Category Archives: Quantum mechanics

On Schrödingers cat & an example known as the envelope problem.

Today the Youtube channel SciShow had one more video out on quantum mechanics and as such the famous cat of the Schrödinger cat in a box problem comes along once more.

As usual we are told the cat can be in a super position of being alive and dead at the same time. I wonder why people think that this can be true, as far as I know history the Schrödinger guy came up with this example as an antidote as being everything into a super position…

I suppose you already know what the cat in the box setup is. The cat dies if just one radioactive atom decays yes or no. If you are outside of the box it makes sense to use a probalistic model of the situation, but does this mean that in reality inside the box the cat is dead and alive at the same time? After all the cat will be the very first to observe if radio active decay has happened because as soon as it does the state of the cat goes from alive to dead. So inside the box there is at least one observer present and as such all quantum states we are interested in (radio active decay yes or no) is constantly measured all of the time.

For myself speaking I use the fact that a cat cannot be in a super position of being alive and dead as an example that an individual atom cannot be in a state where radio active decay has passed yes or no.

That does not mean quantum particles cannot be in super positions, for example photons behave often like they took all possible paths to arrive somewhere. But as soon as there are all kinds of different energy levels involved this becomes more and more problematic. For example can a particle be in a super position of being a neutron and a proton? Can a particle be in a super position of being an electron and a positron? Can a particle be in a super position of being a hydrogen ion (a proton) and a plutonium atom?

Energy is at the heart of the quantum measurement problem: In order to measure a quantum particle some kind of interaction with the particle must be there. This interaction changes (or not) the state of the particle. It is a bit like this: Suppose I am sitting in my home country and I have to measure the length of some grassfield in Germany or Belgium but I can only use atom bombs for that. No matter how smart I craft my grass length measuring device, the giant explosions from the atom bomb will bring a great uncertainty in the outcome of the measurements… Here is the video:

The cat is also an observer…

Ok, now for the lesser known but rather interesting envelope exchange problem. In a nutshell it goes as next:

You can choose one of two closed invelopes and they contain money. The only thing you are told is that the amount in one of the envelopes is double that of the other envelope.

Now you play the game and you choose one of the envelopes, let’s say it contains 100€. You are asked by the quiz master if you want to keep those 100€ or that you want to change your choice and go for the other envelope.

You think about that for a few seconds and you figure out: If this envelope has 100€ and given the rules of the game, the other envelope contains 50€ or 200€ with equal probability of 50%. Suppose I want to swap to the other envelope, what is my expectation for the amount of money? That is simple, both 50€ and 200€ have 50% probability so the expectation of swapping becomes 0.5*50 + 0.5*200 = 125€. Therefore it makes sense to swap and choose the other envelope.

But hey, whatever envelope you choose at first and you find X money in it, isn’t it weird to swap that always? If you would have chosen the other envelope you would also swap…

This envelope swap problem or paradox has a relative simple solution: You assume equal 50% probabilities for having double or half the amount of money you found in the first envelope. But in that case the whole thing crashes because you are now calculating with three outcomes: the 100€ from the first envelope and two other amounts 50 and 200 Euro while there are only two enveloples. It is unwise to calculate the expectation values because the 50€ and 200€ exclude each other: if the outcome 50€ is observed all of the time the 200€ was non existant. And as such the expectation value makes no sense for an individual experiment.

Ok, let me end this post with a standard wiki around the two envelope thing: Two enveloples problem. https://en.wikipedia.org/wiki/Two_envelopes_problem

End of this post.

Calculation of the 4D complex number tau.

It is about high time for a new post, now some time ago I proposed looking at those old classical equations like the heat and wave equation and compare that to the Schrödinger equation. But I spilled some food on my notes and threw it away, anyway everybody can look it up for themselves; what often is referred to as the Schrödinger equation looks much more like the heat equation and not like the classical wave equation…

Why this is I don’t know.

This post is a continuation from the 26 Feb post that I wrote after viewing a video from Gerard ‘t Hooft. At the end of the 26 Feb post I showed you the numerical values for the  logarithm of the 4D number tau. This tau in any higher dimensional number system (or a differential algebra in case you precious snowflake can only handle the complex plane and the quaternions) is always important to find.

