tag:blogger.com,1999:blog-78972354238122776832018-02-25T13:30:29.722+00:00Ramblings of a Short Fat Failed PhysicistChris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.comBlogger165125tag:blogger.com,1999:blog-7897235423812277683.post-36336774622929963892018-02-11T21:42:00.001+00:002018-02-11T21:55:08.095+00:003 bloggers Just a short post to highlight 3 blogs which are useful as a guide to what is going on in the world of physics today.<br /><br />The first is run by Peter Woit<br /><br /><a href="http://www.math.columbia.edu/~woit/wordpress/">http://www.math.columbia</a><a href="http://www.math.columbia.edu/~woit/wordpress/">.edu/~woit/wordpress/</a><br /><br />Peter Woit first came to attention after writing his book <i>Not Even Wrong</i> debunking the pretensions of superstring theory.<br /><br /><a href="https://www.amazon.co.uk/Not-Even-Wrong-Continuing-Challenge/dp/0224076051">https://www.amazon.co.uk/Not-Even-Wrong-Continuing-Challenge/dp/0224076051</a><br /><br />His blog amongst other things continues to fight the battle. Most recently attacking the move by some superstring theorists to remove the concept of falsifiability as a criteria for assessing physical theories especially by people such as Sean Carroll.<br /><br /><a href="http://www.math.columbia.edu/~woit/wordpress/?p=9938">http://www.math.columbia.edu/~woit/wordpress/?p=9938</a><br /><br /><br />This blog has been going for a while now<br /><br />A more recent blog which shares the same aims is run by Sabine Hossenfelder<br /><br /><a href="http://backreaction.blogspot.co.uk/">http://backreaction.blogspot.co.uk/</a><br /><br />One of her key beliefs is that we may have to live with the apparent ugliness of the Standard model as it seems that at present we have no clues as to what lies beyond it<br /><br /><br />(Yippee I can focus on the standard model and cosmology) and ignore supersymmetry, grand unified theories and of course superstrings π means I might actually understand physics before I die)<br /><br />She has a book coming out in the early summer<br /><br /><a href="https://www.amazon.co.uk/Lost-Math-Beauty-Physics-Astray/dp/0465094252/ref=sr_1_1?s=books&ie=UTF8&qid=1518383480&sr=1-1&refinements=p_27%3ASabine+Hossenfelder">https://www.amazon.co.uk/Lost-Math-Beauty-Physics-Astray/dp/0465094252/ref=sr_1_1?s=books&ie=UTF8&qid=1518383480&sr=1-1&refinements=p_27%3ASabine+Hossenfelder</a><br /><br />Which I intend to buy<br /><br />As an antidote (and quite amusing if you can get beyond the way in which he attacks his critics or anyone who disagrees with him) is the blog by Lubos Motl<br /><br /><a href="https://motls.blogspot.co.uk/">https://motls.blogspot.co.uk/</a><br /><br />Unlike Peter Woit or Sabine Hossenfelder he defends quite vigorously those who work in superstring theory and regards the above two blogs of being antiscientific and science haters. An opinion which I do not shate. However once one gets beyond the name calling and ad hominen attacks he does put the case for research in superstring theory quite eloquently and so is worth reading an eloquent defence is here<br /><a href="https://www.blogger.com/goog_1286442893"><br /></a><a href="https://motls.blogspot.co.uk/2004/10/beauty-of-string-theory.html">https://motls.blogspot.co.uk/2004/10/beauty-of-string-theory.html</a><br /><br />However the most interesting posts are where Lubos defends the orthodox interpretations of quantum mechanics and explains that all the foundations were developed by the founders especially by the Born Interpretation.<br /><br />Here is Lubos debunking the idea that the violations of the Bell Inequalities involve superluminal communication<br /><br /><a href="https://motls.blogspot.co.uk/2017/09/why-vanishing-commutators-imply-theres.html">https://motls.blogspot.co.uk/2017/09/why-vanishing-commutators-imply-theres.html</a><br /><br />and there are plenty more where that came from π<br /><br />So three blogs to keep in touch with developments in physics and I do find Lubos's attacks on his critics quite amusing<br /><br /><br /><br /><br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-91174586042400199192018-01-28T16:16:00.000+00:002018-01-28T19:14:27.433+00:00Calculations for 2018 and beyond <div style="text-align: justify;">This may be a bit ambitious but I thought I would outline the key calculations that I would like to do in both Cosmology and particle physics over the next few years. Now that I do not have the distraction of the Open University to deal with I can hopefully concentrate on these calculations (We'll see)</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">I have grouped them by year and topic and I aim to do at least 4 calculations a year</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><u>2018 Particle Physics</u><br /><u><br /></u>In the calculations that follow I shall take a fairly intuitive approach to the derivation of the Feynman rules and avoid as far as possible any attempt to justify the calculations rigorously from quantum field theory. The aim is to understand the actual calculations, for that purpose all that is needed is relativistic particle physics and Fermi's golden rule. </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><u>1) Deep Inelastic scattering part 1 (By end March )</u></div><div style="text-align: justify;"><u><br /></u></div><div style="text-align: justify;"> This concentrates on the early experiments at Stanford carried out in the late 1960's which concentrated on the inelastic scattering of electrons from protons. These experiments showed that the proton could be considered as made up of point like constituents of spin 1/2 initially called partons but conjectured to be the quarks of Gell Mann also that there were other non charge like constitutents present which were later identified with the carriers of the Strong Interaction in a manner similar to that of photons in the electromagnetic interaction. These are called gluons.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><u>2) Deep Inelastic scattering part 2 (By end June )</u></div><div style="text-align: justify;"><u><br /></u></div><div style="text-align: justify;">The development of the parton model and the structure of the proton was further clarified by scattering of neutrino's off the proton, These experiments were able to distinguish between quarks and anti-quarks and gave evidence that the partons had fractional charge thus strengthing the identification of the partons with the static quark model of Gell Mann and also thevgluons. A brief overview of the weak interaction will also be given.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><u>3) The Lagrangian of the Standard model (End of 2018)</u></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Taken together 1 and 2 give evidence for the development of our modern theory of the strong Interaction namely quantum chromodynamics, Also the fact that the weak interaction involves interactions between quarks and leptons. Concurrently with the work outlined above the idea that electromagnetism, the weak interaction and the strong interaction could be see as a gauge theory became prominent. However in order to correctly account for the masses of the carriers of the weak interaction the Higgs mechanism had to be invoked. All this will be outlined also it will be pointed out that when it comes to quantising the theory, the beautiful symmetry of Gauge theories is no longer present, mainly becasuse the propagators for the photons and the gluons are ill defined classically, However it is possible to correctly account for the quantisation rules by invoking an extended symmetry called BRS symmetry (Which I have mentioned before </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><a href="https://www.blogger.com/blogger.g?blogID=7897235423812277683#editor/target=post;postID=8991275106266049243;onPublishedMenu=template;onClosedMenu=template;postNum=0;src=postname">https://www.blogger.com/blogger.g?blogID=7897235423812277683#editor/target=post;postID=8991275106266049243;onPublishedMenu=template;onClosedMenu=template;postNum=0;src=postname</a> )</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">This involves the introduction of ghost particles Normally in most quantum field theory books these are introduced in a highly convoluted manner using path Integrals when by imposing the BRS symmetries right from the start it is possible to obtain the correct quantisation procedure right from the start. Amazing (or at least I think so π). It will be shown in a fairly informal manner how to write down the appropriate Feynman rules for the Standard Model </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><u>2019 "The year of the loop"</u></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The calculations above have so far only dealt with the first order of perturbation theory the so called classical level. However relativistic particle physics only becomes interesting when one goes beyond the tree level to the so called loop level as the Feynman diagrams involve loops these calculations established two amazing facts </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><u>a) Quantum electrodynamics </u><br /><br />For quantum electrodynamics, the corrections to the anamolous magnetic moment of the proton first carried out by Schwinger, and even more amazing the Lamb shift. It was these two calculations that put quantum electrodynamics calculations on the map. However until Non Abelian theories were developed it was not clear how to do extend quantum field theory to other interactions such as the weak and the strong interaction</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><u>b) The Asymptotic Freedom of the Strong Interaction. </u><br /><br /></div><div style="text-align: justify;">Prior to about 1973 attempts to apply quantum field theory to the strong interaction were stymied as it was not clear that perturbation theory could be applied in a satisfactory manner. However a remarkable property of Non Abelian gauge theories showed that at high energies the coupling constant decreased thus making it feasible to apply perturbation theory to the strong interaction. This calculation (which is quite long to say the least) will show how this works at the one loop level. </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><u>2020 and beyond Radiative Corrections to particle physics calculations </u></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">I would hope after the basics of loop calculations has been mastered in 2019 to demonstrate how real calculations at the one loop level are performed. For starters I would like to attempt the 2 research projects in Peskin and Schroeder. </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><a href="https://www.amazon.co.uk/Introduction-Quantum-Theory-Frontiers-Physics/dp/0201503972">https://www.amazon.co.uk/Introduction-Quantum-Theory-Frontiers-Physics/dp/0201503972</a></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The first project at the end of the first section calculates the scattering cross section for electron positron annhilation and involves the handling of of High energy Divergences (Ultra Violet) and Infra Red Divergences which miraculously cancel.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Then the culmination of my calculations in Quantum Field theory will be the last project in Peskin and Schroeder chapter which is a summary of the predictions of the decay rates of the Higg's boson.<br /><br />Other calculations and experiments leading to say the discovery of the W and Z bosons and the top quark may follow.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">If I were to tackle these purely by myself then I would probably get discouraged and give up fortunately there are many sources on the internet where clues as to how the calculations are done can be found. Indeed the first project is described in some detail in Schwartz's book </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><a href="https://www.amazon.co.uk/Quantum-Field-Theory-Standard-Model/dp/1107034736">https://www.amazon.co.uk/Quantum-Field-Theory-Standard-Model/dp/1107034736</a></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">So I won't just be on my own. </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Concurrently with the Quantum Field theory caclulations I want to look at Cosmology in particular the Peebles calculation </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><u>Homogeneous Cosmology (2018 to 2020) </u></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The aim of these set of calculations is to reproduce the calculations of Peebles who predicted the correct ratio of Hydrogen to Helium abundances in the early universe. This involved a synthesis of ideas from Fermi's theory of the weak interaction, Cosmological solutions to Einstein's Field equations, relativistic statistical physics and nuclear reaction physics. He and other people were able to predict the correct abundances of the light elements and it is my aim over the next two years to finally finish the work I started on this over 10 years ago </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><u>Interlude Numerical solutions to Differential Equations (June 2018) </u></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">In order to reproduce Peebles calculation it is necessary to have a robust numerical code which solves differential equations. The standard workhorse for most scientific work is the 4th order Runge Kutta Method and an investigation and derivation of the method will be given along with some examples showing the dependence of the accuracy of the solution on step size will be given.