was a delight to do. 

Question 16   This question was all about calculating various surface integrals and the flux of a field through a surface. It got a bit fiddly but again was relatively straight forward. Another boring topic though I much prefer solving differential equations 

Question 17 This was a boring question on matrices again pretty straighrforward but you have to know the definitions and again there were so many parts to the question. Give me calculus questions any day

Question 18  This was a question on Fourier series you had to find the Fourier series for cosh(x) then differentiate to get the Fourier series for sinh(x). For this topic you really need to be on top of integration by parts. The last part of the question then asked you to use Parsevals theorem to show that the integral of (cosh(x)-sinh(x))^2 over the interval was sinh(2) . I tried this a couple of times but I couldn't get the expansions to cancel out to leave sinh(2) so unfortunately I was unable to complete the paper properly. However if you integrate the function directly it comes out relatively straightforwardly. 

So the two questions for the so called advanced students were as follows 

Question 19 was on Lagrangian multipliers and you had to find the optimum volume of a cylinder the optimum volume of a cone inscrbed in a sphere and then prove that the Arithmetic mean is >= to the geometric mean. I confess to nor really understanding this topic although I can go through the motions and I find it difficult to tell whether I am finding a minimum or a maximum. For the cone inscribed inside a sphere I found it easier to just finding the maximum volume directly. I'll let you solve this question using Lagrangian Multipliers fot your self. 

Question 20  A relatively straightforward question on solving partial differential equations using separation of variables. Two first order ones and a question on the diffusion equaton. This is really a warm up for what comes next year so you aren't asked to solve the differential equations in spherical or cylindrical coordinate systems or use Lagrange Polynomials or Bessel functions or any of the other exotic functions out there. So a bit boring really 

Overall conclusion is that this was an exercise worth doing to remind myself and extend my mathematical knowledge a bit. I think on the whole I preferred the first paper as it seemed to cover slightly more interesting topics. Apart from the two quesitions on differential equations this paper could be described as worthy but dull. 

The second year papers for this year beckon next and I hope that I find them a bit more interesting than this one. Hopefully I can finish them by June next year

I would urge you to have a go for yourself and I hope you find these solutions useful 

Thermodynamics

 The laws of thermodynamics are fundamental for understanding the  structure of matter and the transfer of heat. Remarkably they stand by themselves and have no need of any microscopic underpinning This point is often missed in treatments of thermodynamics which quickly move onto statistical physics and don't encourage physicists to develop their powers of thermodynamic reasoning. 

The best account of thermodynamics I know of is given in Longairs book 

https://www.amazon.co.uk/Theoretical-Concepts-Physics-Second-Alternative/dp/052152878X

Which then goes onto discuss how the attempts to model black body radiation broke down using classical physics thus paving the way for Planck and Einstein to introduce quantum mechanical ideas. It really is a fascinating story and shows that there is more to quantum mechanics than the development of Schrodinger's equation 

In general terms a good overall book on Thermodynamics is the classic by Zemansky 

https://www.amazon.co.uk/Heat-and-Thermodynamics-Fifth-edition/dp/B00X4VPZ0C/ref=sr_1_12?crid=32NLWLWHKDY3O&dchild=1&keywords=heat+and+thermodynamics+zemansky&qid=1633373359&s=books&sprefix=Zemansky+%2Cstripbooks%2C173&sr=1-12

Anyway thermodynamics has many applications Chandresekhar used it to work out the General equations of stellar structure without any need to know the internal structure of a star 

Here is my tribute to Thermodynamics and I would encourage people to study it in it's own right 

                                 Thermodynamics

 

Three laws oh so neat,

Describing the nature of heat.

The first says you cannot win,

You wont get back more than you put in.

 

But if it’s heat, there’s a permanent loss,

That is only regained at greater cost.

Finally there will come a great big chill,

Where all that there is, will stand still.

Lasers

 I was challenged by a work colleague to see if I could write a poem about lasers. The result is given below. Just like supeconductivity lasers are another successful application of quantum mechanics which again no agonising about it's meaning will ever produce a laser Ironically given that Einstein rejected the later formulation of quantum mechanics it was Einstein who worked out the basic theory of spontaneous emission on which the laser is based in 1918  This is a purely statistical argument, which again might surprise people as Einstein is allegedly supposed to have claimed that God does not play dice. Well in the early days of quantum mechanics it was Einstein who used statistical arguments to work out the consequences of the photo-electric effect and applied statistical reasoning to work out the Heat capacity of solids. So the idea that Einstein didn't like statistical reasoning is just incorrect. Indeed in his final years when he surveyed his debates with Bohr, he actually made the statement

The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems.