Informally said, the number tau is the logarithm of the very first imaginary component that has a determinant of 1. For example on the complex plane we have only 1 imaginary component usually denoted as i. Complex numbers can also be written as 2 by 2 matrices and as such the matrix representation of i has a determinant of 1.
And it is a well known result that log i = i pi/2, implicit the physics professors use that every day of every year. Anytime they talk about a phase shift they always use this in the context of multiplication in the complex plane by some number from the unit circle in the complex plane.

In this post, for the very first time after being extremely hesitant in using dimensions that are not a prime number, we go to 4D real space. Remark that 4 is not a prime number because it has a prime factorization of 2 times 2.

Why is that making me hesitant?
That is simple to explain: If you can find the number i from the complex plane into my freshly crafted 4D complex number system, it could very well be this breaks down to only the complex plane. In that case you have made a fake generalization of the 2D complex numbers.

So I have always been very hesitant but I have overcome this hesitation a little bit in the last weeks because it is almost impossible using the complex plane only to calculate the number tau in the four dimensional complex space…

May be in a future post we can look a bit deeper in this danger; if also Cauchy-Riemann equations are satisfied in four real variables, that would bring a bit more courage to further study of the 4D complex number system.

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After the introduction blah blah words I can say the 4D tau looks very beautiful. That alone brings some piece of mind. I avoided all mathematical rigor, no ant fucking but just use numerical results and turn them into analytical stuff.

That is justified by the fact that Gerard is a physics professor and as we know from experience math rigor is not very high on the list or priorities over there…

That is forgiven of course because the human brain and putting mathematical rigor on the first place is the perfect way of making no progress at all. In other sciences math should be used as a tool coming from a toolbox of reliable math tools.

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This post is seven pictures long, all are 550 by 775 pixels in size except for the last one that I had to make a little bit longer because otherwise you could not see that cute baby tau in the 4D complex space.

Here we go:

Just take your time and look at this ultra cute number tau.

It is very very hard to stay inside the complex plane, of course the use of 4 by 4 matrices is also forbidden, and still find this result…

I am still hesitant about using dimensions that are not prime numbers, but this is a first result that is not bad.

End of this post.

I am innocent, I did not do it. I just found the numbers tau in the Schrödinger equation your honour…

Judge: But you were caught red handed placing the number tau in a Schrödinger equation while you do not qualify for being a member of the most bright and enlightened persons in our society: The PHYSISCS PROFESSORS.

Reinko: But Judge I can explain, it was that evil guy that Gerard ‘t Hooft who did it. I can prove that because it is on video.

Judge: Yes you already told that into the statements you made after arrest by the police. So we took the freedom and ask Mr. Gerard ‘t Hooft himself about the evil you have done with molesting the Schödinger equation. Mr. ‘t Hoofd said it had to be Hermitian and although I do not know what that means he said that by using anti-Hermitian matrices you, Reinko Venema, you are nothing more as some sadistic pedophile piece of shit.

Reinko: But judge, it is not Hermitian, that is only a trick. You see if you multiply it by the number 1 like 1 = – i squared you see it is not Hermitian.

Judge: Do you think we get complex analysis in law school? We don’t, we asked some experts and all agreed that Gerard is right and you are wrong and right now rewarded by your own evil deeds to 75 years in prison in a maximum security facility.

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After this somewhat strange introduction I repeat I was innocent. I was just looking at a video of a guy that is just like me old and boring.

And that guy, Gerard ‘t Hooft, was able to give me three nice punches in the face.
That is what this post is about; Three punches in the face as delivered by Gerard.

It is the very first time I observe professional physics professors using the number tau while claiming the stuff has to be Hermitian to make any sense.

I was devastated because in my little world of mathematics it had to be anti Hermitian so at a first glimpse it looks like a simple shootout between Gerard and me: Only one can be right…

Let me first show you the Youtube video where right at the start Gerard succeeds to bring my small sack of human brain tissue into an exited state and after that I am rewarded with finding the number tau into the famous Schrödinger equation.