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><u>Classical Cosmology and the Concordance Model (End 2018) </u></div><div style="text-align: justify;"><u><br /></u></div><div style="text-align: justify;">This calculation will show how General Relativity can be used to derive the Friedmann equations and I have already completed this part, (and a heart breaking calculation it was too π’ ) however I have yet to show how the current model of the universe involving matter, dark matter and dark energy explains the acceleration of the universe and it is possible for a particular combination of matter, dark matter and dark energy it is possible to estimate the age of the universe and other parameters that cosmologists are interested in. The code developed above will be used to calculate the present age of the universe and also demonstrate the rather surprising conclusion that the Galaxies are actually moving away from us at speeds greater than the speed of light. </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><u>Relativistic Statistical Physics (2019)</u></div><div style="text-align: justify;"><u><br /></u></div><div style="text-align: justify;">As a prerequisite to calculating the Abundances of the light elements of the early universe it is necessary to derive expressions for the number density, the entropy and the pressure of the universe as a function of time. This involves expressions not usually found in undergraduate text books on statistical physics, but again a judicious internet search will uncover details usually glossed over. The culmination of this stage will be a code which calculates these properties as a function of the temperature of the universe. </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><u>Calculation of the light element abundance in the universe (2020)</u></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Using estimates of the likely nuclear reactions taking place in the early universe Peebles was able to estimate how the plasma of electrons, neutrinos protons and neutrons were able to combine to give the current ratio of Hydrogen and Helium currently observed. As this contradicted the ideas of people such as Hoyle and Bondi who thought that the remnants of stellar explosions could account for this abundance and Peebles ratio was shown to be correct this put the big bang on the map. Peebles early work just concentrated on a few reaction pathways and it will be the aim of the first part to simply reproduce these calculations. However over the years a sophisticated understanding of about 90 reactions was added to improve the accuracy of the calculation. This work is summarised in two reports by Kawano at Fermi Lab </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><a href="https://ntrs.nasa.gov/search.jsp?R=19920015920">https://ntrs.nasa.gov/search.jsp?R=19920015920</a></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">He also wrote a code Nuc123.for which I managed to down load a while back which calculates the abundances of the light elements and I hope eventually to update his code to something a bit more modern such as MatLab.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">After the work on homogeneous cosmology if I have enough energy left I will look at inhomogenous cosmology with the aim of understanding the anisotropies of the Cosmic microwave background. As ideas about this are still speculative (although some people would say they are not) then I won't be too concerned if I don't complete this work soon. The above calculations should be more than enough to understand how current ideas in particle physics and cosmology relate to the world around us. Fortunately given the internet it is a lot easier for a lone worker outside academia to understand the calculations in some detail and I hope that even though the work is not original putting all this together in some coherent form that is understandable for those who have an undergraduate degree in either physics or maths, will still be useful for those who want to understand contemporary physics.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Needless to say I shall probably not look at music or philosophy in any great depth until this work is completed that can come later. </div><div style="text-align: justify;"><br /></div><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-42081022583420279442018-01-01T16:23:00.001+00:002018-01-02T00:28:15.340+00:00The Rosenbluth Scattering Cross SectionHi everyone and a Happy new year<br /><br />I am pleased to report that I have completed my work on the calculation of the Rosenbluth Cross section. This is a calculation of the elastic scattering cross section for an electron off a proton. It is one of the standard calculations left as an exercise for the student in many books on quantum field theory. However the actual calculation is far from straightforward and difficult to find so I have written it out in full<br /><br />I have enclosed a link to the document here (hopefully there are not too many typos)<br /><br /><a href="https://drive.google.com/file/d/1qA2keWm01SSdoU84jQx-HqeOZroaQYF3/view?usp=sharing">https://drive.google.com/file/d/1qA2keWm01SSdoU84jQx-HqeOZroaQYF3/view?usp=sharing</a><br /><br />I have also enclosed a brief account of the first measurements of the cross section by Hofstadter at a proto type of what would eventually become the Standford Linear Accelerator. (SLAC). I was able to reproduce one of the early graphs from this work.<br /><br /><div style="text-align: justify;">One slightly unusual feature of this work is that it does not require the use of quantum field theory and a simple derivation of the Feynman rules based on a relativistic generalisation of Fermi's Golden rule is all that is needed. The first half of the notes describes how the Feynman rules for QED can be derived. Once one has the Feynman rules then one can go onto to derive expressions for the scattering cross section or particle decay rate of whatever process you are interested in, in terms of the modulus squared of the matrix element. This stage is the same as in other allegedly more rigorous methods based on quantum field theory. It is an interesting question as to whether or not quantum field theory is as necessary to particle physics as it is sometimes made out.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">As a final note the Rosenbluth cross section is expressed in terms of the electromagnetic and magnetic form factors of the proton which can only be determined from the experimental data. Once one has the form factors one can go on to derive expressions for the mean radius of the proton. This work after suitable extensions was carried out was shown to be compatible with other estimates based on the Lamb shift calculated from electron hydrogen spectroscopy. One would have thought that that would have been the end of the matter but in 2010 a series of experiments based on muon spectroscopy has come up with a radius of the proton that is about 4% smaller. Thus the proton has shrunk in size. This is shown below by this whimsical cartoon from the New York times showing the proton worried about it's shrinking waistline</div><div style="text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-Be3MeWG2jzk/WkpdyKqbz3I/AAAAAAAAAD0/W06We5M4PR8wOMNckSCNj6qdcixy7nZXQCLcBGAs/s1600/shrinking_proton.jpe" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="219" data-original-width="230" src="https://3.bp.blogspot.com/-Be3MeWG2jzk/WkpdyKqbz3I/AAAAAAAAAD0/W06We5M4PR8wOMNckSCNj6qdcixy7nZXQCLcBGAs/s1600/shrinking_proton.jpe" /></a></div><div class="separator" style="clear: both; text-align: justify;">There is a flurry of activity currently taking place to try and resolve the discrepancy, One collaboration (MUSE) is going to look at the scattering cross section from muon proton scattering to see if the discrepancy can be resolved. Details of which are given here </div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><a href="http://www.physics.rutgers.edu/~rgilman/elasticmup/">http://www.physics.rutgers.edu/~rgilman/elasticmup/</a></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">It is of interest that calculations performed almost 70 years ago are still of relevance today. </div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">When I next revisit this work (don't hold your breath π) I shall look at the inelastic scattering of electons from protons. This led to the somewhat surprising fact that at high energies the proton could be considered as made up of point like constituents which at the time were called partons. These were late identified as the quarks of Gell Mann (and others) and led to our current understanding of the strong interaction namely QCD. Hopefully by this time next year I will have finished describing this work. The current work provides the requisite background and I hope eventually to provide my followers with a working knowledge of the standard model of particle physics with nothing more than a smattering knowledge of relativistic quantum physics and Fermi's golden rule instead of the more arcane and somewhat mysterious knowledge of quantum field theory in it's many incarnations be it canonical formalism or the path Integral formalism. </div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;"><br /></div><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com7tag:blogger.com,1999:blog-7897235423812277683.post-60969335573533635382017-12-28T13:08:00.001+00:002017-12-28T22:18:06.176+00:00Particle Physics Reading list update <div style="text-align: justify;">This is an update to a previous post written many years ago for those who want to understand modern particle physics. This is aimed at those readers who have a basic knowledge of undergraduate physics such as those who have completed the Open university quantum physics course SM358 and are comfortable with the marhematical techniques given in that course. Unfortunately the Open University doesn't go much further and so students of the Open university are left high and dry if they want to understand the excitement of particle physics. </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Particle physics can be understood at a basic level by extending Fermi's Golden rule for time dependent perturbation theory to relativistic equations of motion in particular the Dirac equation. This is some what surprising given that most postgraduate courses in particle physics launch the student in the deep end by starting with quantum field theory. So that the impression is given that one must wade through many pages and pages of abstract formalism before getting to the stage where one can actually study particle physics. Whilst eventually understanding quantum field theory is important for those who want to do actual research or at least understand what is going on. As a first step I would reccomend the reader to stick with the approach based on Fermi's Golden rule. To that end my current reccomendation for the best introduction to particle physics is given by the first two books in the list. These two books will give the reader a basic understanding of what a Feynman diagram is, how to use Feynman diagrams to calculate the quantities of interest to particle physics such as particle decay rates and scattering cross sections and how numerous experiments and calculations led to the development of the Standard Model of particle physics. </div><div style="text-align: justify;"><br /></div><br /><div class="MsoNormal"><span style="text-indent: -18pt;"> 1)</span><span style="font-size: 7pt; font-stretch: normal; line-height: normal; text-indent: -18pt;"> </span><i style="text-indent: -18pt;">Modern Particle Physics</i><span style="text-indent: -18pt;"> Mark Thomson Cambridge University Press 2013</span></div><div class="MsoListParagraph" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;"><o:p></o:p></div><div class="MsoListParagraph" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;"><br /></div><div class="MsoNormal"> There is an associated web site with summaries of each chapter given to his Cambridge students and also hints and solutions to many of the problems. </div><div class="MsoNormal"><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><a href="http://www.hep.phy.cam.ac.uk/~thomson/MPP/ModernParticlePhysics.html">http://www.hep.phy.cam.ac.uk/~thomson/MPP/ModernParticlePhysics.html</a><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">Another (older book) which covers some topics in more detail at the same level as Thomson is <o:p></o:p></div><div class="MsoNormal"><i style="text-indent: -18pt;"><br /></i><i style="text-indent: -18pt;">2)<span style="font-size: 7pt; font-stretch: normal; font-style: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;"> </span></i><i style="text-indent: -18pt;">Quarks and Leptons: An Introductory course in Modern Particle Physics </i><span style="text-indent: -18pt;">F Halzen and A Martin John Wiley and Sons 1984</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal">It is a testimony to the robustness of the Standard Model of particle physics that a book published 33 years ago is still relevant to today.