Anti-Ode to a Seagull

 In contrast to the cute squirrels I wrote an Ode to last week Edinburgh is plagued by Sea Gulls who make a llot of noise and scavenge amongst the bins. Mind you if the council got it's act together and made bin collections more regularly the problem wouldn't be as great as it is. Anyway here is an Anti-Ode to a seagull

                                      

                                     Anti-Ode to a Seagull

                            Stomping on the roofs

                                    With your great big hooves

                                    What a noise you make

                                    It is more than I can take

 

                                    Rummaging around the bags

                                    Looking out for scrags

                                    Why not let us be

                                    And return back to the sea. 

And here is not such a pretty picture 

Hope this gave you a laugh anyway. Wonder if I have invented a new genre of poetry 

Ode to a Squirrel

During lockdown I have been working from home. However from my living room window there are some trees close by  I often see squirrel's running about and I was inspired to write this little ode which I hope brightens your day 

                                                    Ode to a squirrel 

                                            Little squirrel running around 

                                            In the trees above the ground

                                            What a pleasant sight to see 

                                            You bring such delight to me.

                                            O what fun it must be 

                                             Running around so gay and free

                                              Sometimes with a friend or two

                                             Now the trees belong to you 

And here is a pretty picture 

Take care soon this nightmare will be over 

Solutions to Cambridge 1st Year NST Maths paper 1A 2019

Hi every one I have finally finished my first set of solutions to a Cambridge mathematics paper the paper in question was the first paper of 2019 which was given to the science students. The solutions are given here and I hope you enjoy reading through them

https://drive.google.com/file/d/1vLNuqpxyomuiHsR8nYJ1P-AGPG-XTSTh/view?usp=sharing

Some general comments 

On the whole I thought this paper was quite challenging certainly because the questions were a lot more tricky and involved than say a typical OU paper such as MS224 (although this years paper by all accounts seems to have been really difficult as the OU are still experimenting with the format for remote exams. My sympathies to those who found it difficult and when it eventually comes out I will have a look at it myself and make comments on it) 

So what was in the paper the first set of questions was a set of 10 which were essentially A level topics and on the whole they were relatively straightforward. Although you do have to have integration techniques and trig identities at the tip of your fingers. There is really no time for much thought and one question involved a tricky integration by parts. Something which this paper seems to love testing people on I think there must have been about 6 or 7 times that I had to use this technique. As a result my grasp of this technique improved a lot and I can almost just write down the answer now whereas before I would struggle. Candidates have to answer all of these questions and a bit like multiple guess questions in the MS224 papers no marks are given for showing the working

The next section is a set of 10 questions from which the candidates can choose up to 5 the last two questions being reserved for those who took a more advanced option. The elite of the elite as it were it must be a bit disappointing to be told right at the outset of your undergraduate career that you are not part of this special group. I can see many potential traitors to our country arising from this slight 😅   Anyway the system is what it is and for us working through these papers we don't have to worry about such distinctions. Unfortunately there was one part of this section which I just could not see what the question was asking and hence I just have to accept that I will remain a pleb for the rest of my life 😅 I digress 

Question 11 had four parts on complex variables. I really enjoyed the first part which involved using De Moivre's theorem to reduce a complex trigonometric identity to a product of powers of cos a and sin b. In the course of this I found a quick way of writing the expansion of cos(nx) or sin(nx) in terms of powers of cos(x) or sin(x) using Pascal's triangle. I have described this in detail in an earlier post and now feel I can tackle this sort of problem with ease. The other parts were straightforward enough but one involved a set of trigonometric identities for the  Tan Function something I am not sure I would remember in an exam/ The final part was a bit difficult to decode what the question was asking but it involved the mapping between two complex variables and I got it out eventually. 

Question 12 was a long tedious question on volume integrals with tricky boundary conditions, It went ok but it is not the sort of question I enjoy and it took me two or three attempts to arrive at the correct answer.