Let me also temper the enthousiasm a little bit because at present date 26 Feb in the year 2018 I only know of one example where three quantum states are rotated into each other:
That is the transport of the color charges as it is found on the quarks inside the proton and neutron…

Here is the video, after that the nine pictures that make up the mathematical core of this new post:

Gerard ‘t Hooft – How Quantum Mechanics Modifies the Space-Time of a Black Hole (QM90)

Let me spare you a discussion on the entire video but only look at what you can find on the very introduction as shown above because all of the three punches at my face are already found there.

Here are the nine pictures for this new post:

For readers who have found themselves lost on what a Hermitian matrix is, here is a wiki:

Hermitian matrix
https://en.wikipedia.org/wiki/Hermitian_matrix

And for readers who have found themselves lost on finding an ‘analytic handle’ about how to calculate matrices like in picture 09, a good starter would be about the calculation of the 7D number tau:

An important calculation of the 7D number tau (circular version).

That’s it, till updates.

Oops; CERN did not find magnetic monopoles.

It has to be remarked that the physics folks are very persistant to keep on trying to find the so called Dirac monopole. How this has come to be is still a miracle to me. After all if the electron has one electric charge and for the rest it is a magnetic dipole, it would look naturally to look for a particle that is a magnetic monopole and an electric dipole at the same time…

But I have never heard about such an investigation, it is only the Dirac magnetic monople and that’s it.

Here is a quote from sciencenews dot org:

If even a single magnetic monopole were detected, the discovery would rejigger the foundations of physics. The equations governing electricity and magnetism are mirror images of one another, but there’s one major difference between the two phenomena. Protons and electrons carry positive and negative electric charges, respectively, but no known particle has a magnetic charge. A magnetic monopole would be the first, and if one were discovered, electricity and magnetism would finally be on equal footing.

Source:

Magnets with a single pole are still giving physicists the slip
https://www.sciencenews.org/article/magnetic-monopoles-single-pole-physics

Comment on the quote: Because in my view I consider the electrons having one electrical charge and one of two magnetic charges, I think we have a nice equal footing of electricity and magnetism… (End of the comment.)

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Back to CERN and stuff. Last month it came out that the MoEDAL experiment has failed in the sense that no magnetic monopoles were observed. Here is a small screenshot from the preprint archive stuff:

Comment: No idea what these people are talking about when they talk about 68.5 times the electric charge… Are they talking about electric charge or magnetic charge?
(End of comment)

Source of the content of the picture above:

detector in 2.11 fb−1 of 13 TeV proton-proton collisions at the LHC.

https://arxiv.org/pdf/1712.09849.pdf

After a bit of searching I found back this beautiful video, coming from CERN, explaining how to find magnetic monopoles. It is clear they never ever studied the electron.

Yeah yeah my dear average CERN related human; what exactly is a magnetic monopole?

Does it have electric charge too and why should that be?

In my view where the electrons carry both electric and magnetic charge, a magnetic monopole with zero electric charge just does not exist.

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Ok, let me bring this post to an end by observing that at CERN they were not capable in the year 2017 of detecting the magnetic monopole as it should exist following the lines of thinking like Paul Dirac once did.

So that is a good thing because after thinking about four years about magnetism it would be horrible for me to find that at CERN they had a major discovery about magnetic monopoles…

Sorry CERN folks, your failure to find magnetic monopoles your way does not prove that electrons are indeed carrying magnetic charge. It just makes it a little bit more plausible that they do…

So my dear CERN folks, thanks for publishing your failure because for me it is another tiny quantum move into the direction of accepting the electron as it is.

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End of this post.

More on the Majorana equation.

Yesterday I finally looked into the so called Majorana equation and it is easy to find where the Dutch universities have gone wrong. At the technical universities in Delft and Eindhoven they use electrons together with a hole that supposedly has a positive electrical charge so that the overall combination of electron and hole is electrically neutral.

And it is very easy to explain: If I am in the right and electrons also carry magnetic charge, the above constellation of an electron and a hole is not magnetically neutral like, for example, the Cooper pairs of electrons in super conductivity.