So the only way to make sense of quantum mechanics according to Einstein is to endorse a statistical interpretation.

Anyway lasers are extremely useful devices but of course in the wrong hands can be used as terrifying weapons. Also idiots shine laser pens in pilots eyes, these people should be forced to face the consequences of their actions. In the right hands of course lasers are a benefit to mankind and a testimony to the ingenuity of scientists all over the world. Here is my tribute to them.




                                     Lasers

 

 Purest light that shines so bright,

 All because a photon takes flight.

Channelled by some clever means,

 Into a set of very intense beams.

 

Once the stuff of science fiction,

 It’s now part of our jurisdiction.

The wise will put you to good use,

But we must guard against abuse

.

James Bond concerned that Goldfinger’s Laser might destroy his manhood. I’ll leave you to decide whether or not that would have been a good thing  😀

Super Conductivity

 One of the most amazing applications of the formalism of quantum mechanics was the explanation by Baarden, Cooper and Schrieffer (BCS) of the phenomenon of Superconductivity. At low temperatures roughly about 4K it was noticed by Onnes that some metals appeared to have a dramatic reduction in their resistance. Whilst classical models were developed describing this phenomenon it wasn't until the 1950's that a microscopic version of the theory was developed. At first sight given the Pauli exclusion principle the phenomenon would seem impossible as it implies many electrons are occupying the same state. Which electrons being Fermions was impossible. Cooper realised at low temperatures there was the possiblity of an interaction between the lattice of the solid and the electrons causing them to effectively pair off. At low temperatures this interaction would be stable as the lattice vibrations would be relatively low. If electrons pair off their total spin becomes zero and they now behave like bosons for which it is possible for many bosons to occupy the same state. Thus the resistance of the metal is lowered BCS quickly realised that the properties of superconductors could be explained and they were awarded the Nobel prize for this work in 1972. 

Compared to the endless debates about the meaning or not of quantum mechanics which are going nowhere. This gives us a real insight into how nature works and ia a triumph of Mankinds ability to understand nature, something that will never come from discussing the meaning of the wave-function. As an interesting foot-note it was discovered in 1986 that some cuprates exhibited Superconductivity at much Higher temperatures than the 'normal ones' As yet there is no convincing explanation for this phenomenon so if you want a Nobel prize get cracking 😅

Here is my poem 

Super Conductivity

 

When it becomes very cold,

Nature becomes extremely bold.

All resistance suddenly dies,

An electric current really flies.

 

Electrons interact with the grid,

Combined in pairs they are hid.

This really ingenious tactic,

Was explained by a quantum mechanic¹⁾

1)    Someone who uses their knowledge of quantum mechanics to explain a feature of nature. In this case it was Cooper (what a clever fellow 😅😅 ).

 

  

Quantum Debates part II

 Ok I realise in the last post some people will say well what about the collapse of the wavefunction and the violation of the Bell inqualities don't they show that quantum mechanics is weird. I have covered this in many posts before but to save you looking here is a recap 

Let's take the so called collapse of the wavefunction first. This is usually illustrated by the so called cat paradox. If a quantum system can occupy a number of possible states then prior to measurement the sysrem is said to be in a superposition of states and on the usual story this means that when a measurement is made the so called wave function of the system collapses into one of the possible states with a given probability. Those who see the so called wave function as something more than the square root of a probability density function then make the leap to say that the measurement has caused the system to change from it's superposition to a single state. So there is something special about measurement and indeed quantum mechanics via this so called collapse vindicates idealism. So that we create reality by acts of measurement something which is alleged to be part of the Copenhagen interpretation which in fact it isn't. This should really be called the 'California Interpretation' and if it were true would indeed be weird and mysterious and all those who like to link Buddhism or Transcendental meditation with quantum mechanics would be vindicated. 

But this is totally unnecessary, lets take the classical situation for any statistical event if I know the underlying probability density function which summarises all the possiblities and I pick a given sample then I can calculate the probability that that person has a given height or a particular number will turn up if I throw a dice or spin a roulette wheel. Prior to the outcome I did not know what the outcome would be after the outcome I do. Thus the probability density function has 'collapsed' to give the particular outcome but all that is saying is that for classical systems the probability density function summarises all the possible outcomes and after the event has occured the outcome was realised with the given probability. I would argue the same is true of quantum mechanics before a measurement is made I do not know what the result would be but only the total outcomes with a given probability which if I know the appropriate solution to Schrodinger's wave equation I can calculate via the Born rule. Thus the superposition is not a real superposition but just a summary of possible outcomes of a measurement with the appropriate outcomes. The cat is definitely alive or dead before I open the box all I have done is updated my knowledge of the situation before hand. Opening the box hasn't caused the cat to be alive or dead but whether the radioactive poison was released or not. Something you could estimate if you know the half life of the radioactive material. All of this is consistent with quantum mechanics and how it is applied to calculate the probabilities of certain things happening and there is no need to invoke the collapse of the wavefunction to explain this process. 