Question 13  was a question on first order differential equations and exact differentials. I generally prefer  second order differential equations but  that will have to wait till the next paper Two parts involved using the integrating factor method and one involved a not so obvious substitution in order to get the final integral out. The last part involved a lot of partial differentiation to show that a function obeyed Laplace's equation in spherical polar coordinates. So just like question 12 above this would have been tedious and I doubt whether I would have got it out in time 

Question 14 was divided into two parts the first part asked you to calculate the variation in the relativistic formula for energy in terms of the variations for mass and energy and then given the velocities of two particles work out the relative mass of these particles wrt to each other. This was OK if a bit novel to me and it was difficult to tell whether I had got the signs correct. The second part of question 14 involved a really tedious exercise in partial differentiation of functions of functions. Whilst in principle this type of question is relatively straightforward in practice ir can get extremely tedious so best avoided in an exam and I found myself losing the will to live

Question 15 was a question on Taylor series and expansions. Again relatively straightforward and tedious you were allowed to use known Taylor expansions but could anyone off the top of their head remember if they ever knew the Taylor expansion for sech(x). I had to derive this myself. 

Question 16 was an unusual one about probability involving the various probabilities of drawing a blue ball from a  bag and then deciding to return it or not with a certain probability and then drawing another ball. Why anyone would think this is something useful to do is beyond me. It certainly wouldn't help you win the lottery😅   I hadn't really done much of this type of question before so it was useful to learn the various formulae for conditional probability and how to draw a probability tree using MATLAB. So whilst the actual activity described in the question is quite pointless, it was quite instructive to work out the various probabilities involved again this would have been really difficult to complete in the time given

Question 17 was a whole bunch of questions on integration some of which seemed impossible at first sight but you just have to persist and I strengthened my integration skills a lot by doing this question

Question 18 was a long boring question on matrices the first part solving a matrix equation for 2x2 matrices. The second part at first sight was quite intimidating as it involved a long introduction to set the scene for testing a hypothesis about the size of worm lengths. It turned out that in it's long winded way the question was taking the examinee through the method of least squares fit. So it turned out to be relatively interesting but it really is not the sort of question that any normal human being can just rattle off under exam conditions

Now we come to the questions for the elite. 

Question 19 This was split in two parts the first question being a question on the convergence or not of various series. This was quite straightforward if you are up to speed with regards to the techniques but this is really the sort of question that belongs in a pure paper not a mathematical methods one. I suppose the idea is to get their students to think a bit about real analysis. But they are better off doing a full blown course on it rather than spending just a couple of lectures or so. I would recommend the OU course M208 as giving a far better understanding of real analysis than this sort of rapid overview.

The second part was my nemesis for this paper it involved relating the Newton Rhapson method to a Taylor expansion and estimating the errors with each iteration. I was unable to understand what was going on. However given that the function was a quadratic equation for which the roots are known exactly I was able to provide an estimate of the error at each stage but the resulting expressions looked nothing like the expressions in the paper. If anyone knows the answer to this question I would be grateful. I can't see the point of applying the Newton Rhapson method to a quadratic equation anyway 

Question 20 This was the final question Yippee and was on Integration the first part was showing how the area of a parabola could be approximated by a series of rectangles under the parabola and showing how in the limit the exact expression could be obtained. I got this question out but again it involved a lot of tricky algebra the thing to do here  as Corporal Jones from Dad's Army would say is 'DON'T PANIC' and just see it through even if at first sight your answer appears to be going nowhere.

The last two parts of the question involved something which I hadn't really come across before namely differentiating functions under an integral sign involving variable limits. Apparently Leibniz derived a formula for this many years ago (what a clever fella that man was 😅 ) and once you know this formula the question was quite straightforward. The final part involved yet another tricky substitution making use of the obvious fact  😅that sin(x) = tan(x)/sec(x)  and the derivatives of both sec(x) and tan(x). 