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The above two books should be enough for most people who want to gain a working knowledge of the Standard model of particle physics, without getting bogged down in the intricacies of quantum field theory. Having said that there is no doubt that quantum field theory is one of the foundations of modern physics, so for those who want to begin a lifes time journey to understand quantum field theory then the best starting point (IMHO) is <o:p></o:p></div><div class="MsoNormal"><span style="text-indent: -18pt;"><br /></span><span style="text-indent: -18pt;">3)</span><span style="font-size: 7pt; font-stretch: normal; line-height: normal; text-indent: -18pt;"> </span><i style="text-indent: -18pt;">Student Friendly Quantum Field theory</i><span style="text-indent: -18pt;"> Robert D Klauber Sandtrove Press 2013</span></div><div class="MsoListParagraph" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;"><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">Whilst this book only covers quantum electrodynamics, and the so called canonical formalism it does so in great detail, filling in alll the gaps (or at least most of them) and includes an introduction to renormalisation theory and how to do 1 loop calculations (the meat of quantum field theory). Again there is an associated web site <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal" style="margin-left: 36.0pt; text-indent: 36.0pt;"><a href="http://www.quantumfieldtheory.info/">http://www.quantumfieldtheory.info/</a><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The next step which covers both QCD and the Weinberg Salam Model from a field theory perspective in some detail is this book <o:p></o:p></div><div class="MsoNormal"><i style="text-indent: -18pt;"><br /></i><i style="text-indent: -18pt;">4)<span style="font-size: 7pt; font-stretch: normal; font-style: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;"> </span></i><i style="text-indent: -18pt;">Quantum Field Theory 2<sup>nd</sup> Edition </i><span style="text-indent: -18pt;">Franz Mandl and Graham Shaw Wiley 2010</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal">Finally for those who want graduate level textbooks which will take the reader up to the stage of postgraduate research then my two current favourites are <o:p></o:p></div><div class="MsoNormal"><span style="text-indent: -18pt;"><br /></span><span style="text-indent: -18pt;">5)</span><span style="font-size: 7pt; font-stretch: normal; line-height: normal; text-indent: -18pt;"> </span><i style="text-indent: -18pt;">An introduction to Quantum Field theory</i><span style="text-indent: -18pt;"> Peskin and Schroeder Perseus Books 1995</span></div><div class="MsoListParagraph" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;"><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">This book has 3 research projects at the end of each section, culminating in an investigation of the properties of the Higgβs boson (It is my ambition to try and attempt these some time) <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">A more modern text book is <o:p></o:p></div><div class="MsoNormal"><span style="text-indent: -18pt;"><br /></span><span style="text-indent: -18pt;">6)</span><span style="font-size: 7pt; font-stretch: normal; line-height: normal; text-indent: -18pt;"> </span><i style="text-indent: -18pt;">Quantum Field theory and the Standard Model</i><span style="text-indent: -18pt;"> M D Schwartz Cambridge 2014</span></div><div class="MsoListParagraph" style="mso-list: l0 level1 lfo1; text-indent: -18.0pt;"><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The above two books, unless one is really dedicated, are probably best seen as reference books, as it will take the reader pages and pages of algebra to fill in the details. With such books it is probably best to focus on repeating one or two calculations in detail and accept the rest on trust. <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">Finally a good survey of the experimental foundations of particle physics which includes many original papers is<br /><i style="text-indent: -18pt;"><br /></i><i style="text-indent: -18pt;">7)<span style="font-size: 7pt; font-stretch: normal; font-style: normal; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal;"> </span></i><i style="text-indent: -18pt;">The Experimental Foundations of Particle Physics </i><span style="text-indent: -18pt;">R Cahn and G Goldhaber Cambridge 2009</span><br /><span style="text-align: justify; text-indent: -18pt;"><br /></span><span style="text-align: justify; text-indent: -18pt;">I hope this helps I am near the end of completing a particle physics calculation on elastic electron proton scattering and its experimental confirmation, as a preliminary to umderstanding the inelastic scattering which led to the development of the quark model and our current understanding of the theory of the strong interaction. I will post details of this calculation in the next post. Unfortunately Drop box is no longer supporting public acccess to files which means that some of the links I have posted to my work, no longer work. I will try and find a work around. If not then I can post the files to anyone who contacts me via my e-mail address </span><a href="mailto:chrisf19572002@yahoo.co.uk" style="text-align: justify; text-indent: -18pt;">chrisf19572002@yahoo.co.uk</a><span style="text-align: justify; text-indent: -18pt;"> It only remains for me to wish all my readers and followers a happy new year and good luck if you are studying for any Open University or other course.</span><br /><br /></div>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-74691781851011578122017-10-15T18:11:00.002+01:002017-10-15T19:07:53.092+01:00Quantum Biology much ado about noting First apologies for not posting for a while I did not finish S383 the third TMA was a nightmare all waffle and no maths. A very disappointing course my reccomendation for those who want a mathematical based course is to avoid this one like the plague. Anyway I am now on restricted status which means under the new regulations I can't take any more OU courses unless I apply for removal from my restricted status. I did manage to salvage something namely I had built up enough credits for a second open degree without honours. If I want an honours degree in mathematics then I would have to take a new 1st level course in Statistics, another 1st level course and the second level course in mathematical methods. I can't really see me taking this so that would appear to be the end of my open university studies<br /><br />Anyway I digress the main focus of this post is on the allegedly new science of quantum biology the claim has been made that biological systems exhibit all the mysterious effects associated with quantum mechanics such as non-locality, entanglement and bose Einstein condensation a short overview of these claims is given by Jim Al Khalili (Who I would have thought known better but I guess he has to get his money from somewhere and how better than to jump on a bandwagon)<br /><br /><a href="https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwjp2K27hvPWAhWIChoKHYr6Dz4QtwIIKDAA&url=https%3A%2F%2Fwww.ted.com%2Ftalks%2Fjim_al_khalili_how_quantum_biology_might_explain_life_s_biggest_questions&usg=AOvVaw2SeqAMJK9gfXHM97NcUGMU">https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwjp2K27hvPWAhWIChoKHYr6Dz4QtwIIKDAA&url=https%3A%2F%2Fwww.ted.com%2Ftalks%2Fjim_al_khalili_how_quantum_biology_might_explain_life_s_biggest_questions&usg=AOvVaw2SeqAMJK9gfXHM97NcUGMU</a><br /><br /><br />I have severe doubts about the whole thrust of this (as those who have read my earlier posts on quantum mechanics might have guessed)<br /><br />Just a few points<br /><br />a) It is highly surprising that phenomenon such as Bose Einstein condensation occur at room temperatures usually such effects manifest themselves at very low temperatures. The wet noisy enviroment associated with biological systems would seem to mitigate against this<br /><br />b) The fact that experiments seem to have established long range electron coherence in photo-synthesis and other similar processes should not necessarily be seen as evidence for non-locality, the conditions in which so called entanglement occur and violation of the Bell inequalities are usually quite specific involving the emission of two particles from a common source such that their net angular momentum is zero. In photosynthesis there would appear to be no such initial conditions<br /><br />c) Khalili and others make the common fallacy of assuming that the so called wavefunction is a physical entity and not as the Born interpretation would have effectively the square root of a probability density function or probability amplitude whose modulus squared gives rise to a probability density function. Thus the pictures that Khalili show of quantum tunneling are pictures of a probability density function not of an electron or other particle spread out over all space.<br /><br />d) Quantum superposition is not an actual superposition but a superposition of possible states. The idea that some people have that before a measurement is made a quantum system is in a state of limbo which collapses to one of the possible states on measurement is misleading to say the least. Before measurement the observer does not know which one of the many possible states the system will be found in. He or she can only assign certain probabilities in accordance with the physical setup On measurement the system will be found to be in one of those states and so the superposition of possibilities 'collapses' to the one found on measurement. If the experiment is repeated a sufficiently large number of times then the system will have been found in each one of the states in accordance with the probabilities initially assigned. However because of the random nature of quantum events I can never tell by a single measurement what state the system will be found in.<br /><br />e) The above interpretation simplifies all the agonising over the Aspect experiment. On the above interpretation a given observer A at his or her station will either measure a particle to have spin up or down. However once A has done this he or she will immediately know the result of B's measurement But of course as B has no way of knowing what A's measurement is then as far as he is concerned his particle could still be measured with spin up or spin down. There is no faster than speed of light communication caused by A's affecting B's particle.<br /><br />So for all those reasons even if it were true that biological systems are exhibiting the so called mysterious aspects of quantum mehcanics. There is a perfectly rational explanation based on a purely statistical interpretation of quantum mechanics. Particles do not split in two when passing through slits, Particles do not traverse all possible paths at once and there is definitely no faster than light communication between separated quantum systems.<br /><br />However recent experiments have caused doubt on whether or not the long correlation times observed in photosynthesis are due to quantum effects.<br /><br /><a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5559008/">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5559008/</a><br /><br />A good overview of how photosynthesis is modelled is given by this MSc thesis which again shows that there is no need to invoke the mysterious aspects of quantum mechanics to explain the electron coherence.<br /><br /><a href="https://arxiv.org/abs/1503.03277">https://arxiv.org/abs/1503.03277</a><br /><br />Also this web site is the blog of a condensed matter physicist who has been very sceptical about the quantum biology bandwagon<br /><br /><a href="http://condensedconcepts.blogspot.co.uk/">http://condensedconcepts.blogspot.co.uk</a>/<br /><br />So for all the above reasons I think it is safe to say that the Quantum biology bandwagon is truly much ado about nothing<br /><br /><br /><br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-54621100440043011602016-09-10T16:09:00.002+01:002016-09-10T16:19:57.081+01:00And we're off S383 First impressions So the website for S383 opened this week and I got to look at the TMA's and the pdf of the course units.