In contrast to other realist interpretations such as those invoking hidden variables or the Many Worlds interpretaton nothing is added to the formalism. There is no need to invoke many worlds in a desperate attempt to maintain realism. My solution is robustly realist because the entities to which quantum mechanics is applied electrons, atoms and all the various exotic particles and there interactions are seen as real. Once one accepts that the so called wavefunction isn't anything physical but related to the probability density function via the Born rule, then there is no need to agonise whether or not the wave function is a physical object defined in (3N+1)*S space-time-spin dimensions for an N body system it makes no sense as a physical object but as a probability density function it makes a lot of sense. So by sticking to the statistical interpretation of Born then we can retain a fairly robust realism about the entities to which quantum mechanics is applied to and there is no need to worry about any form of idealism or mysticism. 

I'll talk about the Bell inequalities in a later post. But I defy anyone who disagrees with my perspective to show that my position is inconsistent with quantum mechanics. One can never observe a physical superposition of states because by definition any observation would collapse the wavefunction into one of the possible eigenstates. Those who claim otherwise are simply collating the reuslts of various measurements and claiming that this represents a real superposition. 

 

Quantum Debates

 Ok this poem is a rant against the endless debates about the alleged meaning of quantum mechanics. A debate I regard as totally pointless as the main issues were all settled by the following two rules of interpretation

1) The Borm rule which says that the modulus squared of the solution to Schrodinger's equation, when suitably normalised gives rise to a probability density function from which wc can calculate the expectation value of any variable. Conversely this means that the solution to Schrodinger's equation is effectively the square root of a probability density function not a real wave.

2) The link between the eigenvalues of the solution to Schrodinger's equation and the energy levels of the system under consideration. 

These rules are readily extendible to relativistic equations 

That really is all you need. plus the ability to use Schrodinger's equation or it's relativistic generalisation to understand the results of experiments and from which all the great successes of quantum mechanics have been applied to and which help us understand how nature works. 

 The rest whether or not the solution to Schrodinger's equation is a real wave in 3N+1 dimensional space time or merely a mathematical object which enables the probability that certain events will happen to be calculated or the energy levels of a system to be obtained. Whether there are multiple universess and so forth have nothing to do with physics at all. To coin a phrase quantum mechanics isn't mystical it is just statistical. 

Once the implications of this are accepted then much of the so called debate just dissolves. If there are no hidden variables then we can only have a statistical solution. But this does not mean that particles or their interactions are not real. just that their so called wave functions are not real, but only as real as say the Gaussian Distribution describing the distribution of heights of a collection of human beings in a sufficiently large enough sample.

 Instead of worrying about the meaining of the wave function try and apply the well trodden path of discovering the correct Hamiltonian to describe particles and how they interact with each other once this has been obtained just get on and calculate the consequences. If your Hamiltonian agrees with experimental results, then brilliant you have understood a part of nature. No amount of agonising about whether or not the solution to Schrodinger's equation is a real wave,. or signals travel faster than the speed of life is going to add one iota to the new knowledge. So let's celebrate all those physicists and chemists who daily apply quantum mechanics to understanding the world around us. Rather than the parasitic journalists and philosophers and I am afraid some prominent scientists such as Jim Al Khalili who jump on the band wagon of claiming that quantum mechanics is difficult to understand or mysterious. 

If you want to understand how quantum mechanics works, learn how to solve Schrodinger's equation or any of the relativistic equations don't waste your time reading what is essentially gibberish. Anyway here is my rant 

Quantum Debates

 

                                                  Many thoughts about you abound,

                                            Some of them seeming quite profound.

                                            But it’s all an illusion,

                                            Just adding to our confusion.

 

                                            It’s not mystical,

                                            It’s just statistical.

                                            Time to end this silly debate,

                                            And get on and calculate. 

Example of a silly debate about the meaning of quantum mechanics, that is going nowhere IMHO and there are plenty more around unfortunately. 

Spherical Harmonics

Todays poem is about Spherical Harmonics, just as any function in rectangular coordinates can be expressed as a Fourier Series in Trigonometric functions and in Cylindrical coordinates we can expand any function in terms of Bessel Functions. So in spherical coordinates it is possible to express any function as a series in Spherical Harmonics. Naturally then this expansion  has been applied to spherical waves, the calculation of the fields outside a sphere which leads to a series of Polynomials called Legendre polynomials. One of the most exciting application is to the calculation of the angular dependence of the wave function of the hydrogen atom. It is often thought that because the "waves" are quantised that this is unique to quantum systems however this is mistaken. Any spherical wave be it classical or quantum is quantised in the sense that depending on the size of the sphere only a certain set pf standing waves are available. 