So what would it take to do really well at this exam essentially a real fluency in techniques of integration, trigonometric identities, hyperbolic functions, their derivatives and identities, Taylor series, differentiation using the product and quotient rules, Being able to partially differentiate functions of functions using the chain rule very quickly and a good deal of persistence. Not for the faint hearted but an exercise well worth doing. I certainly got a lot from this exercise even though I am not particularly fluent at the techniques. I guess like good musicians you have to just practice practice until the techniques just become routine. I look forward to doing the second paper in this series which will cover such topics as vector calculus surface and line integrals, and second order differential equations (yippee my favourite technique at this level) I am not very confident with things like the divergence theorem and surface and line integrals I hope by the time I have finished the second paper I will become more so and no doubt I will learn a whole load of new techniques as well I encourage any one reading this blog to have a go themselves. The last paper took me about 6 months to complete and write up I hope the second one wont take so long 

 

Now for Something Different (Shakespeare again )

 Hi about 10 years ago I signalled my intention to watch all the Shakespeare Plays However this project stalled. Recently however a friend of mine asked me to list my favourite Shakespeare plays and give a recommended order of what ones to watch. Whilst I am no expert I have loved Shakespeare since my student days over 40 years ago when I was enthralled by the BBC Shakespeare Series especially Derek Jacobi playing Hamlet. It was the first time I had really understood a Shakespeare play. Anyway as a bit of a distraction I will attempt to resume my project but narrow the focus down to concentrate on the more well known ones following the list below then maybe fill the rest later. Anyway as a start here is the list

Comedies 

A MidSummer Nights Dream

Twelfth Night. We did this at school and I played Sir Toby Belch, the drama teacher obviously thought it would bring me out of myself as I was one of the school swots. 

There are of course plenty more such as As you like it or Loves Labours lost, but these are representative and whilst the Comedies are entertaining, they aren't really what Shakespeare is about. 

Dark Comedies (Problem plays)

The Merchant of Venice. This is a truly problematic play one wonders why it was classed as a comedy it raises all sorts of issues to do with the Anti-Semitism of the time. In mitigation Shylock has the best speech where enraged by the taunts of two Viennese citizens he utters the immortal lines 'If you prick us do we not bleed...'; Which can be seen as Shakespeare endorsing the common humanity of all people. On the other hand Shylock is definitely a caricature of the Jews as they were perceived in late Elizabethan England. I will explore the issues raised in a later post. 

Measure for Measure 

Another problematic play, in the light of the Harvey Weinstein scandal, it shows up the abuse of those in postions of power have those in power with regard to those in more vulnerable positions especially women. Probably my favourite of the problem plays. 

The so called Histories 

Shakespeare obvously knew what side his bread was buttered  on and wrote these plays essentially to justify the Tudor line. Thus Richard III is made to be a villain rightly deposed by the Duke of Richmond, who later became Henry VII and who wasn't exactly free from Tyranny himself. However the plays whilst having dubious claims to historical accuracy make good drama. I have just invested in the Hollow Crown and look forward to watching it. Despite their obviously sympathetic view of the Tudors it is interesting that Richard II was staged just before the Essex rebellion against Queen Elizabeth in her later years.

Shakespeare wrote his histories in two installments the first set written in his early career as a playwright involved the three parts of \Henry VI followed by his greatest history Richard III, then a few years later he wrote the series starting with Richard II then the two parts of Henry IV and culminating in Henry V. These plays are far more mature than the parts of Henry VI and also introduce a bunch of rogues Bardolph, Pistol and Shakespeare's greatest comedy character John Falstaff, These characters are seen as drinking companions of the young Prince Hal and often make Falstaff the butt of their jokes. Having said that for all Falstaff's buffonery he probably has the most pragmatic attitude to war and the dubiousness of military glory enshrined in his speech which includes the lines 

"Who hath honor then, Why he who died on Wednesday, doth he feel it No, doth he hear it No 'Tis insensible then. Yea to the Dead ... I'll have none of it" 

Needless to say despite those sentiments, it doesn't prevent Falstaff from falsely claiming that he killed Hotspur to get a bit of glory for himself.

Of course when Prince Hal becomes King Henry V he has to disown himself from Falstaff and his companions which he does most brutally. Henry V relates how he won the battle of Agincourt and defeated the French, It borders on bombastic Nationalism but is mitigated by scenes before the battle of Agincourt where soldiers debate the fact that whilst they know nothing of the rights or wrongs as to why they are fighting, they will bear no responsibility but the responsibility lies with the Monarch who has led them into battle. 