<br /><br />S383 Unit 1 is a pretty good summary of General Relativity and the Schwarzschild metric and the Friedmann equations. It skimps on some details for example there is not a full blown derivation of the Friedmann equations from General relativity but enough is given so that you could in principle derive the results for yourself (Good luck with that one I would estimate a fairly intensive month to get it exactly right if you don't die of boredom :) )<br /><br />S383 Unit 2 is a summary of observational cosmology starting with the big bang and then an overview of modern ideas in cosmology centering around inflation. From what I can tell there is not a great deal of detail here. However the author seems to think inflation is still quite speculative and wants to concentrate on more tangible things. The course then goes on to look at Galaxies, the evidence for black holes gravitational lensing and the Lyman alpha forest. All interesting and active areas of research and good to get an overview<br /><br />S383 Unit 3 is called extreme environmental astrophysics and is concerned amongst other things with the formation of acretion discs near a black hole and many other things which I will cover in more detail as time goes on.<br /><br />Looking at the TMAs that are available it has to be said that they don't really do justice to the depth of material covered in the course I dare say the electronic tmas will cover more topics. This is based only on 3 of the TMAs the TMA for the second block being an extended one and so not issued yet.<br /><br />Ok so initial impressions are that this course covers a lot of interesting material but not necessarily at a mathematical depth that would satisfy more mathematically minded people. I guess one would have to look elsewhere for that.<br /><br />One place to look is the Cambridge Part III courses examples sheets<br /><br /><a href="http://www.damtp.cam.ac.uk/user/examples/indexP3.html">http://www.damtp.cam.ac.uk/user/examples/indexP3.html</a><br /><br />Scroll down to the bottom and you will find example sheets for three of the core courses in relativity that are offered. Also most generously Dr Baumann has published his lecture notes on Cosmology<br /><br /><a href="http://www.damtp.cam.ac.uk/user/db275/Cosmology.pdf">http://www.damtp.cam.ac.uk/user/db275/Cosmology.pdf</a><br /><br /><br />More than enough to satisfy those wanting a more mathematical approach I hope to do at least some the examples associated with Cosmology and maybe some of the General Relativity example sheets alongside my work on S383.<br /><br /><br />As an aside I decided to leave MST326 for this year, I may as I have 2 years left do it in conjuction with the new MST327 course<br /><br /><a href="http://www.open.ac.uk/courses/modules/ms327">http://www.open.ac.uk/courses/modules/ms327</a><br /><br />next year and round off my second open degree with M303 the year after or vice versa Who knows<br /><br /><br /><br /><br /><br /><br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com2tag:blogger.com,1999:blog-7897235423812277683.post-42029930378728096932016-08-29T21:39:00.003+01:002016-08-29T21:40:28.594+01:00Decisions again MST326 0r Not MST326 As well as doing S383 I have a week to decide whether or not to do MST326 again<br />I hadn't realised that if you fail as I did you can retake the whole module again at some point and not get your marks downgraded<br /><br />Pros<br /><br />I have revised my interest in partial differential equations<br /><br />If I get say grade 2 along with a grade 2 in M303 I could start the MSc in 2 years time<br /><br />If I decide not to do it then I will have to wait 2 years before taking it again I can't see me doing this alongside M303 next year so its now or never<br /><br />Cons<br /><br />Money and time as always :)<br /><br />It is a tough one<br /><br />Speaking of Partial Differential equations I am currently trying to understand the solution of Schrodinger's equation in parabolic coordinates. I have seen it mentioned in a few books but the solutions are only sketched out. It turns out that it is possible to solve Schrodinger.s equation in 11 of the orthogonal coordinates but most courses just concentrate on the well known spherical polar coordinate system. It would be an interesting exercise over the next few years to see how feasible it is to solve Schrodinger's equation in all eleven coordinate systems.<br /><br /><a href="https://en.wikipedia.org/wiki/Orthogonal_coordinates">https://en.wikipedia.org/wiki/Orthogonal_coordinates</a><br /><br />and with the power of computer graphics plot out the solutions Watch this space (but don't hold your breath)<br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-80186472836464485312016-07-03T15:33:00.001+01:002016-07-03T15:33:10.128+01:00Update S383 and M303 Hi sorry for not blogging for a while other priorities and so forth<br />Anyway I got bored with linear statistical modelling so couldn't motivate myself to complete it My track record for dropping out of OU courses isn't looking so good.<br />However I have decided to register for S383 Relativistic Universe<br /><br /><a href="http://www.open.ac.uk/courses/modules/s383">http://www.open.ac.uk/courses/modules/s383</a><br /><br />which starts in September<br /><br />I then intend to do M303 which I have been lucky enough to get the materials for<br /><br /><a href="http://www.open.ac.uk/courses/modules/m303">http://www.open.ac.uk/courses/modules/m303</a><br /><br /><br />I will then have enough credits to get my second Open university Open degree<br />After that who knows I believe it is possible to study modules separately rather than as part of a degree as I have no inclination to do a whole load of level 1 courses for the sake of a degree<br />I am tempted to essentially build up a portfolio of second level courses followed by a third level course<br /><br />Topics I would like to study include Chemistry, Economics and Biology and I dare say completing the third level philosophy course would be useful as well. I think it would be quite satisfying to do a second level course followed by a third level course I'll see how things pan out after M303<br /><br />In the mean time almost by coincidence there is a coursera course on Galois theory starting up looks quite intimidating but I have the OU M303 books to give me background in ring and field theory<br /><br /><a href="https://www.coursera.org/learn/galois#syllabus">https://www.coursera.org/learn/galois#syllabus</a><br /><br />At any rate I shall get some idea of what the topic is all about.<br /><br /><br /><br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-65738693255952645482015-12-03T20:06:00.001+00:002015-12-16T19:16:47.092+00:00M346 Linear Statistical Modelling TMA01 So finished the first TMA01 for the linear statistical modelling course M346<br />This course is dull as dishwater but a necessary evil for me to complete my maths and statistics degree<br /><br />The first question covered the usual stuff t tests, various distributions and chi squared and F tests and a bit of hypothesis testing.<br /><br />All quite mind numbingly tedious you look up the formulae in Unit 1 and apply them<br /><br />Question 2 covered how to use Gen Stat to test for normality and how to use the tools to calculate various probabilities<br /><br />Question 3 covered linear regression the first a straightforward application of linear regression to a data set and the other requiring a transformation of the variables. You are taught how to interpret the linear regression output from GenStat<br /><br />Really am going to die of boredom from this course but wont give up as I need the qualification<br /><br />Not even sure what this is doing in a maths course as there are no derivations or proofs in the course<br /><br /><u>Addendum 16th DEC </u><br /><br />Got the tma back on Monday 4 marks short of distinction<br />lost marks on the transformation question as I didn't think of transforming the x variable and made a silly mistake in the linear regression question as I used the prediction of mean instead of future prediction. Still my tutor has provided a really useful summary of the correct answers<br /><br />Probably heading for grade 2 if I don't die of boredom before I aim to do 1 questions from the TMA's per week, Might even finish the 2nd TMA over the Xmas holidays<br /><br /><br /><br /><br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-68272354877151601832015-11-30T20:52:00.001+00:002015-11-30T20:52:15.333+00:00Grade 6 result So got the percentage mark for my Grade 6 theory result it was a respectable pass 72% but way short of a merit so glad to have passed but slightly disappointed the mark wasn't higher. I will know in 2 weeks just what the breakdown in marks was and round about the end of January ABRSM will publish the 2014 papers along with the model answers. So I will be able to learn from the experience<br /><br />Grade 7 next depending on how it goes I will try and do it in June but at least by November of Next year<br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-47551423510940811042015-11-04T16:46:00.001+00:002015-11-04T19:33:20.422+00:00Grade 6 Music Theory Debrief Well yesterday I braved the gloomy Fog of Edinburgh and went to the darkest frontier of Sighthill to sit my grade 6 Music theory exam. I think it went reasonably well although there were one or two tricky bits. The exam consists of 5 sections and I had prepared myself by working through quite a few past papers upto 2014 over the past year or so. Also with the help of Victoria Williams my music theory website and Udemy videos<br /><br /><a href="http://www.mymusictheory.com/">http://www.mymusictheory.com/</a><br /><br /><br /><br /><a href="https://www.udemy.com/u/vickywilliams/">https://www.udemy.com/u/vickywilliams/</a><br /><br />A hint for those doing Udemy courses if you invest in one then practically every week they will have course offers between Β£8 and Β£15 pounds. For grade 6 I did the Grade 6 melody course and the Figured bass course. Definitely worth investing in if you are contemplating doing the Grade 6 exam<br /><br />So to the exam itself there are 5 questions<br /><br />1 Given a melody can you work out an appropriate chord sequence or alternatively given the start of a melody with figured bass can you complete it.<br /><br />I had got into the habit of doing the first question. To start with it is usual to start a melody with either an anacrusis leading onto the tonic chord or just usually the tonic chord so the first chord is either the tonic (Ia or ia) or if an anacrusis is used (Va-1a)<br /><br />Usually the piece ends with a perfect cadence (Va-Ia) if the melody ends with a descending scale figure ending on the tonic of the piece then it is possible to use the cadential 6-4 formula (Ic-Va-Ia)<br />So that is usually 3 or 4 chords that you can write down almost automatically.<br /><br />The rest of the chords are worked out by first writing each possible chord that would fit (at this level there are only 3 that would work) and seeing if a logical sequence can be fitted in<br /> For example if in the middle of the piece 3 notes form a descending scale passage leading to the tonic then a good fit is a sequence in the opposite direction ascending upwards<br /><br />Suppose we have (in the key of C major) the notes E D C then in the bass we would have C D E and the chords would be Ia -viiob - Ib or vice versa if the melody was ascending<br /><br />Then it is usually possible to find a sequence which follows part of the circle of 5th's progression namely<br /><br />I-IV-vii-iii-vi-ii-V-I<br /><br />So to some extent this question is a bit like a Sudoku puzzle you have certain clues and you have to fill in the rest of the grid. There are certain rules which one must follow, Never have the bass line or any of the other lines move in parallel fifths or parallel octaves with each other, So if the soprano line is G and you have chosen to use the tonic chord C if the next note in the soprano line is A do not make the note in the bass line D (a ii chord) One way to avoid this as far as possible is to ensure for the most part that the bass line moves in contrary motion to the soprano line.