They want unpaired electrons because the Cooper pairs live there in the nano wire where the super conductivity is so they do not consider an electron pair together with two holes because that is both magnetic and electrically neutral…

No, I do not think that in Delft they found the elusive Majorana fermion. But time will tell because if this way of quantum computing will keep on failing or never get anywhere, I can use that as a future reason of why electrons cannot be magnetic dipoles.

Here is a wiki about the Majorana equation, already at equation number 2 I am lost in the woods because the mass m suddenly goes to the other term in the equation.

Majorana equation
https://en.wikipedia.org/wiki/Majorana_equation

And here is a short video from Youtube where the technical university of Eindhoven explains how they will try to prove the existence of the Majorana fermion as a quasi particle. The video is from 23 August 2017, that is only four and a half months ago.

From the wiki we have this information, what the differential operator with the ‘Feynman slash’ does is actually not important at all. The nice thing here is to understand what they try of find here:
A particle (or a collection of particles, the quasi particles) where all charge is compensated. Apperently the mass related to charge comes in with opposite charge and indeed if you can find solutions to such a wave equation you might hope to find it one day.

Yet in Delft and Eindhoven they hang on to the opinion that electrons are magnetic dipoles and as such they never had a need to put the ‘anti part’ of the magnetic dipole into the problem…

That was more or less what I had to say about the Majorana equation.
Of course I also wish you a happy new year! Till updates.

Prediction for 2018 and beyond: The Delft quantum computer attempts will fail.

Already for a few years the folks at the university of Delft are trying to make a quantum computer. They even teamed up with Microsoft and as memory serves the Dutch government is investing about 100 million € over the course of 10 years.

Only recently I dived into that Delft stuff and the spokeswoman from Microsoft was even talking about a Nobel prize for Leo Kouwenhoven because he seemed to have discovered so called Majorana fermions.

And I just felt sooooo proud that my fellow Dutch guy Leo who is sooooo ultrasmart would have a chance of winning such a prestigious prize like the Nobel prize. I will never get a Nobel prize for my stupid finding of the magnetic monopoles, come on that is not important because I am not a university person and Leo is a full blown physics professor.

After having said that it is nice to observe that the Delft team is trying to craft quantum computer with qubits made from Majorana fermions. So what are Majorana fermions because they have never been found since a guy named Ettore Majorana speculated about stuff like that in 1937? Well these are fermions that are their own anti particle.

It is well known that when you have a particle with a particular charge, the anti particle must have the opposite charge. Now our Leo Kouwenhoven genius from the Delft university is putting an electron into entanglement with an electron hole and as such it has no electrical charge if the electron hole has a positive electrical charge.

Furthermore since an electron entangled with a hole is only like half a fermion they cannot exist on their own so our genius folks from Delft figured out that two of those quasi particles would form a Majorana fermion.

Here is a Youtube video of about one hour long where our super hero Leo explains it all:

Majorana Fermions: Particle Physics on a Chip- Leo Kowenhoven – May 28 2015

Anyway, to make a long story short:

The Majorana particles as found by the heroic members of the Dutch university of Delft have a tiny problem: the electrons carry also magnetic charge beside the electrical charge. So a quasi particle made up of an electron and an electron hole cannot have the Majorana property of being it’s own anti particle…

So my estimation is rather simple: As long as the Delft hero’s keep on ignoring that electrons carry also magnetic charge, they will not succeed. On the contrary they will fail and very likely they will keep on failing because they are university people.

Too much money and too much titles & prestige, why should they change and get a more realistic view on quantum computing?

Before we split, here is a wiki on Majorana fermions. For me it is new that when a fermion is it’s own anti particle the wave function is real valued and not complex valued. As a take away you can also conclude that the Delft hero’s also got the wave function of the electron and electron hole completely wrong. Just like all those people in the science of chemistry who cannot model even the hydrogen molecule properly. So the chemistry people say ‘We need quantum computers’ and Leo Kouwenhoven says ‘I have great ideas in topological quantum computing!’

In my view these people are all crazy, but here is the wiki stuff on Majorana fermions:

Majorana fermions
https://en.wikipedia.org/wiki/Majorana_fermion

Till the next post.

Electron spin as explained by the Scientific American.