 As a break between the end of Henry IV part II and Henry Vth, the Merry Wives of Windsor provides comic relief where Falstaff short of money (as always) attempts to persuade two rich women Mistress Page and Mistress Ford to marry him even though they are already married, Needless to say his attempts fail badly. In itself this play isn't that significant but as light relief to the history plays it is worth watching now and again 

Jumping ahead chronologically to Richard III then, Shakespeare portrays him as a villain. but attributes this to his jealousy compared to his other brothers that he is not attractive to women due to his physical deformities. In one of the most brilliant opening speeches, Richard III decries the fact that whilst he is not made for 'sportive tricks' his brother Edward IV, currently King, capers nimbly in his mistresses chamber. Thus long before Freud, Shakespeare has identified a psychological motive for a persons evilness. Of course historically it's all tosh. The recent finding of the body of Richard III in Leicester has established that he had no real deformity, but of course the Tudors needed someone to justify their somewhat dubious ascension, to the throne. Despite the lack of historical accuracy, no one can deny the powerfulness of the play as drama, just take it with a pinch of salt. 

So in my humble opinion I would suggest that those wanting to familiarise themselves with the Shakespeare historical plays should start with Richard II then work systematically through the plays culminating in Henry V. Then jump to Richard III. The parts of Henry VI can come later to fill in background to the character of Richard III, It is interesting to note that the Hollow crown has reduced the three parts of Henry VI to two so I suspect that there will be many cuts I will let you know 

The Roman Plays 

Julius Ceasar and Cleopatra, Whilst these plays are interesting they aren't exactly mainstream but still worth watching or reading. I will give more details as to what I think in future posts 

The Tragedies 

Now we come to the core of Shakespeare,  In comedy despite a temporary disruption everything returns to so called normal reinforcing the status quo however unjust. Thus Midsummer nights dream poses no real challenge to the patriarchal view that daughters should marry men whom the father chooses. The Merchant of Venice ends with a conversion of Shylock to Christianity and no challenge to the Anti-Semitism of Venice at the time and so forth. 

Tragedy blows this consensus apart, there is no normality to return to as practically everyone is killed at the end. Thus if the Arts are to be more than entertainment, tragedy has the potential to make people question the status quo. Hence in my opinion tragedy, speaks more powerfully to us, than comedy ever will. Shakespeare's tragedies provide a powerful critique of society as it was without the sycophancy of the history plays or the facile resolutions in many of the comedies. In recommended order I would single the following 

Romeo and Juliet - Shakespeare's first big hit, but fairly youthful it has inspired many composers such as Berlioz and the theme formed the basis for Leonard Bernstein's West side story 

Macbeth Written in 1606 just after the gunpowder plot and the ascension of James VI of Scotland at first sight it might seem surprising that Shakespeare managed to escape censure for this until one realises that James claimed to be descended from Banquo and not Macbeth. So another sycophantic play. 

Othello We have seen Shakespeare as Anti-Semitic now he is racist. Having said that Shakespeare shows how jealousy arises from a persons basic insecurity and the tragic consequences that follow In Iago we have along with Richard III Shakespeare's greatest villain. There is a good production from the RSC with Iain McKellan as Iago 

Hamlet. It is a cliche but this is probably Shakespeare's greatest play the psychological depth of the characters prefigures Freud by almost 300 years. My favourite version is that by Kenneth Branagh with a close second being the BBC production that got me into Shakespeare in the first place with Derek Jacobi. The advantage of the Brannagh Version as opposed to the one starring Laurence Olivier is that it doesn't cut the political scenes involving Fortinbras. Plus some absolutely gorgeous music.

King Lear Probably the bleakest of all Shakespeare's tragedies. The BBC production has Michael Horden playing King Lear, There is also a Russian Version which emphasises the bleakness by the director Kozintsev. Just as Hamlet is something that anyone who aspires to be a leading actor must pass through King Lear is for actors at the end of their career. 