<br /><br />Also never double the third of a major triad and a few other rules all explained brilliantly by Victoria Williams in her notes, sticking to these and learning a few basic chord progressions, usually gets you through. For this question you don't have to fill in the actual chords but just give the letters. One thing I find helpful in these exercises is to fill in the bass line making sure it doesn't give rise to parallel fifths or octaves with the Soprano line. If I were doing a full harmonisation I would always write the bass line first. Then fill in the other notes. One thing that is slightly odd about the ABRSM exam is that you can only label the chords by their letters or fill in the notes, So if you do leave the bass line in you will get marked down, So you end up rubbing out the bass line. A bit pointless in my opinion<br /><br />The actual question worked out OK I think but I was a bit repetitive using far more Tonic and Dominant chords than I would normally expect to and also repeating the Ia-viib-Ib chord progression twice, I'll wait and see what the examiner thinks,<br /><br />The second question was figured bass a topic which can be quite confusing but again explained brilliantly by Victoria in her notes and video. This is even more like a Sudoku puzzle in that the figures on the bass line tell the arranger or accompanist the type of chord to use, So example if there are no figures underneath the chord then the chord is in root position, If the note has a 6 below it is in first inversion and finally if it has 6-4 then it is in second inversion<br /><br />Thus if the bass note were C with no figures under it would mean a chord of C major in root position C - E -G. If a figure of 6 were below it would be an A minor chord (A-C-E) in first inversion i.e with the note C as the bass note. Finally if 6-4 were the figures under the note then it would be the F major chord (FAC) in second inversion. Also there may be an accidental included. So the first thing to do is to work out which notes the figures are telling you to use. After this is done write out a soprano line which is interesting again trying for contrary motion with the bass line as much as possible and again making sure that the third of a major triad is not used.<br /><br />Then comes a slightly trickier part filling in the Alto and tenor parts making sure that there are no parallel fifths or octaves between all 4 of the other parts. The only way to fully check this is to systematically work out the intervals between all the other parts. In the context of exam pressure this is really quite a tedious exercise to say the least. I generally trust to judgement and intuition hoping that provided I have no parallel fifths or octaves between the Bass and Soprano lines no others will creep in. Generally speaking provided the alto and tenor parts just move to the nearest available note in the next chord then things should work out, But I have in my practice been known to let the odd parallel fifth creep in. I suspect the same would happen here. In order to make the Soprano part a bit more interesting then it is possible to fill in gaps with quavers so that suppose in the soprano part you had a descent from D to B then make the D a quaver and put in another quaver before the B.<br /><br />The question was slightly odd in that there were quite a few quavers in the bass line needing a different chord. However provided I got the correct notes and didn't have to many parallel fifths or octaves then I should be OK (fingers crossed anyway)<br /><br />The third question is to complete a melody of which the first few bars are given again Victoria explains how to go about doing this very well in her Grade 6 video. This often involves a modulation in the last few bars. If you plan out in advance the key points then it helps give a structure. Generally speaking a melody will have 8 - 10 bars and will move from the tonic chord to an imperfect cadence V on the 4th bar and then back to the tonic of either the same key if there is no modulation or to the new key. So again the first note of the 4th bar will be a note which fits the Vth of the given key and the last note will be the tonic of whatever key you have ended up with.<br /><br />The examiner is looking for you to take what ever is given and use it again with some possible variations in rhythm. I generally try and sequence the second bar in the third bar by raising it up a tone and ensuring a movement by step to the dominant note in the first half of the 4th bar. Then using a logical sequence such as the circle of fifths to get to the modulation section usually with an inversion of the movement in the first phrase so if the first phrase has an ascending passage I will have a descending passage in the second phrase, If there is a large upwards movement then will have downwards leap and so forth finally if there is a modulation ending with a pivot chord from the old key to the new key and then V7-I in the new key. Where in the penultimate bar or half bar the notes of the new key are emphasised.<br /><br />The question was a bit odd in that it involved a rapid scale like figure occupying 4 hemi-semi quavers and also a modulation from a minor key to the dominant of that minor key. Also despite being in 6/8 time the harmony of the second bar was static just oscillating about 1 note. The modulation caused me a bit of trouble but I managed to scrape something together hopefully it will be enough to get me through,<br /><br />The last two questions are usually a bit more straightforward than the first 3 and I usually do them first to get them out of the way. They involve asking questions about 2 extracts of music the first one being a piece for piano or with piano and another instrument or voice. and the second a short orchestral extract.<br /><br />In these questions you are generally asked to identify various chords and intervals the meaning of certain terms such as Etwas Geschwind (Somewhat agitated, which I was before the exam), Also can you spot certain things such as a triad in first inversion and so forth.<br /><br />The orchestral score question usually involves asking you transpose the music written out for horns in F or clarinets in Bb to how they would actually sound. If a horn in F plays what is notated as C it actually sounds like F so you would transpose down a 5th,(7 semitones), Similarly for clarinets in Bb a notated C sounds like Bb you transpose down a 2nd (2 semitones). So the easiest way to do this is to write out the notes of the chromatic scale of C major and underneath write out the notes of F major underneath it for horns in F or Bb for clarinets. Then write out explicitly the notes for each part as given and then under them the corresponding transposed notes. This can be a bit time consuming, also and I never got the hang of this, if the original note has a flat attached do you transpose with the original note or with a sharp instead. Technically the notes would sound the same so Db is equivalent to C# but an interval with either C# or Db would have a different name. Thus Db - A is a 5th but C# - A is a 6th. As the interval questions on the orchestral score often involves one of the transposing instruments. It's not clear whether or not after you have made the correct transposition that you will get the correct interval<br /><br /><br />Anyway that was that I think I got most of the last two questions correct and hopefully will have done OK on the first 3. We'll just have to see. I await the examiners report with interest.<br /><br />Next year provided everything went reasonably well I hope to do grade 7. For now upto Xmas I need to focus on my statistics course.<br /><br />If you took the grade 6 exam or anyother grade theory exam I hope it went well for you as well<br /><br /><br />As a final comment it may seem to the average person that this approach takes the creativity out of music. However just as physicists or mathematicians can be creative within the context of a given framework (i,e the laws of physics or axioms of mathematics) so in music the laws of harmony provide a framework whereby some one can be creative. Of course if you are a Mozart or Beethoven then you can use the framework to construct quite different types of pieces but the idea that is often floated that composers don't use the laws of harmony but just write whatever comes into their head is quite a misconception. They will usually have an idea of a melody either by picking notes out on a piano or something that they hear in their head, But those fragments will be developed into complete pieces only by following a given structure. I find it quite remarkable that provided you follow the basic laws of harmony and form you can write down a competent piece of music and that this craft can be taught.<br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-14065922375618151322015-10-07T19:31:00.000+01:002015-10-12T15:30:33.125+01:00A bit more on Bell Some readers of the last post on Bell may claim that I have missed the point. What my last account fails to do they would say is take into account the effect of the polarisation settings and the fact that they can be changed during the time of flight. So that as the usual story goes if the polarisation of the left detector is changed then the polarisation of the photon (or electron) will change in accordance with that polarisation but because the other photon must have opposite spin then it will change its polarisation accordingly.<br /><br />I think from a statistical point of view this is misguided. A polariser acts as a filter and for a given polarisation angle only a fraction of the particles in the beam will be able to pass the polariser in a accordance with Malus's Law the probability of passing through the filter being proportional to the square of the cosine of the angle between the polariser setting and the photons polarisation angle. On emergence the photons that pass through the filter will have the polarisation of the polariser.<br /><br />However some photons will not pass through. The polariser has destroyed any correlation there might have been with an EPR pair. Similarly at the other end a different number of photons will also pass through. The number of photons that pass through both ends being dependent on the polariser settings All this can be calculated from the basic laws of quantum physics. But given that polarisers will destroy any prior correlation. I can't see why it is inconsistent with the idea that before 1 or both of the photons hit the polarisers they were emitted from the source with perfect anti-correlation.<br /><br />Clearly if one thinks the Bell State applies to single pairs alone then one would have to invoke some spooky action at a distance, But from a statistical point of view all we have is the probability of the photons electrons or whatever emerging through the polarisers. One has to imagine the pairs being randomly emitted from the source with all sorts of polarisation angles, each pair being anti-correlated especially as the Bell state is rotationally invariant. However it is not clear, if one is just concerned with the overall transmission probability of photons through the polarisers which is all one can measure, that the photon that passes through the left polariser is one that is paired with the photon that passes through the right hand polariser. Because the polariser obviously destroys any correlation. Imagine if for example there was only 1 polariser say the left side then clearly the polarisation of the other photon would not change.<br /><br />Given that in order to verify the statistics many events have to occur then there is no way of ever detecting a single pair of photons. <br /><br />On that basis then I still feel justified in denying the need for any form of super-luminal signalling between a single pair of photons.<br /><br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-294939829813455572015-10-05T20:26:00.002+01:002015-10-07T20:40:49.560+01:00Long term plans So my (current) long term plans are as follows<br /><br />2015 - 2016 M346 Linear Statistical Modelling<br />2016 - 2017 M347 Mathematical Statistics and possibly M140<br /><br />So hopefully I will then have a BSc in Maths and Statistics<br /><br />Then I shall build on what I have already to convert this to a Maths and Economics degree<br />For which I will need the 2 economics courses<br />2017 -2018 DD209 Running the economy<br />2018-2019 DD309 Doing Economics People Markets and Policy<br /><br />and also the Maths Methods course MST224 which I would do alongside DD209 probably<br /><div><br /></div><div>Then having got the economics courses under my belt I could go on to do the Politics Philosophy and Economics degree by doing the 2nd level politics course,</div><div><br /></div><div>2019-2020 DD211 Understanding politics: ideas and institutions in the modern world </div><div><br /></div><div>Then a third level politics course and finally the third level philosophy course. So that would complete 3 Open university degrees within the next 7 years or so. </div><div><br /></div><div><br /></div><div>Concurrently I would be continuing with my music studies Grade 6 theory beckons in a months time </div><div>then Grade 7 and<br />Grade 8. Also by the middle of next year I would hope (money permitting to embark on the Open College of Arts composition course with the aim of writing an extended piece of music within the next 5 years or so.