In a nice article there are three people explaining, for example, electron spin. The reason to post this here is because they are in climbing order of stupidity and explainer number three gives a total retarded explanation.

Recall once more that the name electron spin is one hundred percent misleading because of what we know of the size of the electron it should certainly be rotating much faster than the speed of light even if all electrical charge was concentrated on the equator of the electron.

I hope that by now my dear reader you know that I think electrons carry beside electrical charge also magnetic charge and as such they come in two flavours:
1) Electrons with a negative electrical charge and a north magnetic charge and;
2) Electrons with a negative electrical charge and a south magnetic charge.

Because particles with mass cannot mover faster than the speed of light, all explanations based on the electron spinning are wrong by definition. Therefore it is often said that electrons (and also protons and neutrons) have so called intrinsic spin so the rotation problem can be avoided.

It has to be remarked once more that this is about the fourth year I am writing about electrons having magnetic charge and that as such they are the long sought magnetic monopoles, but until now I have zero reactions from only one of those professional physics professors… That abundantly shows how dumb they actually are and that there is little use in trying to write a real publication because it is still totally impossible to pass the so called ‘peer review barrier’. I mean; read the quotes I will post from these three people as found in the Scientific American and suppose they would be the ones that do the peer review of my article. What would happen?
Very simple: It will be rejected.

Let’s get started, here is the title and link to the small article in the Scientific American:

What exactly is the ‘spin’ of subatomic particles such as electrons and protons? Does it have any physical significance, analogous to the spin of a planet?
https://www.scientificamerican.com/article/what-exactly-is-the-spin/

The first quote is from Morton Tavel, quote:

“Unfortunately, the analogy breaks down, and we have come to realize that it is misleading to conjure up an image of the electron as a small spinning object. Instead we have learned simply to accept the observed fact that the electron is deflected by magnetic fields. If one insists on the image of a spinning object, then real paradoxes arise; unlike a tossed softball, for instance, the spin of an electron never changes, and it has only two possible orientations. In addition, the very notion that electrons and protons are solid ‘objects’ that can ‘rotate’ in space is itself difficult to sustain, given what we know about the rules of quantum mechanics. The term ‘spin,’ however, still remains.”

Comment: From the macroscopic world we do not observe much ‘deflection’ of, let’s say, bar magnets in the presence of other magnets and magnetic fields. If electrons really were magnetic dipoles, because electrons are so small all magnetic forces would cancel out and we would never observe deflection.
And if Morton Tavel would have done some calculations or estimations, it is extremely hard for electrons to get deflected by non-constant magnetic fields. On the contrary, you need magnetic fields with a gradient of millions of Tesla’s per meter in order to accelerate the electron with only one meter per second squared…
No idea is smarthead Morton Tavel will ever read these words I write about him, but in reason number 50 I did such an estimation. Here is the link:

14 Oct 2017: Reason 50: A calculation on electron acceleration by a magnetic field.
http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff02.htm#14Oct2017

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Let’s proceed with the second physics professional, his name is Kurt T. Bachmann and here is the quote from the wisdom he has to share:

“Starting in the 1920s, Otto Stern and Walther Gerlach of the University of Hamburg in Germany conducted a series of important atomic beam experiments. Knowing that all moving charges produce magnetic fields, they proposed to measure the magnetic fields produced by the electrons orbiting nuclei in atoms. Much to their surprise, however, the two physicists found that electrons themselves act as if they are spinning very rapidly, producing tiny magnetic fields independent of those from their orbital motions. Soon the terminology ‘spin’ was used to describe this apparent rotation of subatomic particles.

“Spin is a bizarre physical quantity. It is analogous to the spin of a planet in that it gives a particle angular momentum and a tiny magnetic field called a magnetic moment.

Comment: It is important to know that the original SG experiment was done with evaporated silver ions, this beam of silver ions was split in two parts by just a few unpaired electrons. If the professionals would do the calculations they would find this cannot be explained by inhomogeneous magnetic fields. The fact that no one says this makes clear they have never done the calculations needed…
That is the same as a carpenter that refuses to use the handsaw when needed or simply states: I do not need a screw driver, I just talk to these screws until the matter is resolved. Normally the carpenter would get fired but all those physics professors are glued to their seats living in ‘academic freedom’.