Pastoral Plays 

This is a genre that Shakespeare came too late in life, and represent the autumn of his working life the two main ones are a Winter's tale and the Tempest. In the Tempest Prospero can be seen as Shakespeare leaving the stage. I will write more about them in later posts 

So there we have it a rough guide to the greatest sexist, racist, anti-Semitic, royalist sycophantic, bombastic nationalistic playwright that ever lived 😊😊

Cambridge MAFFS

How does one make sure one's mathematical skills are up to date and relevant?  One way would be to aim to do past papers from relevant mathematics courses once a year as they come out.  Such a set of papers and resources for study of these papers is provided by the Cambridge University mathematics department. This post collects together a few resources for self study of the  Cambridge Mathematical Methods courses for Scientists (NST). I am continually amazed at the generosity of the Cambridge Maths site in offering to the public these resources

The past papers  are collated here. 

https://www.maths.cam.ac.uk/undergradnst/pastpapers

 The syllabus for both years is given here 

https://www.maths.cam.ac.uk/undergradnst/files/misc/NSTschedules.pdf

There are two courses one for first year students and one for second year science students who want to study mathematics in more depth. 

The recommended text book is Riley Benson and Hobson 

https://www.amazon.co.uk/Mathematical-Methods-Physics-Engineering-Comprehensive/dp/0521679710

Which has a solution manual which covers the odd problems in great detail so ideal for self study 

https://www.amazon.co.uk/Student-Solution-Mathematical-Methods-Engineering/dp/0521679737/ref=pd_lpo_14_t_0/262-4198048-8655769?_encoding=UTF8&pd_rd_i=0521679737&pd_rd_r=c25e3e37-5d85-425d-ae5f-e2b662f55a36&pd_rd_w=5uBAB&pd_rd_wg=upWCR&pf_rd_p=da0677f5-a47b-4543-8b54-10be576b8f26&pf_rd_r=2N0058F97ARMT25NE1M6&psc=1&refRID=2N0058F97ARMT25NE1M6

Investing in these two books would cover most mathematical methods that a physicist is likely to need in their careers. However it would probably take a good few years to do all the examples in Riley Benson and Hobson so it is more for reference than anything. 

Buried in the vaults of the Cambridge Mathematics department are the following resources 

Answers to first year past papers can be found from this website 

https://www.robinson.cam.ac.uk/iar1/teaching/index.html#nst1a_maths

Unfortunately I have not been able to find any lecture notes or the example sheets. However if you have Riley Bence and Hobson you should be able to find the techniques that you don't know already covered there. However having the answers to the exam questions is obviously helpful 

Year II is much better served the lectures occur in three parts 

The first part is covered by a set of lectures from  Dr Simon Cowley who from his web site seems quite an engaging person

http://www.damtp.cam.ac.uk//user/sjc1/index.html 

Lecture notes for the various courses he has taught over the years can be downloaded from here 

http://www.damtp.cam.ac.uk//user/sjc1/teaching/

The ones of main interest are the notes for the first part of the Cambridge NST Part IB 

http://www.damtp.cam.ac.uk//user/sjc1/teaching/

This covers things such as vector calculus, matrices partial differential equations Fourier Transforms and solution of differential equations,  by series including a survey of Legendre Polynomials, Green's Function and a bit on real analysis (for physicists not mathematicians 😅). This is more or less equivalent to MS224 and the mathematical methods part of MST326 but also includes Green's Functions and goes into more detail about the solution of differential equations by Series solution especially the Frobenius Method. 

Notes for the second lecture course in Part IB are given by Dr Hunt 

https://www.damtp.cam.ac.uk/user/reh10/lectures/

Scroll down to the Mathematical Methods part 

He also gives the examples sheets that were part of the course and hints for their solution 

This second part covers variational principles, solutions to Poisson's equation and a summary of the basic techqniques of Complex Analysis again taught as Physicists would use them rather than a Pure Mathematician. There is a bit of overlap with the OU course MS327 but neither MST326 or MS327 cover inhomogeneous partial differential equations in any depth. 