Of course the biggest misconception about the so called wavefunction of the hydrogen atom is the fact that many people think they are really waves. Well as I have argued many times in quantum mechanics the so called waves are really probability density functions. Despite the misleading pictures of orbitals of the hydrogen atom the electron is not spread out over all space as they would seem to imply. Like any probability density function they just represent the probability that an electron of a certain energy will be seen at a certain place with a given probability. The surfaces represent the boundary at which the electron is likely to be found upto 95% of the time. But an electron is essentially a small point like particle with a definite mass, intrinsic spin and charge. Quantum mechanics does not change this picture. Also it should be remembered that the probability density functions are three dimensional functions so the electron is not in an orbit at all. But can be anywhere within that three dimensional region. Calling these three dimensional probability density functions orbitals is just so misleading. Matters are not helped by energy level diagrams showing the electron jumping from one energy level to another. But it must be remembered that these are energy level  diagrams not positional diagrams. An electron does not jump from one energy level to another. It just changes energy and with each energy level the form of the probability density function changes. 

Anyway having said that Spherical Harmonics just like Bessel Functions are truly amazing functions. There are a whole load of other 'Special Functions' associated with the differential equations of mathematical physics, but anyone who masters the mathematical properties of Spherical Harmonics and Bessel functions will be well placed to understand a lot of physics. 

Finally it is of interest to note that the Spectrum of the Cosmic Microwave Background radiation has been expressed in terms of Spherical Harmonics. Also via the Spherical Harmonic addition theorem there is a close link between Spherical Harmonics and Bessel Functions another example of the interconnectedness of mathematical ideas. Here is my tribute to them 

                                                        Spherical Harmonics

 

                                        Many uses we have found,

                                    For your functions so profound.

                                    The fields inside a sphere,

                                    Have become very clear.

 

                                    You describe the spectrum,

                                    Of an atoms angular momentum.

                                    And the background radiation,

                                    That comes from natures creation. 

        Probability density funtions of an electron outside a hydrogen atom 

                                              

 

Ode to Bessel Functions

Ok here is what I believe is the first ever poem to be published on Bessel Functions. Those who studied partial differential equations will know that if a function is expressed in cylindrical coordinates the separation of variables technique will usually end up with one of the equations resulting in Bessel's equations, named after Bessel who first came across it when investigating planetary ,motion. If the equation is solved by assuming a series solution by the Froebenius method a set of orthogonal functions is obtained called Bessel Functions. These are essentially similar to cosines and sines in rectangular coordinates and any function in cylindrical coordinates can be expressed in terms of them. 

Since their discovery (or invention) they have been used for all sorts of purposes to model the acoustic waves in a cylindrical tube, the electromagnetic waves in a cylinder, heat conduction in a cylinder and many other things so it is important to become familiar with them, Unfortunately the Open University Applied  maths and physics courses only mention them in passing and their properties are hardly developed. A good little book although out of print but easy to find on Amazon which discusses Bessel functions and their properties is by Sneddon 

https://www.amazon.co.uk/Special-functions-mathematical-physics-chemistry/dp/B00316KJ0Q

 Perhaps the most amazing application of Bessel Functions was their use by Crick to deduce the diffraction pattern of the Double Helix. He realised that if DNA was a helix then it's diffraction pattern could be expressed in terms of Bessel Functions. Resulting in a 'Saltire' like pattern as shown in the photograph below. What is even more amazing was the fact that if there were two helices intertwined with each other then the 4 peak would be minimised and again this can be seen in the photograph below. So there we have it almost definitive proof that DNA the molecule of life is a double helix and ultimately dating back to Bessel's Functions

I have to add a sorry note to this marvelous tale, the woman who did all the hatd work in obtaining the diffraction photographs was a French woman called Rosalind Franklin, Being a women she was treated really appallingly and Watson in his book makes some quite sexist remarks about her. Alsu unfortunately she died at a fairly young age too late to get any credit when the Nobel prize was awarded. So just as in the case of Jocelyn Bell, sexist males got all the credit for a woman's work. There was a very good Horizon programme about this made in the mid 1980's I don't know if it is still available if you can get a copy it is definitely worth trying to track down. 

Anyway here is the tribute to Mr Bessel and his amazing set of functions 

                                                        Bessel

                                            What an amazing function,

                                            We can use it without compunction.