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The third person is truly 100% crazy, the term ‘intrinsic spin’ for the electron was used in order to avoid the problems with the spinning of an electron. And what does this weirdo named Victor J. Stenger make from this? Quoting this idiot:

“Spin is the total angular momentum, or intrinsic angular momentum, of a body. The spins of elementary particles are analogous to the spins of macroscopic bodies. In fact, the spin of a planet is the sum of the spins and the orbital angular momenta of all its elementary particles. So are the spins of other composite objects such as atoms, atomic nuclei and protons (which are made of quarks).

“In classical physics, angular momentum is a continuous variable. In quantum mechanics, angular momenta are discrete, quantized in units of Planck’s constant divided by 4 pi. Niels Bohr proposed that angular momentum is quantized in 1913 and used this to explain the line spectrum of hydrogen.

Comment: This is so utterly stupid it is hard to comment upon. By talking about intrinsic angular momentum he only shows that he thinks the electron is spinning. So he is a nutjob for sure.

 

Ok, end of this post without pictures but with three idiots as found in the Scientific American. Now some people might think I better be a little bit more diplomatic but from 1992 until 2012 I was very very diplomatic about higher dimensional number and thought that if you give people time enough that in the end they will do the right thing.

Two decades of diplomacy are gone, now I know that when confronted with idiots you better explain why they are idiots…

See you in the next post my dear reader.

On a way to find more equations so that the 1D existence of exponential curves in all possible dimensions is assured.

In part this post picks up where I left the stuff of the missing equations back in the year 2015. The missing equations are found inside the determinant equation; for this to succeed we must factorize determinant of the matrix representations of higher dimensional numbers. A well known result from linear algebra is that the determinant is also the product of the eigen values; so we need to craft the eigen value functions that for every X in our higher dimensional number space give the eigen values.

These eigenvalue functions are also the discrete Fourier transform of our beloved higher dimensional numbers and these functions come in conjugate pairs. Such a pair form two factors of the determinant and if we multiply them we can get rid of all complex coefficients from the complex plane.

A rather surprising result is the fact that if we subtract a cone equation from a sphere equation we get a cylinder…

This post is also a way of viewing the exponential circles and curves as an intersection of all kinds of geometric objects like the unit sphere, (hyper) cones, (hyper) planes and (hyper cylinders. Usually I represent it all as some analysis but you can take a very geometric approach too.

I have no idea if the shape of the higher dimensional curves is studied as a geometrical object; I suspect this is not the case since the use of complex numbers outside the complex plane is very seldom observed. The professionals just want their tiny fishing bowl (the complex plane) and declare it an Olympic swimming pool…
Well, let it be because these people will never change.

All in all this post is 20 pictures long (size 550 x 775) so it is a relatively long read.

                                     

The pictures of the graphs were all made with an applet named Animated drawing, here is a link and there you can find it under ´Online calculators and function plotters´±

https://wims.sesamath.net/wims.cgi

For example you can cut and paste the next five dimensional equations that represents a hypercone going through all the coordinate axis:

((1/5)*sin(5*x)/sin(x))*((1/5)*sin(5*(x-2*pi/5))/sin(x-2*pi/5)) +
((1/5)*sin(5*(x-2*pi/5))/sin(x-2*pi/5))*((1/5)*sin(5*(x-4*pi/5))/sin(x-4*pi/5)) +
((1/5)*sin(5*(x-4*pi/5))/sin(x-4*pi/5))*((1/5)*sin(5*(x-pi/5))/sin(x-pi/5)) +
((1/5)*sin(5*(x-pi/5))/sin(x-pi/5))*((1/5)*sin(5*(x-3*pi/5))/sin(x-3*pi/5)) +
((1/5)*sin(5*(x-3*pi/5))/sin(x-3*pi/5))*((1/5)*sin(5*x)/sin(x))

The above thing should give identical zero for all x.
An important feature of exponential curves in spaces with an odd number of dimensions is that they all are inside a hyperplane. The hyperplane says the sum of the coordinates is always 1. If you cut and past the next sum of the five coordinate functions you see that you always get one for all x:

((1/5)*sin(5*x)/sin(x)) +
((1/5)*sin(5*(x-pi/5))/sin(x-pi/5)) +
((1/5)*sin(5*(x-2*pi/5))/sin(x-2*pi/5)) +
((1/5)*sin(5*(x-3*pi/5))/sin(x-3*pi/5)) +
((1/5)*sin(5*(x-4*pi/5))/sin(x-4*pi/5))

At last the link to the original update from 2015 where I found the missing equations for the first time. But all I knew they were hidden inside the determinant. A few weeks ago I decided to take a better look and the result is this post.