Finally the third part covers group theory and representation theory for physicists including an analysis of multi-mode oscillations. MS327 covers multi-mode oscillations but do not relate it to group theory. 

https://www.damtp.cam.ac.uk/user/examples/N23L.pdf

The examples sheets for the whole year are also given scroll down to the bottom 

Ok  I aim to have generated a complete set of solutions to the 2019 papers by the end of the year. This is a nice way IMHO to extend ones mathematical ability and keep it topped up. Overall the topics covered here in terms of mathematical knowledge overlap with MS224, MS327 and MST326 but also extend it to cover topics such as

1) Solution by Series of Differential equations including the Frobenius Method

2) Green's Functions and their application to Partial Differential equations 

3) Complex Analysis (although to be fair if you do M337 you will do far more than is given here) 

4) More detail on Special Functions including Legendre Polynomials 

5) Group Theory 

Also I suspect the questions will be harder than the OU exam questions 

Anyway I hope this post will encourage you to extend and revise your mathematics skills. 

De Moivre's Theorem and Trig Identities

 Hi Guys and Gals Hope you have a great New Years eve party if you are having one. Hopefully 2021 will see the end of Covid and life can get back to normal. IMHO the restrictions have been unnecessarily punitive and the shut down of the pubs, theatres and concert halls has been totally unnecessary anyway I'm not here to gripe about the restrictions laid down by Mrs Oliver Cromwell (aka Nicola Sturgeon) up in Scotland during the lockdown. I have been amusing myself to see if I can do some Cambridge Maths past papers set for their Natural science students and available here 

https://www.maths.cam.ac.uk/undergradnst/pastpapers/2019

In doing the 2019 1st paper I came across this question which at first sight looks nigh impossible the question was 

Use De Moivre's theorem to write out the expression 

$$ 8 cos 6\theta + 15sin 4\theta sin 2\theta $$ in terms of $ cos \theta$ and $ sin \theta $ 

However I discovered for myself a simple method to write out the expansion of $cos (n\theta)$ or $sin (n\theta) $ Almost instantly from Pascal's triangle 

First let us recall De Moivre's theorem it states that 

$$exp(in\theta) = [exp(i\theta)]^n $$ using 

$$ exp(in\theta) = cos(n\theta) + i sin(n\theta) $$ 

it is seen that 

$$cos(n\theta) = Re [cos(\theta)  + i sin(\theta)]^n $$ and 

$$sin(n\theta) = Im [cos(\theta) + i sin(\theta)]^n $$ 

Ok that looks really messy how do I remember the coefficients in a binomial expansion and how do I pick out the Real and Imaginary parts of each coefficient and all in about 5-10 mins which is all I have time to answer the question

Well a short cut is as follows Recall that the binomial coefficients in a binomial expansion are given by the rows of Pascal's triangle . Namely 

                                n = 0                                 1

                                n  = 1                              1     1

                                n  = 2                         1     2    1

                                n  =  3                     1     3     3    1

                                n  =  4                   1     4     6     4    1

                                n =  5                1     5    10   10   5     1

                               n   = 6            1    6    15     20    15  6      1

where each number in a given row is given by adding the two numbers above it and the coefficient corresponds to the following expansion 

$$ (a + b)^n = C_1  a^n + C_2  a^{n-1} b + C_3 a^{n-2} b^2 + ...........C_{n-1}.a b^{n-1} + C_n b^n$$

The coefficients are just the corresponding element in the nth row of Pascal's triangle. So for example 

$$(a + b)^5 = a^5 + 5 a^4 b + 10 a^3 b^2 + 10 a^2 b^3 + 5 a^1 b^4 + b^5 $$

Now to apply this to De Moivre's theorem we would have 

$$[(cos(\theta) + i sin(\theta)]^n = C_1 cos^n (\theta) + C_2 cos ^{n-1}(\theta) (i sin(\theta)) + $$

                                                   $$   C_3 cos^{n-2}(\theta) (i sin(\theta))^2 + ..... $$

So the odd coefficients in the expansion will correspond to the real part and the even coefficients will correspond to the imaginary part of the expansion, Furthermore as $i^2 = -1$ and $i^4 = +1$ the coefficients will alternate in sign 