                                            Any field that’s in a tube,

                                            You have put it in the groove.

 

                                            Aided by your Function,

                                            Franklin, Crick and Watson,

                                            After a long, long trial,

                                            Deduced life’s double spiral.

 

Rosalind Franklin's photograph of DNA, note the missing peak this indicates that there are two helices interwined with each other. 

Mr Fourier and his series

 One of the marvels of mathematics is the Fourier series which shows how to either decompse a complex function into a sum of sines and cosines or vice versa build up a complex function by summing over various sines and cosine functions. This idea has many applications in partial differential equations and any function expressed in rectangular co-ordinates can be expressed in terms of them. The separate sine and cosine terms can be seen as vectors in a function space. In that they are orthogonal to each other thus there is a an amazing analogy between geometric vectors and vectors in function space. This can be extended to other coordinate systems as we shall see in the next two poems. One of the most amazing application of Fourier series is the ability to synthesize complex sounds and make new music. An application that I bet Fourier never thought of. So next time you are struggling to solve a problem in Fourier series just remember how useful they can be. Here is the tribute. 

Mr Fourier

 

You discovered something quite profound,

How to create a complex sound.

From it’s component bits,

To make a wave that fits.

 

All these waves combine,

Into a sound that’s fine.

The possibilities are endless,

To create sounds quite stupendous.

A Moog synthesiser must be much more fun playing around with one of these than the digital synthesisers which you can generate using a computer A concrecte application of Fourier series. 

  

 

Ode To Calculus

 The next few posts will be a tribute to mathematics or more strictly the branches of maths that are useful to physicists. For physicists as a whole mathematics is secondary to nature. It is a moot question whether maths is discovered or created. I tend to the view that it is essentially created which no doubt will surprise people after all I am a scientitst and I use mathematics every day and it works so it must be true. Well to some extent, but mathematics can only ever provide a good approximation to an underlying reality. Anyone who has solved a differential equation will know that of all the possible solutions only one will represent the reality that one is trying to model. Certainly it is amazing how given a few basic facts about the world mathematics can extend the relationships and maybe even uncover relatiohnships that we didn't think were there. Maxwell's equations for example. 

But let's not get carried away there are certain constraints on the world. Signals cannot travel faster than the speed of light, there are only 4 space time dimensions, there is only one universe. Any theory such as superstrings or the many worlds interpretation of quantum mechanics which contradict those basic facts is just fantasy and should be called out as such.Also no amount of mathematical reasoning will explain why the Gravitational coupling constant is what it is or the mass of the electron or any fundamental particle. These are givens and any mathematics which claims otherwise is just over reaching itself. There is no need to invoke as some people do God or the anthtopic principle to explain why the world is the way it is, ii just is. Of course once we know some basic facts we can use mathematics to deduce inter-relationships between those facts and that is amazing. One of the reasons it works so well is that nature is approximately linear and calculus exploirs this linearity

Without the invention (discovery) if you prefer of calculus there would be no explanation of the world as it is.  As it was when  Newton and Leibniz invented calculus the laws of motion were explained and the fact that planets moved in ellipses was also deduced thus being one of the greatest intellectual achievements of mankind. The Open university used to include a couple of units in the old course MST209 which has been superseded by MST210 from which this has been dropped. I think that is a pity if anyone reading this blog would like a copy of the missing units please contact me on 

chrisf19572002@yahoo.co.uk 

as it is one of the greatest intellectual achievements of mankind. If you want to understand how physics works then you really must learn some calculus I think it a shame that the currrent Open university course in physics S217 minimizes the use of calculus in it's units. It is often claimed by Arts students that you are ignorant if you haven't read a certain book or watched say a certain Shakespeare play, Well conversely I would argue that if you don't know calculus to at least A level maths and why physicists use it to explain how nature works you are equally just as ignorant. Anyway here is my tribute to one of the greatest inventions of mathematics 

Calculus

You explain how nature works,

It’s a task we should not shirk.

Integration and Differentiation

Gives me such exhiliration¹⁾

 

All the laws of motion,

Are explained without commotion.

When we’ve mastered you,

There is so much we can do.