From 14 July 2015: The missing equations.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff03.htm#14July2015

Ok, that is what I had to say. Till updates.

The pull back map applied to the coordinate functions of the 3D exponential circle.

In this post, number 50 by the way, I am trying to use as elementary math as possible in order to use the pull back map from the 3D circular number system to the complex plane.

With this the pull back map and the 3D circular number system are treated so basic that with only high school math and a crash course in the complex plane students can understand what I am doing.

So for reading this post number 50, what do you need in mathematical knowledge?
1) Understand how to write cos(a + b) and sin(a + b) in terms of cos a and sin b.
2) Understanding of e to the power it in terms of cos t and isin t.
3) Understanding of the roots of unity as found inside the complex plane, in particular being able to calculate all three roots of unity when we take the third root of the number 1.

That’s all, so basically all first year students in math, physics and chemistry could understand this post at the end of their first year on a local university.

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The words above are only one reason to write this post; to be honest for me it took a long time to write down for the first time the coordinate functions for the 3D exponential circle.

And I never did give much solid proof for that these coordinate functions have indeed the properties as described. It all more or less came out of the sleeve as some kind of monkey trick.

Therefore for myself speaking, this post giving the results in it also serves as a proof that indeed there is only one class of coordinate functions that do the job. They can only differ in the period in time they need to go around, if you leave that out the triple of coordinate functions becomes unique.

All in all the goals of this post number 50 are:

1) To do the pull back of an exponential circle as simple as possible while
2) In doing so give some more proof that was skipped years ago.

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This update is seven pictures long, each 550 x 775 pixels in size.
Hit the road Jack:

03nov2016_pull_back_map01

03nov2016_pull_back_map02

03nov2016_pull_back_map03

03nov2016_pull_back_map04

03nov2016_pull_back_map05

03nov2016_pull_back_map06

03nov2016_pull_back_map07

 

I think I have nothing more to say, so see you around my dear reader in post number 51.

Till updates.

Wirtinger derivatives and the factorization of the Laplacian.

This post could have many titles, for example ‘Factorization of the Laplacian using second order Cauchy-Riemann equations’ would also cover what we will read in the next seven pictures.

The calculation as shown below is, as far as I am concerned, definitely in the top ten of results relating to all things 3D complex numbers. Only when I stumbled on this a few years back I finally understood the importance of the so called sphere-cone equation.

The calculation below is basically what you do when writing out the sphere-cone equation only now it is not with variables like x, y and z but with the partial differential operators with respect to x, y and z. In simplifying the expressions we get I use so called second order Cauchy-Riemann equations, if you understand the standard CR equations these second order equations are relatively easy to digest.

Have fun reading it.

05Aug2016_factorization_of_the_Laplacian01

05Aug2016_factorization_of_the_Laplacian02

05Aug2016_factorization_of_the_Laplacian03

05Aug2016_factorization_of_the_Laplacian04

05aug2016_factorization_of_the_laplacian05 05aug2016_factorization_of_the_laplacian06

 

05Aug2016_factorization_of_the_Laplacian07This post is also categorized under Quantum Mechanics, the reason for that is that the wave equation contains the Laplacian operator and the more you know about that rather abstract thing the better it is in my view.

I would like to close with a link to a wiki on Wirtinger derivatives, originally they come from theory with several complex variables. That explains why in the wiki the Wirtinger derivatives are written as partial derivatives while above we can use the straight d´s for our differential of f.

Here is the wiki: Wirtinger derivatives
https://en.wikipedia.org/wiki/Wirtinger_derivatives

Till updates.