So this gives us a nice quick method of writing out $cos(n\theta)$ and $sin(n\theta)$ in terms of $cos(\theta)$ and $sin(\theta)$

For a given n work out the corresponding row in Pascal's triangle then the expansions of $cos(n \theta)$ and $sin(n \theta)$ are as follows 

$$ cos(n\theta) = C_1 cos^n(\theta) - C_3 cos^{n-2}(\theta) sin^2(\theta) + $$

$$ C_5 cos^{n-4}(\theta) sin^4(\theta)+ ...$$

$$ sin(n\theta) = C_2 cos^{n-1}(\theta) sin(\theta) - C_4 cos^{n-3}sin^3(\theta)+ .... $$ 

Hence given a horrendous expression involving $cos(n\theta)$ and $sin(n\theta)$ the expansions of these functions in individual powers of $cos (\theta)$ or $sin(\theta)$ can be written down by sight 

So for our problem we have 

$$sin(2\theta) = 2 sin(\theta) cos(\theta) $$ as the second row of pascals triangle is 1 2 1 and the even coefficient is 2 

$$sin(4\theta) = 4 cos^3(\theta) sin(\theta) - 4 cos(\theta) sin^3(\theta) $$ as the fourth row of Pascal's triangle is 1 4 6 4 1 and the even terms are 4 and 4 

Finally 

$$cos(6\theta) = cos^6(\theta) - 15 cos^4(\theta) sin^2(\theta) + 15 cos^2(\theta) sin^4(\theta) - sin^6(\theta) $$ 

as the 6th row of Pascal's triangle is 1 6 15 20 15  6 1 and the odd numbered coefficients are 1 15 15 and 1 

So 

$$8 cos(6\theta) =8[ cos^6(\theta) - 15 cos^4(\theta) sin^2(\theta) + 15 cos^2(\theta) sin^4(\theta) -$$ $$sin^6(\theta) ]$$ 

and 

$$15sin4\theta sin 2\theta=15 [4 cos^3(\theta)sin(\theta)-4cos(\theta)sin^3(\theta) ] $$

$$ \times 2 sin(\theta)cos(\theta)$$

This simplifies to 

$$ 15 \times  8 [cos^4(\theta)sin^2(\theta) - cos^2(\theta) sin^4(\theta)] $$ 

and it is seen that the mixed terms in $cos(\theta)$ and $sin(\theta)$ cancel leaving 

$$ 8 cos 6\theta + 15sin 4\theta sin 2\theta = 8(cos^6(\theta) -sin^6(\theta)) $$

So next time you are faced with a seemingly horrendous expression involving $sin(n\theta)$ or $cos(n\theta)$ just remember Pascal's triangle and De Moivre's theorem.

Hope every one reading this has a great new year I will write another post over the weekend summarising my plans for 2021 and hopefully achieving them :) 

Foundations of Quantum Mechanics Norsen a review

Hi Sorry I haven't posted for a while (over 2 years) I have not been as focused as I should have been so apologies. I hope you are all surviving the COVID lock down about which the least said the better and the sooner it is over the best for all of us.

I found myself reviewing a book on the Foundations of quantum mechanics for Amazon which is fine as far as it goes but I do think it is fundamentally flawed

The book is

https://link.springer.com/book/10.1007/978-3-319-65867-4

and you can download it for free for now via the springer website which is really helpful anyway here are my thoughts on the book

On the whole this is a pretty good survey of the foundations of quantum mechanics and the alleged problems associated with it's Interpretation and the ways out of it. However it is arguable that all one needs to interpret quantum mechanics successsfully are the Born Rule and the link between the eigenvalues of the Schrodinger equation and the energy levels of the system under consideration

If the Born rule is taken seriously, then it implies that the solution to Schrodinger's equation, the so called wave function is in effect the square root of a probability density function or probability amplitude. However unlike most square roots of a probability density functions, in some cases this may be a complex function and one can only get a measurable quantity by taking the modulus squared of the solution to Schrodinger's equation. If there is more than one possibility then to get the overall probability for a given situation one must add theprobability amplitudes for the two or more possibilities together before taking the modulus squared of the wavefunction to get the overall probability density function for the situation. As these can be complex functions then there is interference between the two or more possibilities and this interference is seen in many quantum systems. All of this is explained (or described) in Feynman's lectures on quantum mechanics, which surprisingly the author does not mention.