 

1)    I realise not everyone feels the same way J

Jocelyn Bell

 This poem pays a tribute to Jocelyn Bell who with the aid of her phased array antenna the she built and maintained  (Radio telescope) discovered Pulsars  At first she noticed a small amount of persistent noise in her plots which she jokingly referred to as LGM (little Green men). The pulses were regularly and quite rapid. For a while there was speculation that they may indeed be signals from outer space and of course the media latched on to it. However it was soon realised that they could in fact be neutron stars and this was later confirmed. A neutron star is a much more compact object than a white dwarf and represents another stable state of stellat collapse. Oppenheimer and his student Volkoff had worked out that one of the implications of General relativity was indeed this new stable state. One of the signals of a neutron star is the fact that it rotates rapidly (due to conservation of angular momentum) hence the short bursts that Jocelyn Bell measured. In one of the great scandals despite having done all the work it was her supervisor that got the credit and the subsequent Nobel prize. This is typical of the sexism that prevailed in those days and probably still lingers on today. Jocelyn Bell continued to develop her career as an Astrophysicist and was for a time Professor of Astronomy and Astrophysics for the Open University. As I am an antenna engineer it would be interesting if I can find out the details of Jocelyn Bell's antenna and plot some typical radiation patterns watch this space

Anyway her is my tribute to her 

Jocelyn Bell

 

What was that strange pulse from the stars,

Could it be little men from Mars.

No it was a star so dense,

Made of neutrons with mass immense.

 

Spinning round so fast,

Emitting pulses in a blast.

This marvel you found,

With your antenna on the ground. 

  

Chandrasekhar

 This poem is a tribute to Chandrasekhar, probably one of the greatest astro-physicists ever. He was a prolific Astro-physicist probably an Astrophysical equivalent of Ramanujan the Indian mathematician. Like Ramanujan Chandrasekhar came to Cambridge to Study amongst other people Eddington who had demonstrated that Einstein's theory of relativity was correct by measuring the bending of light by the sun during an eclipse in 1919. Also he amongst others worked out the basic equations governing stellar structure which are still used today and was able to reproduce the Herzsprung Russell diagram. However Eddington did not have any idea what happened to a star when it died out. Chandresekhar using the relatively new ideas of statistical physics as applied to fermions and the Pauli exclusion principle worked out that there was a tension between the radiation pressure of a collection of fermions and their graviational attraction. he worked out that if the mass of the star was less than about 1.4 solar masses the resulting configuration would be stable and it's radious would much smaller than the radius when it was active. The matter would be ejected in a glorious explosion giving rise to a supernovae. The final state would be a white dwarf. All this was fine, but the consequence of this was that if the mass was greater than 1.4 solar masses the star would continue to collapse indefinitely and become a black hole (although that name hadn't been used back then). Chandrasekhar presented this idea at a meeting of the Royal Astronomical society. Somewhat unfairly even though Eddington knew about Chandrasekhar's results he waited until Chandrasekher had given his presentation to raise his objections. Of course Eddington's objectons, being who he was, prevailed and as a result the idea of black holes was suppressed for about 30 years. Chandrasekhar after he got his PhD moved to Chicago and continued to do work in Astrophysics and contributed to the ideas of stability in Fluids leading to some early investigations of phenomenon that would be later recognised as prefiguring chaos theory. In the end of course Chandrasekhar was vindicated. He wrote a book on Stellar structure which is still relevant today 

An introduction to the Study of Stellar Structure 

https://www.amazon.co.uk/Introduction-Study-Stellar-Structure-Astronomy/dp/0486604136/ref=sr_1_1?dchild=1&keywords=An+introduction+to+Stellar+structure+Chandrasekhar&qid=1630760779&s=books&sr=1-1

And hid definitive text on black holes including a full derivation of the relativistic equations governing the Kerr metric (one day I'll get round to understanding this :) )

https://www.amazon.co.uk/Mathematical-Theory-Classic-Physical-Sciences/dp/0198503709?asin=0198503709&revisionId=&format=4&depth=1

Although Chandrasekhar does include the health warning that at times the line of reasoning will be quite obscure as there are many leaps in some of the derivations. Perhaps with the aid of a symbolic manipulator such as Maple or Mathematics it might be possible to understand these derivations. 

There is also a summary of his work on instabilities in fluids 

https://www.amazon.co.uk/gp/product/B00C59C7ZA/ref=dbs_a_def_rwt_hsch_vapi_tkin_p1_i0

Finally a good general overview of the history of the development of Black holes and stellar stability is given by Kip Thornes book 

https://www.amazon.co.uk/Black-Holes-Time-Warps-Commonwealth/dp/0393312763

Anyway here is my tribute to this fine Astrophysicist 

                                   Chandrasekhar

 

                                            When a star collapses,

                                            There are no relapses.

                                            If it’s mass is really great,

                                            There is no final resting state.

 

                                            This idea made Eddington frown,

                                            After his great big put down.

                                            In science there was a great lack,

                                            Until people found holes that are black. 