<br /><br /><a href="http://www.oca.ac.uk/courses/music-courses/">http://www.oca.ac.uk/courses/music-courses/</a></div><div><br /></div><div>So that should keep me busy for a while. </div><div><br /></div><div><br /></div><div><br /></div>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-53223784025954231582015-09-26T16:02:00.002+01:002015-09-26T18:59:46.366+01:00The Bell State a defence of naivety<div style="text-align: justify;">In this post I wish to defend a naive interpretation of the Bell State against some of the more wilder claims made for it in the literature</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The Bell state (for those who have been asleep for the past 40 years) applies to a pair of particles emitted from a common source in opposite directions. The Bell State describes the joint spin of the two particles which because of the conservation of angular momentum must be zero. But until measurement occurs we do not know the spin of a given particle so the probability amplitude takes the form</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">$$|B> = \frac{1}{\sqrt{2}}(|u1>|d2>-|u2> |d1>)$$</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">where u represents spin up and d represents spin down and the naive interpretation I want to defend is that this simply represents the conservation of angular momentum, so that once the spin of 1 particle is known by measurement, then the other spin is automatically known and must be equal and opposite. All the Bell state represents is the 50 % probability on measurement that the particle a person observes at one detector will either be measured with spin up or spin down. Repeated measurements will confirm the fact that at a particular detector the spin will be measured up or down 50% of the time and at the other detector the spin will be equal and opposite. It is well known that the Bell state is rotationally invariant so if one the particles are emitted at an angle $$\theta$$ with respect to the polarisation axis between the particles, the other particle will be emitted at an angle $$-\theta$$ with respect to the polarisation axis . Applying a bit of Pauli spin algebra to the Bell state is well known to give the correct correlation function as was spectacularly confirmed by Aspect in the early 1980's</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">So why all the fuss, apart from the intrinsic skill in being able to measure the relative spin states of two particles separated a long distance away, from the naive perspective I am offering this is just another routine success of quantum mechanics.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">As many people will know, the fuss has arisen because it is claimed that this experiment encapsulates the essence of the debate that Bohr and Einstein had in the early 1930's about the interpretation of quantum mechanics and which culminated in the so called EPR paradox.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">To cut a long story short the claim is made that because we don't know the spin state of a given particle before measurement, The Bell state represents a physical state of superposition so that the particles spin state is undetermined until measurement which then collapses the Bell state into one of the two possibilities. So that it is claimed the other particle instantly knows what spin take to take as soon as the first particles spin state has been determined. As the particles can in principle be millions of miles away, it is claimed that this means that some signal faster than the speed of light has been sent to the other particle. Thus there would appear to be a prima facia contradiction between special relativity and quantum mechanics.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Furthermore in his paper</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><a href="http://www.drchinese.com/David/Bell_Compact.pdf">http://www.drchinese.com/David/Bell_Compact.pdf</a></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"> It is claimed that Bell showed that the 2 assumptions</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">a) that there were hidden variables that predetermined the spins of the particles</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">b) the assumption that the two detectors were independent of each other</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Led to a contradiction with the predictions of quantum mechanics.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The failure of assumption a) would imply that quantum measurements cause reality to happen giving rise to the 'California Interpretation' and the failure of assumption b) gives rise to the idea that a measurement of the spin of a particle at one detector affects the other. This is the so called 'Spooky action at a distance' On this reading (which is the one that seems to pervade the popular literature) The Aspect experiment would imply that quantum mechanics does either involve spooky action at a distance (non-locality) or a form of non realism in which measurement determines properties of particles. What is worse in the popular literature, both claims are made about quantum mechanics when it is clear that only one of the two need be made. So that ever since the Aspect experiment quantum mechanics is seen in many circles as being both non-realist and non-local. </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">I want to challenge not the assumptions that Bell made but the interpretation of their significance as they have become known in the popular literature.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">First it should be stressed that Bell himself did not think quantum mechanics in itself showed non-locality or spooky action at a distance. If one reads the conclusion ( section VI) Bell states explicitly in the first sentence.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><i>In a theory in which parameters are added to quantum mechanics to determine the results of <u>individual</u> measurements without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover the signal must propagate instantaneously, so that such a theory could not be Lorentz invariant. </i></div><br /><div style="text-align: justify;"><span style="font-style: italic;"><br /></span></div><div style="text-align: justify;">So what Bell thought he had proven was that<u> hidden variable</u> theories had to have some form of non-locality or spooky action at a distance not quantum mechanics. Indeed if one goes through the actual calculation of the correlation function measured in the Aspect experiment using quantum mechanics no mention is made of any form of signalling at all. All one does is apply a bit of Pauli Algebra to the Bell state and out pops the result almost by magic. It seems odd to claim non-local signalling to explain an experimental result when the correct quantum mechanical prediction makes no mention of it.</div><br /><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Lets consider the situation in a bit more detail</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Quantum mechanics is essentially a statistical theory, the solution of Schrodinger's equation for a given problem gives rise to a probability amplitude the modulus squared of which gives rise to a probability density function via the Born rule. The solution to Schrodinger's equation is not a real wave existing in (3N+1)*S configuration space where N is the number of particles and S is the product of all the possible spin states of the particles involved. But effectively the square root of a probability density function and the (3N+1)*S configuration space is simply the probability sample space not a real physical space.<br /><br />That being so the Bell state is nothing physical it is (as is any quantum superposition) a superposition of possible outcomes with coefficients representing the probability of one of the possible outcomes. When a measurement is made the so called collapse of the probability state vector is not a physical process, but a realization of one of the possibilities, just as say I can assign a probability state vector to a coin or a dice or any situation which results in a number of possible outcomes N</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">This is simply</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">$$ |Final> =\frac{1}{\sqrt{N}}{a_1|1a_>+.........a_n|N>} $$</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">where the a's represent the square roots of the probability of one of the outcomes being realized.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">But the system, is not before measurement physically in some hybrid state of all the possibilities. All one can say is that one of the possibilities will be realized on measurement, not that measurement causes the system to jump from it's limbo state into one of the possible outcomes. The probability amplitude can be seen as akin to a Bayesian prior representing our best guess as to what the possible outcomes will be. Given that one can by definition only ever see a quantum system in one state speculation as to what happens before the act of measurement occurs is totally meaningless.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Furthermore if as the standard story has it the spin states of the particles are only determined at measurement then that would make a nonsense of the conservation of angular momentum. It would imply that the conservation of angular momentum only applies at the point of measurement. On the naive view I'm defending the relative orientations of the particles spin states is fixed at the point of emission but we wont know what spin state we will measure at a given detector. So whilst a given individual particles spin state is not determined, the relative orientation with respect to the other is, If that is not so then the conservation of angular momentum is meaningless and we would not be able to predict the outcome of experiments such as those at CERN. I hope then to have dispelled the myth that the Aspect experiment or any other such experiment involves spooky action at a distance.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Ok so what does entanglement or non-locality actually mean? Again there is a fairly straightforward interpretation. If the joint probability state vector of a system of many objects is entangled that means that it cannot be written as the product of the individual objects probability state vectors. This is well known in statistics, If the probability density function of two or more variables cannot be written as the product of the probability density functions of the individual variables, then that is is an indication that the two variables are not independent of each other. For the Bell state we know that the individual particle spin states are not independent of each other, as they are equal and opposite because of the law of conservation of angular momentum. Again there is no need to invoke spooky action at a distance and I have to say Bell has misled a whole generation of physicists by invoking this in his conclusion. </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Is my interpretation consistent with the quantum formalism undoubtedly yes as I am using the Bell state to make the correct predictions as measured in the Aspect experiment. The question has to be why invoke spooky action at a distance or collapse of the wave packet when a simple statistical interpretation in accordance with the Born interpretation is all one needs to successfully predict the outcome of quantum mechanical experiments.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">As a final point if Bell has really disproved the existence of hidden variables then we cannot go beyond the statistics to find out "what is really going on". The hope of going beyond the statistical predictions of quantum mechanics to predict the results of individual measurements is an illusion and "Nature really does play dice." </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">"</div>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-47922185165677194002015-09-18T18:35:00.001+01:002015-09-18T18:37:11.086+01:00More quantum madnessYet another episode in the quantum madness game<br /><br /><div style="background-color: white; color: #222222; font-family: Arial, sans-serif; font-size: 14px; line-height: 19.6px; margin-bottom: 1em; padding: 0px;"><a class="_blanktarget" href="http://www.theguardian.com/science/2015/sep/16/experiment-to-put-microbe-in-two-places-at-once-quantum-physics-schrodinger" style="color: #1f536b;">http://www.theguardian.com/science/2015/sep/16/experiment-to-put-microbe-in-two-places-at-once-quantum-physics-schrodinger</a></div><div style="background-color: white; color: #222222; font-family: Arial, sans-serif; font-size: 14px; line-height: 19.6px; margin-bottom: 1em; padding: 0px;"><a class="_blanktarget" href="http://arxiv.org/abs/1509.03763" style="color: #1f536b;">http://arxiv.org/abs/1509.03763</a></div>This time the claim is the rather somewhat ludicrous claim that a microbe before it's position is measured is in two places at once.