The author does a good job in describing the ignorance interpretation of Born, Einstein and Schrodinger which essentially claims that the wave function is not a physical entity but more a reflection of possible outcomes on measurement. He also goes onto show problems with interpreting the wave function as something physical. However for reasons which he does not really clarify goes onto reject the ignorance interpretation. This is despite the fact that it avoids all the problems with the collapse of the wavefunction, if taken literally as a physical phenomenon and one does not need to invoke spooky action at a distance to explain the violation of the Bell inequalities. Indeed the latter half of the book is the most disappointing, having shown the problems with interpreting the wave function as a physical entity he then embraces a literal interpretation of Bohm's version including the rather tendentious claim that in situations such as the Aspect experiment, really does involve a causal influence from one part of the system to another at speeds faster than the speed of light.

This is not really plausible despite the claims of many people in the popular literature. From the ignorance interpretation, the explanation is really quite simple. Prior to making a measurement observer A will have a 50% of measuring a photons or electrons spin in one direction but does not know which, so his wave function is constructed in such a way to reflect this 50% chance. However after his measurement he will know that his spin is in one direction, and he instanteanously knows that another observer will measure the spin of the other particle to be in the opposite direction. But until B makes his or her measurement B will not know the outcome of his measurement/ There is no signalling, the other particles spin was set at the moment of emission due to the law of conservation of angular momentum and to deny otherwise is to make a mockery of the conservation of angular momentum which plays such a major part in understanding particle physics today. As there are no hidden variables A or B will never know on a given measurement what he or she will measure, only that if a sufficient number of measurements are made 50% of the time A will measure the spins to be up and 50% of the time they will down and vice versa for B, It does puzzle me greatly that this simple solution is claimed to be ruled out by Bell's analysis by many books including this one. But does this interpretation of the wave function for situations actually violate the quantum mechanical predictions. of course not as it is using the wave function that makes the correct prediction.

 Looking at the assumptions behind the proof of Bell's inquality as many people do, doesn't shed any light on why the quantum mechanical explanation works as we know Bell is analysing the phenomenon in a classical manner. Despite the overblown rhetoric by Bell and others, all he managed to show was that the joint probability distribution for the particles spins could not be separated into a product of the probability distributions of the individual particles spins. But it is well known in statistics that if a probability distribution for a multivariable system cannot be factorised into a product of the individual variables probability distribution functions then the variables are not independent. But we know this for the particles spin components as the total spin must obey the conservation of angular momentum. If there were signalling as Bell claims why doesn't any signalling (a Green's function say) term appear in the actual maths which predicts the measured correlation. 

Bell himself at the end of his 1964 paper claimed to have shown that it is only if one wishes to go beyond a purely statistical intepretation of quantum mechanics then superluminal signalling must be involved in situations such as the Aspect experiment. This may or may not be true, however if one wishes to avoid invoking superluminal signalling then one is stuck with a statistical interpretation to which there has been no experimental refutation

For this reason I can only give the book 3 stars despite the fact that it is extremely well written and probably is one of the most accessible accounts of the various positions informing the current debate today

I am currently working on completion of my calculation of neutrino proton scattering which was the final piece of the jigsaw in establishing the quark model and I will hopefully publish my results by the end of June. Even though no one has yet liberated a quark the results from these experriments carried out in the early 1970's gave evidence that quarks had fractional charge which is really quite amazing. 

Also I have some thoughts on the two slit experiment which I will publish soon. One of the things that most accounts miss is that the characteristic interference pattern is in the far field, However in the near field there is no interference, thus the idea that a photon or electron passes through both slits simultaneously is clearly incorrect. 

In the mean time keep sane