 

Alan Turing

 This is a short poem about Alan Turing, who as many people will know played a part in helping decode the Enigma machine and thus helping to save many lives in the war of the Atlantic. He became interested in the whole idea of computing and whether machines could think just at the right time. It is still an open question whether or not a machine can think but my chess computer regularly thrashes me at chess and I get computers to perform long tedious calculations not just involving numbers but stuff like integrals as well. I believe you can get computers to evaluate Feynman Integrals so for many well defined problems computers definitely can think even if it is not in the way that we do. Has this solved the problem of consciousness well not exactly but it is at least a step in the right direction. 

Unfortunately whilst Turing was good at decoding secrets he wasn't able to hide from the world that he was Gay and indulged in sexual activity which at that time was considered illegal/ Had he just kept in within a Cambridge college or at public school he would have probably got away with it. But it was the fact that he indulged in sex with working class men that seemed to alarm the authorities. So despite being a war hero Alan Turing was convicted and forced to take oestrogen which had the effect of making him impotent. As he couldn't live with this he committed suicide a tragic loss both to the country and the mathematical community. The narrow mindedness of the authorities really deserves condemnation although I suspect that there are still a substantial number of people Evangelical Christians in particular who would want to make Homo-Sexuality illegal again. Well let's hope they don't succeed. 

Alan Turing

 

Build a machine that can think,

Otherwise the ships will sink.

Many lives were saved by you,

And your talented Bletchley crew.

 

Yes you cracked Enigma,

But because of societies stigma,

The secret you could not hide,

Led ultimately to your suicide.

James Clerk Maxwell

 Ok as the first tribute (of many) this poem is about the greatest physicist of the 19th century namely James Clerk Maxwell. Whilst his main contribution was in unifying electromagnetism (the first unification theory) he also developed the Kinetic theory of gases and the theory of colour. When he was 17 he deduced that the rings around Saturn could not be solid but had to consist of small bits of rock. For this he was awarded the Adams prize essay.

 Anyway his main achievement is of course summarising Faraday's work on electromagnetism into 4 equations and then he realised that applying some mathematical transformations using quaternions that electromagnetic waves had to exist. He missed a trick though in that it was thought like all waves there had to be a medium for which the waves had to be propagating through. It wasn't until Einstein came along when it was realised that the so called Ether was not necessary. Indeed electromagnetic waves are self generating in that an oscillating electric field generates an oscillating magnetic field which in turn generates an oscillating electric field so the wave self propagates without any underlying media. Also whilst Maxwell used quaternions it wasn't until Heaviside reformulated them in terms of vector calculus that the structure of Maxwell's equations became apparent. 

Whilst Maxwell's equations are seen as part of classical physics, they are relativistically invariant and are still used today in many fields such as antenna engineering which is my specialist field of study. Furthermore despite the alleged importance of quantum mechanics in understanding the two slit experiment there is no quantum mechanical explanation of the two slit experiment. Indeed it is difficult to see how there could be as Schrodinger's equation is a scalar equation whereas Maxwell's equations involve vector quantities such as Electric and Magnetic fields. There is no concept of the near field from Schrodinger's equation I will expand on these points later. In the mean time here is a verbal summary of the four Maxwell equations 

1) The electic field of a source is proportional to the total charge enclosed by a surface surrounding that source 

2) There is no magnetic charge 

3) If I wiggle a magnetic in front of a conductor there will be an electric current associated with it and the faster the magnetic is wiggled then the greater the electric current will be 

4) An electric current passing through a wire generates a magnetic field 

Put 3 and 4 together and you end up with a wave equation the velocity of which is the same as the speed of light so light is in fact an electromagnetic wave. Truly magical 

Anyway here is the poem hope you enjoy it 

                             James Clerk Maxwell

 

Four basic facts all combine,

To show how fields intertwine.

Static or Dynamic,

It really is magic.

 

A wiggly field generates another,

Which in turn regenerates the other.

With this great insight,

We now understand light.

 

 

Tribute to science

Ok as promised this is the first of many poems which pays a tribute to science and various scientists, The first poem is a general tribute to how science has played an important part in changing life for the better and challenging vested interests. I find it quite depressing that even today there are people who dispute the basic facts of science because it challenges pre-conceived ideas about say creation as in fundamentalist evangelical circles creating a suspicion of science. 

Far closer to home are those people who deny climate change and who are given a platform from the media in the interests of so called balance. Well I am sorry if 95% of scientists have strong evidence for the fact that mankind is accelerating climate change then there is no way a person such as say Nigel Lawson or Donald Trump should be allowed equal media coverage because it suits their interests in order to do so. Their views are minority ones and have nothing to do with science. 