<br /><br />This really is such gibberish that it is hardly worth bothering with. But at the heart of it lies a misunderstanding of the nature of quantum superposition<br /><br />The claim is that a quantum superposition is real so that particles really can be in two places at once until measured in which case the 'position wavfunction' collapses into one of the two positions.<br /><br />Of course the more pragmatic point of view would see quantum superposition as a superposition of possibilities with a given probability of the object being in one position after measurement.<br /><br />The latter interpretation is much more sensible and still consistent with the formalism of quantum mechanics. I really do despair why such a view isn't propagated more widely.<br /><br />I don't understand the need to see quantum mechanics as something mysterious or weird has taken hold on so many people.<br /><br />One can only sympathise with David Hume when he says regarding the tendency of rationalist philosophers to overstate their case<br /><span style="line-height: 200%;"><br /></span><span style="line-height: 200%;">"Though the chain of arguments which conduct us to it were ever so logical, there must arise a strong suspicion β¦that it has carried us quite beyond the reach of our faculties when it leads us to conclusions, so extraordinary, and so remote from common life and experience, </span><i style="line-height: 200%;">We are got into fairy land.</i><span style="line-height: 200%;">β</span><i style="line-height: 200%;"> </i><span style="line-height: 200%;"> (An Enquiry Concerning Human Undestanding VII ) </span><span style="line-height: 200%;">.</span><br /><span style="line-height: 200%;"><br /></span><span style="line-height: 200%;"><br /></span><br /><div class="MsoBodyText" style="line-height: 200%;"><o:p></o:p></div><br /><br /><br /><br /><br /><br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-79988203480393842472015-08-28T17:05:00.000+01:002015-08-28T17:40:08.840+01:00Two Songs Here are two songs that I have set to poems written by my girlfriend Angela<br /><br />The first comfort me was written just after I completed A224, The lyrics are as follows<br /><br /><br /><br /> <u>Lullaby by Angela Brown</u><br /><br /> Sing me a lullaby as I drift to sleep<br /> Hold me so close, whilst my dreams secrets keep.<br /> Lie still beside me, feel the bond that's deep.<br /> Sing me a lullaby as I drift to sleep.<br /><br /> Whisper how precious worlds that collided now reap. <br /> Kiss me so gently for I dare not weep.<br /> Sing me a lullaby as I drift to sleep.<br /><br /> Tell me your sorrows for I am not weak.<br /> Trust in a future we both dare to seek.<br /> Sing me a lullaby, as I drift to sleep.<br /><br />The Scorch file is here and for those who don't want to download Scorch I've included the midi file.<br /><br /><a href="https://dl.dropboxusercontent.com/u/16049029/comfort_me_final.htm">https://dl.dropboxusercontent.com/u/16049029/comfort_me_final.htm</a><br /><br /><br />Midi File<br /><br /><a href="https://dl.dropboxusercontent.com/u/16049029/comfort_me_final.mid">https://dl.dropboxusercontent.com/u/16049029/comfort_me_final.mid</a><br /><br />Musically the accompaniment is based on a figure from Bach's C Major Prelude. The song starts in Eb major and by the end of the first phrase has modulated to Bb major. For the second verse a musical interlude faciliates the moduluation to C minor and then for the final verse the song finds its way back to Eb major. A iib-V7-I cadence in Eb major finishes the song off. For those who managed to down load the Scorch file the roman numerals under the Bass clef give the chords.<br />Each phrase more or less follows the Harmonic Scheme I - - ii vi - IV -V -1 and modulation is effected by pivot chords usually vi - ii for the modulation from Eb major to Bb major and for the transition to the minor key<br /> <br />The second one Angel Man went through a number of iterations and was finally completed earlier this week. Angela envisages some one who is troubled when an Angel visits them whilst he\she is walking through a garden or park giving them reassurance that things will work out ok<br /><br /> <u> An Angel Man by Angela Brown</u><br /><br /> Let's take a walk he said to me<br /> Through a garden filled with love<br /> I will help you set your mind free<br /> To help you hear all those from above<br /> This journey is from me to you<br /> I will give you knowledge rare<br /> Inner peace and blessings true<br /> Gifts for in abundance for you to share<br /> He kissed me gently and said it is done<br /> For your work now has just begun.<br /><br />The accompaniment is that of a stride (appropriate for a walk) and I have used what might be called the magic formula of music, namely the circle of 5ths harmonic progression. That is each chord follows the harmonic scheme I-IV-vii-III-vi-ii-V-I, with two chords to a bar.<br /><br />The song starts in D major modulates to A major then to B minor and back to D major again using pivot chords.<br /><br />One thing that is quite remarkable is how following a basic chord progression enables pieces of music to be created quite easily. Almost algorithmic in fact.<br /><br />The Scorch file is here<br /><br /><u><a href="https://dl.dropboxusercontent.com/u/16049029/NewAngelMan6c.htm">https://dl.dropboxusercontent.com/u/16049029/NewAngelMan6c.htm</a></u><br /><br /><br />and the midi file is here<br /><br /><br /><a href="https://dl.dropboxusercontent.com/u/16049029/NewAngelMan6c.mid">https://dl.dropboxusercontent.com/u/16049029/NewAngelMan6c.mid</a><br /><br /><br />I hope you enjoy both the words and music<br /><br /><br /> Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-8664269730789397832015-08-18T20:53:00.000+01:002015-08-18T21:13:59.055+01:00Test scorch link 2 Minuet in C This is a little minuet I composed (constructed) a while back<br /><br /><a href="https://dl.dropboxusercontent.com/u/16049029/MinuetinC.htm">https://dl.dropboxusercontent.com/u/16049029/MinuetinC.htm</a><br /><br />Hope you enjoy it I'll publish the recipe for constructing a minuet on another post<br /><br />This won't work if you are trying to access it from Google Chrome use internet explorer instead<br /><br />Best wishes Chris<br /><br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-16993996687757328832015-08-03T21:32:00.001+01:002015-08-03T21:32:51.665+01:00M303 Alternative (And free) I have been asked my opinions about M303 the new pure maths course which started last year. Given my car crash with the old topology course which I just scraped a pass at, and my dropping out of M381 I have to say my experience of pure maths at the OU has not been a particularly happy one. Looking at the fora it would seem that M303 has had a number of people criticise it. I suspect part of it is the intrinsic nature of pure maths which really does require quite a different mind set to some one like me who likes the calculational side of things and not the endless forest of definition, lemma, proof that seems to be part of pure maths. However M303 does seem to give some one like me a second bite of the cherry. But I can't see me doing it. It would be 60 points of relentless slog. On the other hand it would be nice to learn about rings fields etc and the more advanced parts of group theory such as the Sylow theorems.<br /><br />There is in fact an alternative supplied by Saylor Academy<br /><br /><a href="https://legacy.saylor.org/">https://legacy.saylor.org/</a><br /><br />which has a number of maths courses free of charge. Unfortunately Saylor have changed their support for a lot of courses and the maths courses have been moved to legacy which means that Saylor will no longer give exams on these topics, Nevertheless if one wants a structured guide to maths at undergraduate level then this would appear to provide an alternative.<br /><br />The maths options<br /><br />MA231 Abstract Algebra I<br /><br /><a href="https://legacy.saylor.org/ma231/Intro/">https://legacy.saylor.org/ma231/Intro/</a><br /><br />MA231 Abstract Algebra II<br /><br /><a href="https://legacy.saylor.org/ma232/Intro/">https://legacy.saylor.org/ma232/Intro/</a><br /><br />would appear to cover the algebra and group theory topics of M303 if not more as the two courses go up to Galois theory.<br /><br />The main text is Judson<br /><br /><a href="http://www.saylor.org/site/wp-content/uploads/2011/07/MA231-1.1.1book.pdf">http://www.saylor.org/site/wp-content/uploads/2011/07/MA231-1.1.1book.pdf</a><br /><br />But there are links to you tube lectures and other goodies.<br /><br />Obviously doing this wont give you a real qualification but as it's free and structured, you wont be overwhelmed with having to do all the exercises. At 4 - 5 hours per week it should be possible to do these two courses in a year.<br /><br />Anyway I have started hopefully I will finish <br /><br /><br /><br /><br /><br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com5tag:blogger.com,1999:blog-7897235423812277683.post-91866004710123707752015-08-01T12:00:00.001+01:002015-08-01T12:24:20.287+01:00Normalisation of relativistic wave functions. Is the Born interpretation viable in relativistic quantum mechanics ? I have just been reminding myself about relativistic particle physics with the help of these wonderful lecture notes given to 4th year undergraduates (MSc level) at Cambridge university (scroll down)<br /><br /><a href="http://www.hep.phy.cam.ac.uk/~thomson/partIIIparticles/">http://www.hep.phy.cam.ac.uk/~thomson/partIIIparticles/</a><br /><br />Anyway a point of significance which is normally skated over, is that in relativistic quantum mechanics, the wavefunctions (solutions to either the Klein Gordan equation or the Dirac equation ) are normalised such that the Integral of the modulus squared of the wave function over a volume is eqaul to 2E where E is the energy of the particle, This is in contrast to the case in non relativistic quantum mechanics where the integral of the modulus squared of the wave function is put equal to 1 ie for Non relativistic quantum mechanics we have<br /><br />$$\int \psi^*\psi dV = 1 $$ but in relativistic quantum mechanics we have<br /><br />$$\int \psi^*\psi dV = 2E $$<br /><br />The reason for this normalisation is that in relativity volumes contract in proportion to the energy of the particle and the factor of 2 is conventional. In non relativistuc quantum mechanics the normalisation effectively means that there is 1 particle per unit volume. Whereas in relativistic quantum mechanics there are now 2E particles per unit volume.<br /><br /><br />In non relativistic quantum mechanics the normalisation to unity, leads to the Born interpretation and despite all the hoo hah in the popular literature about how quantum mechanics is not understandable, The so called wave function (solution to Schrodinger's equation) has a fairly simple interpretation as effectively the square root of a probability density function, albeit in order to account for quantum phenomenon this square root of the probability density function is often a complex function and not a real one.<br /><br />However given that probabilities are constant and not functions of energy, the direct link between the solution to Schrodinger's equation and a probability density function is no longer there in relativistic quantum mechanics, Those who spend all their time trying to understand the 'meaning' of the wave function in quantum mechanics are going to have change their understanding when it comes to relativistic quantum mechanics.<br /><br />There is a solution in terms of particle currents which I will post a subsequent blog on but I do find it surprising given all the hoo hah there is about the interpretation of quantum mechanics and focusing on the solution to Schrodinger's equation that this difference hasn't been emphasised at all, At least superficially it would mean that the Born interpretation is no longer applicable to relativistic quantum mechanics,<br /><br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-12060068200079580162015-07-22T20:06:00.001+01:002015-08-18T19:26:45.256+01:00Test of scorch link Hi this is a test of a scorch link<br /><a href="http://www.tonybloomfield.pwp.blueyonder.co.uk/kevock/">http://www.tonybloomfield.pwp.blueyonder.co.uk/kevock/</a><br /><br />Scorch is an Application which enables posting of Sibelius Files to be accessed by anyone. If you want to hear the piece then simply click on tthe link and if you haven't already installed Scorch then you will be invited to do so. It's free a word of warning though if you are using Google Chrome as your browser then it wont let you access it as for whatever reason best known to them. Google Chrome has decided to not support Java Applications any more. So use Internet Explorer or firefox.<br /><br /><br />Anyway hopefully I'll be able to post some of my own pieces here soon.