Friday, 28 August 2015

Two Songs

Here are two songs that I have set to poems written by my girlfriend Angela

The first comfort me  was written just after I completed A224, The lyrics are as follows

                                       Lullaby  by Angela Brown

                                       Sing me a lullaby as I drift to sleep
                                       Hold me so close, whilst my dreams secrets keep.
                                       Lie still beside me, feel the bond that's deep.
                                      Sing me a lullaby as I drift to sleep.

                                       Whisper how precious worlds that collided  now reap.
                                        Kiss me so gently  for I dare not weep.
                                       Sing me a lullaby as I drift to sleep.

                                       Tell me your sorrows for I am not weak.
                                       Trust in a future we both dare to seek.
                                       Sing me a lullaby, as I drift to sleep.

The Scorch file is here and for those who don't want to download Scorch I've included the midi file.

Midi File

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.
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
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

                                              An  Angel Man by Angela Brown

                                               Let's take a walk he said to me
                                               Through a garden filled with love
                                               I will help you set your mind free
                                               To help you hear all those from above
                                               This journey is from me to you
                                                I will give you knowledge rare
                                               Inner peace and blessings true
                                               Gifts for in abundance for you to share
                                               He kissed me gently and said it is done
                                               For your work now has just begun.

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.

The song starts in D major modulates to A major then to B minor and back to D major again using pivot chords.

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.

The Scorch file is here

and the midi file is here

I hope you enjoy both the words and music


Tuesday, 18 August 2015

Test scorch link 2 Minuet in C

This is a little minuet I composed (constructed) a while back

Hope you enjoy it I'll publish the recipe for constructing a minuet on another post

This won't work if you are trying to access it from Google Chrome use internet explorer instead

Best wishes Chris

Monday, 3 August 2015

M303 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.

There is in fact an alternative supplied by Saylor Academy

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.

The maths options

MA231 Abstract Algebra I

MA231 Abstract Algebra II

would appear to cover the algebra and group theory topics of M303 if not more as the two courses go up to Galois theory.

The main text is Judson

But there are links to you tube lectures and other goodies.

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.

Anyway I have started hopefully I will finish

Saturday, 1 August 2015

Normalisation 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)

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

$$\int \psi^*\psi dV = 1 $$ but in relativistic quantum mechanics we have

$$\int \psi^*\psi dV = 2E $$

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.

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.

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.

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,