Similarly for those idiots who refuse to take vaccinations the scare about the possible link between MMR and autism has meant that measles has come back when to all intents and purposes it was eradicated. Those journalists such as Melanie Philips who supported Andrew Wakefiled should hang their heads in shame.  Similarly the only way we have any hope of defeating the Corunna virus is to develop vaccines and if need be take boosters as and when necessary. If you want to end the lockdown restrictions then make sure you take the vaccine. 

Finally I strongly resist the current trend of the transgender lobby to defy science and for men to claim that because they feel like a woman they should be allowed to invade women's private spaces such as toilets or womens showers. Or take part in women's sports or women only short list/ It is biologically impossible for a man to change into a woman no matter how many hormones they take and anyone who claims otherwise is just denying science. I hope this nonsense will stop soon but the way things are going it seems highly unlikely. I have a number of female colleagues who are genuinely worried about this and I hope they win their battles against this unscientific nonsense. 

 Anyway here is my Tribute to science in general I hope you enjoy it. 

                                                         Ode To Science

In the midst of Man’s inanity,

Science brings a voice of sanity.

Getting rid of superstition,

You improve our condition.

 

Vested interests they despair,

When the truth is bought to bear.

Only with you will we find,

What it is that drives Mankind.

Tribute to David Attenborough part 2

 This is the second batch of poems paying tribute to David Attenborough's life series. This time we cover 

1) Reptiles 

2) Birds 

3) Mammals this one focuses on our nearest primate the chimpanzee

I dare say others will follow in the fullness of time But 'Uncle David' for your achievement in helping bring nature close to us. I salute you 

The first poem is on reptiles which are closely related to dinosaurs what happened to the dinosaurs is an interesting scientific problem and as yet there is still no answer. It is amazing to think that they ruled the earth for 150 million years whilst human civilisation has only been around for about 100,000 years and recorded history about 20000 at the most. Anyway here is the poem 

                                     Reptiles

 

                                            Relics of an age long gone,

                                            Nature still allows you to throng.

                                            From the lizard to the crocodile,

                                            You’ll still be here for a while.

 

                                            When Dinosaurs ruled the earth,

                                            You were there, at their birth.

                                            Now they are gone, I wonder why,

                                            What it was, caused them to die.

And here is a picture of a magnificent crocodile best keep your distance 

The second one is on birds very inspiring creatures with their ability to fly, they have always inspired us long may they do so.

Birds

 

From the eagle that soars,

To the talking Macaws.

In variety there are plenty,

Without you life would be empty.

 

When you take flight,

It’s a truly wonderful sight.

Helping us to aspire,

To all that we desire.

Here is a picture of a an Eagle wonderful 

Finally here is a poem about the species closest to us namely the chimpanzee. I hope the days when chimps were used to entertain us via chimpanzee's tea parties are long gone. Never to return 

                                     Chimpanzees

 

We used to laugh at the Chimpanzee,

Making them drink our tea.

But now we know this primate,

Is our species closest mate.

 

Closer to them than we thought,

Let us treat them as we ought.

Open the zoos and let them free,

Swinging from tree to tree.

 

I will start publishing a whole load of poems about science and mathematics soon 

Tribute to David Attenborough Part 1

 Hi these two poems have been inspired by David Attenborough's 'Life series' a truly amazing set of DVD's There are five specialist titles in the series 

Life of Plants

Life in the Undergrowth

Life in Cold Blood 

Life of Birds 

Life of the Mammals 

These two poems are about flowers and insects. Which complement each other nicely. It is normal to stress how nature is essentially about competing species fighting for scarce resources. However plants and insects cooperate with each other in a way that is beneficial to both. One doesn't have to be a creationist or particularly religious to admire the ingenuity in which both insects and plants cooperate with each other. How does a particular orchid know for example how to disguise it's flowers as a female wasp so that it will attract male wasps. Truly amazing and without 'Uncle David' this amazing fact about nature would be hidden from us. So a truly amazing series and if you haven't seen it then I would urge you to get hold of it. Start with life on earth the best introduction to evolution there is even if the photography is a bit dated.

                                                Flowers

 

                                            Your blooms are such a sight,

                                            Giving us all great delight.

                                            But all that, you have planned,

                                            To get insects, on you to land.

 

                                            The little bug might think,

                                            They have had a nice free drink.

                                            But for you, it’s also great,

                                            Now you have a chance to mate.

 

                                   

  

And here is the one on insects generally speaking we avoid insects and indeed David Attenborough shows some pretty gruesome ones including a giant millipede which preys on bats. However without them the soil would not be aereated and of course the flowers would not be pollinated.

                                             Insects

Nasty horrible little things,

That quite often stings.

But if you did not toil,

The earth would have no soil.

 

The flowers would not grow,

They would not put on a show.

On you so much depends,

To help nature meet it’s ends.