<br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-15700563264678884222015-04-26T18:51:00.001+01:002015-04-27T19:41:54.946+01:00M346 Linear Statistical Modelling Well after two years absence from the OU I have decided to complete my BSc in Mathematics and Statistics under the old regulations I need to do two 3rd level statistics modules namely M346 Linear Statistical modelling and M347 Mathematical Statisitics and these need to be completed by 2017.<br /><br />As I'm busy with other things namely music theory and piano and also my own studies I feel it best to do one module at a time. So will do M346 first and then M347. I did start M346 a while back but it got in the way of other OU courses and to be quite frank I wasn't all that impressed with it to say the least. soo I abandoned it<br /><br />Maybe I will find it better 2nd time around however it is a necessary step for a named degree so I shall treat it a bit more seriously than I did last time<br /><br />Apologies for not posting for a while.My immediate focus is on grade 6 music theory and I shall be sitting the exam on June 20th. Then start preparing for Grade 2 piano in November.<br /><br />As far as maths is concerned I have been working through the Cambridge Mathematical methods example sheet on Fourier Series<br /><br /><a href="http://www.damtp.cam.ac.uk/user/examples/">http://www.damtp.cam.ac.uk/user/examples/</a><br /><br />The first half of Sheet 8a<br /><br />The last problem has imtroduced me to a phenomenon which I haven't come across before called the Gibbs phenomenon. Namely that near a discontinuity such as those found in a square wave there will always be an overshoot or undershoot in the Fourier Series no matter how many terms in the series are taken. When I have written up the problem I'll describe it in more detail in my next post.<br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com1tag:blogger.com,1999:blog-7897235423812277683.post-9509594266952101062014-11-29T00:18:00.000+00:002014-11-29T00:18:17.675+00:00True Patriotism or NationalismThis rant is a lot less objective than I usually am but I am getting really sick of politicians and the press appealing to a bogus sense of patriotism and I despair of parties that should know better such as the :Labour party for abandoning core values of the basis of socialism such as that what ever country you come from shouldn't matter as to how you are treated.<br /><br />Dr Johnson a person, who given my instintively left wing sympathies I would not have had any sympathies with, summed the current situation up precisely, when he claimed that patriotism was the last refuge of a scoundrel. It seems to me that both Alex Salmond and his successor Nicola Sturgeon and Nigel Farage are all scoundrels in that respect. Their so called arguments are essentially appeals to the lowest denominator of tribal instincts.<br /><br />None of them appeal to what has actually made <u>Britain </u>great (and I use that word delibrately as opposed to Scotish or English) the Britain that makes me proud is the contribution to the intellectual and aesthetic contribution to the world not the flag waving nationalism of those who thought it right to display a saltire during the referendum or a St George cross in rochester.<br /><br />No, the Britain that makes me proud is the scientific acheivements of Newton, James Clerk Maxwell, Charles Darwin, Crick and Watson, and Fleming and Peter Higgs, the mathematical achievements of Napier, Hamilton, and Wiles, The poetry of Shakespeare,. Dryden, Milton, Pope, Byron Shelly Keats Coleridge, Wordswith and Burns. The novels of Austen,. the Bronte sisters, Charles Dickens, Robert Louis Stevenson, Thomas Hardy and George Orwell. The philosophy of Locke, Berkely Hume, J S Mill, and Bertrand Russell. The music of Purcell, Handel (who spennt most of his compostional life here), Elgar, Vaughan Williams, Benjamin Britten. (And I am sure there are plenty of others who I have missed) <br /><br />How many of those idiots who feel it necessary to drap a Saltire or a St George cross outside their windows even know who most of the people I have mentioned above are or even appreciate what the afoerementioned people have achieved. <br /><br />No it's all a bogus appeal to the lowest common denominator of tribal instinct to enable the scoundrels who dress themselves up in the nationalist flag of whatever country be it Scotland or England without any appreciation of the people in their countries who have achieved lasting achievements which will transcend the vagaries of the current political situation, as opposed to an appeal to the lowest common denominator of flag waving nationalism of the worst kind. <br /><br />If that makes me snobbish I don't give a monkeys.<br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-2571234648828731382014-11-23T12:44:00.003+00:002014-11-23T12:44:41.427+00:00Piano Grade 1 result Pass This is just a short post to let those lknow that I passed my Grade 1 piano exam with 110 marks. A respectable pass but nothing great. I think I'll have to postpone my booking for the Usher hall :)<br /><br />My main problem is that I find it difficult to maintain a steady pulse and two hand co-ordination this is only going to get more difficult as time progresses. I am handicapped by not having a natural sense of rhythm which some people seem to have. I was never good at dancing, I don't buy the idea as some people seem to think that you either have it or you don't. I believe firmly that by training you can gradually improve. Although I do find trying to follow a metronome really tedious and boring. Obviously it's more difficult but I believe you can actually get there. I must have improved my sense of basic pulse from last time as I was able to do the hand clapping example without any problem whereas at my last resit. I was unable to do so.<br /><br />I am now faced with a dilemma, as the ABRSM syllabus for grade 2 has changed but you are allowed to do the past syllabus for the next time round which would be in March. I have been studying the three pieces on and off and am reasonably confident about two of them, so I could put myself forward for the grade 2 exam but unless I get my rhythm sorted out then it would be marginal whether or not I pass. I will make the decision in the next two months or so<br /><br />On another topic I want to make a few remarks about Duncan's comment to my last post that we don't know anything about the nature of dark matter or dark energy. That's not entirely true, we know their equations of state and also their ratios. Also we can use the Friedmann equations to predict the future expansion rate of the universe. In a way the situation is similar to the classical ideas of gravity, electromagnetic fields and entropy. It was never possible to get an idea of what these entities actually were however just like dark matter and dark energy their effects on other objects were able to be predicted and measured. To some extent it doesn't really matter that we don't know what dark energy or dark matter actually are because as inputs to cosmological models their effects on the universe can be calculated. Just as say knowing that the gravitational force between two bodies obeyed an inverse square law enabled planetary motions to be predicted. I would say this is more important, given the lack of experimental evidence, than a fruitless attempt to undestand the true nature given the lack of experimental evidence as to the nature of dark matter or dark energy.<br /><br />It seems to me that physics has made great strides in developing our understanding of nature works, without getting bogged down in trying to understand the actual nature of some of the important entities that it has used. I'll leave the speculation to those cleverer than me in the mean time I want to focus on the uses of physics which is more than enough to keep us busy for a long time. <br /><br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0tag:blogger.com,1999:blog-7897235423812277683.post-71934114767745842192014-11-06T18:03:00.000+00:002014-11-06T18:03:09.476+00:00Grade 1 piano Resit Hi first of all apologies for not posting for a while. I guess without the pressure of OU TMA's there is not that much to post about<br /><br />Anyway today I resat my grade 1 piano exam. I felt a lot more comfortable than before and think I've done enough to pass but it won't be a great one. So how do I feel slightly ambivalent to say the least. I know on a good day I can knock off the pieces set for grade 1 no problem, but in a different environment it's not so easy. I still had problems adjusting to the feel of the piano in the church as opposed to my home one. Maybe I should ask the church if I can spend a few hours practisinng on their piano.<br /><br />So I stumbled on most of the tasks I was asked to do but unlike last time I was able to recover. Is that enough to pass who knows hopefully yes but for me piano playing isn't a natural thing. I'll persist in practice and see how far I can go, until I really do hit a brick wall.<br /><br />As for other stuff I have been trying to understand the cosmological revolution of the late 1990's where it was established that the current universe is<br /><br />a) Accelerating<br /><br />b) Appear's to consist of 70% dark energy and 30% matter of which 26% is so called dark matter and only 4% is baryonic matter<br /><br />c) Flat<br /><br />I hope to finish my write up of this by the start of the new year until then watch this space<br /><br />Best wishes Chris<br /><br />Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com1tag:blogger.com,1999:blog-7897235423812277683.post-74647610067637142062014-08-24T18:57:00.002+01:002014-08-24T19:02:42.425+01:00Orbitals In another round of the 'quantum wars' on the OU fora we have got round to discusssing orbitals and what they mean. It wasn't till I started thinking about it that I realised the usual picture of an electron orbiting the nucleus in an orbit of fixed radius like a mini-solar system, and that if it loses energy it 'jumps' from one to another is totally misleading to say the least. The latter view is definitely true of the Bohr model and I guess most people carry this over to their thinking when (if) they study quantum physics further.<br /><br />However it is just not correct, When you solve Schrodinger's equation for the hydrogen atom what, one ends up with for each energy level is a 3 dimensional probability density function which gives the probability of finding an electron in a given region. This means that the electron in a given orbital can be anywhere in principle allowed by the particular orbital (better called probability density function) so there is a small but finite probability that it could be 1 m, 10m and so on away from the nucleus. Of course it will be closer to the average of the probabiity density function. <br /><br />The crucial point is however that when the electron either gains or loses energy that loss or gain in energy is fixed (quantised). However the electron does not 'jump' from one orbital to another what happens is that the probability density function changes in accordance with the appropriate function for that energy level.<br /><br />Thus the misleading picture given by the Bohr model is totally inadequate to do justice to the picture presented by the solution to Schrodinger's equation. <br /><br />Matter's aren't helped by the depiction of the energy level diagrams with their pictures of arrows going from one energy level to another giving the impression that the particle is actually jumping from one energy level to another. But those 'jumps' are changes in <u>energy </u>not position.<br /><br />Finally it is important to remember that those pictures of orbitals shown in chemistry or physics textbooks are pictures of probability density functions. The usual convention when drawing the boundary surfaces is to draw the boundary marking off the region where the electron is likely to be found 95% of the time they do not represent the fixed distance of the electron from the nucleus of the atom. 95% of the time the electron will be inside the so called 'shell' but there is also a 5% probability that it will be outside the shell. Here are some pretty pictures of hydrogen atom probability density functions for you you to drool over.<br /><br /><a href="https://search.yahoo.com/search?ei=utf-8&fr=ytff1-yff22&p=Pictures%20of%20hydrogen%20atom%20orbitals%20&type=">https://search.yahoo.com/search?ei=utf-8&fr=ytff1-yff22&p=Pictures%20of%20hydrogen%20atom%20orbitals%20&type=</a><br /><br /> And here is the first experimental observation of the probability density function of the hydrogen atom<br /><br /><a href="http://io9.com/the-first-image-ever-of-a-hydrogen-atoms-orbital-struc-509684901">http://io9.com/the-first-image-ever-of-a-hydrogen-atoms-orbital-struc-509684901</a><br /><br /><a href="https://search.yahoo.com/search?ei=utf-8&fr=ytff1-yff22&p=Pictures%20of%20hydrogen%20atom%20orbitals%20&type="><br /></a>Chris Fhttp://www.blogger.com/profile/03530660309554429226noreply@blogger.com0