Another short post, this time again on the totally crazy so called bonding and non-bonding pairs in theoretical chemistry. It is one of the many energy problems that come along if you want electrons to be tiny bipolar magnets. And if you view the electrons as magnetic monopoles, in that case all of a sudden you don’t have these kind of weird energy problems.
Lets dive into it, this post is 3 images long and here we go:
That’s more or less all I had to say today, if you view the magnetic properties of electrons as monopole magnets just like their electric properties the standard electron pair becomes the lowest (potential) energy state. Before I close this post let me quote from a wiki an interesting detail about the non-bonding electron pairs: they have a tendency to be outside the so called ‘bonding region’ between atoms in molecules. Once more this only makes sense when electron carry a monopole magnetic charge.
In theoretical chemistry, an antibonding orbital is a type of molecular orbital that weakens the chemical bond between two atoms and helps to raise the energy of the molecule relative to the separated atoms. Such an orbital has one or more nodes in the bonding region between the nuclei. The density of the electrons in the orbital is concentrated outside the bonding region and acts to pull one nucleus away from the other and tends to cause mutual repulsion between the two atoms. This is in contrast to a bonding molecular orbital, which has a lower energy than that of the separate atoms, and is responsible for chemical bonds.
That’s it, see you in some other future post or enjoy some old posts on say the 4D complex numbers because they are beautiful and that is something we cannot say about the behaviour of the average physics professor with their weird fixation on electrons as tiny magnets…
This is a very short post, in it I even joke this is done in “Tik Tok style”. It is about finding the eigenvalues for an arbitray elliptic complex number. Such numbers have matrix representations and as such they have also eigenvalues that live in the ordinary complex plane. Here the elliptic plane is the same as we always studied the last couple of months, it is ruled by the imaginary unit i via i^2 = -1 + i. If you need the eigenvalues of such a number, instead of going through the calculation for eigenvalues every time, with the eigenvalue functions you just substitute it in and it spits out the two eigenvalues. To be honest I did not explain in detail why it works, I hope it is rather obvious. Take for example two commuting (square) matrices A and B. They have the same eigen vectors (because they commute) and as such it is very easy to find the eigenvalues of any linear combination of A and B.
The post itself is only two pictures long and I included a third picture that I used on the other website. Beside a female robot the third picture contains another factorization of the equation of the ellipse that is the determinant of the matrix representations. So this equation for the ellipse can be factored on the standard complex plane and also on the elliptic complex plane. The interesting detail is of course that on the elliptic complex plane you have integer coefficients while on the standard complex plane this is impossible.
Basically the eigenvalue funtions are both a map from the elliptic complex plane to the ordinary complex plane. If it was made by a professional math professor he or she would likely call it an isomorphism but I name them eigenvalue functions. In the past I also made them for the 3D complex and circular numbers and of course for the 4D complex numbers that were under study years ago. Enough of the introdutionary talk, lets go:
There is a small ‘cut & paste’ error at the top of the next picture.
Now I left a lot of stuff out otherwise it would not be a Tik Tok short kind of math post. But you can also use the elliptic complex plane as your primary source of eigenvalues. For example at the other side of our galaxy there lives an alien race known as the Orcs. And for some kind of religious reason these Orcs just don’t want to use circles because as they all know circles are evil. But they found the elliptic complex plane and they use that for solving eigenvalue problems like eigenvalues from square matrices or even stuff that we humans know as the Hamiltonian energy operator. That should work just as good as we humans do in using the complex plane we have over here where the complex exponential is a circle. So let us now look at the third picture that has both factorizations in it:
This cleary is not for math professors; they won’t understand this conjugate.
Let me leave it with that and as always thanks for your attention.
Likely in the year 1991 I had figured out that the conjugate of a 3D complex number could be found in the upper row of it’s matrix representation. As such the matrix representation of a conjugate 3D number was just the transpose of the original matrix representation. Just like we have for ordinary complex numbers from the complex plane. And this transpose detail also showed that if you take the conjugate twice you end where you started from. Math people would say if you do it twice, that is the identity operation. But for the two 2D multiplications we have been looking at in the last couple of months, the method of taking the upper row as a conjugate did not work. I had to do a bit of rethinking and it was not that hard to find a better way of defining the conjugate that worked on all spaces under study since the year 1991. And that method is replace all imaginary units by their inverse. As such we found the conjugate on 2D spaces like the elliptical and hyperbolic complex planes. And the product of a 2D complex number z with it’s conjugate nicely gives the determinant of the matrix representation. And if you look where this determinant equals one, that nicely gives the complex exponentials on these two spaces: an ellipse and a hyperbole. Now when I was writing the last math post (that is two posts back because the previous post was about magnetism) I wondered what the matrix representation of the conjugate was on these two complex planes. It could not be the transpose because the conjugates were not the upper rows. And I was curious what it was, it it’s not the transpose what is it? It had to be something that if you do it twice, you do the identity operation…
All in all in this post the math is not very deep or complicated but you must know how te make the conjugate on say the elliptic complex plane. On this plane the imaginary unit i rules the multiplication by i ^2 = -1 + i. So you must be able to find the inverse of the imaginary unit i in order to craft the conjugate. On top of that you must be able to make a matrix representation of this particular conjugate. If you think you can do that or if you don’t do it yourself you will understand how it all works, this post will be an easy read for you.
It turns out that the matrices of the conjugate are not the transpose where you flip all entries of the matrix into the main diagonal. No, these matrix representation have all their entries mirrored in the center of the matrix or equivalently they have all their entries rotated by 180 degrees. That is the main result of this post.
So that’s why I named it the “Cousin of the transponent” although I have to admit that this is a lousy name just like the physics people have with naming the magnetic properties of the electron as “spin”. That’s just a stupid thing to do and that’s why we still don’t have quantum computers.
Enough intro talk done, the post is five pictures long and each picture is 550×1200 pixels. Have fun reading it.
That was it for this post, one more picture is left to see and that is how I showed it on the other website. Here it is:
Ok, this is really the end of this post. Thanks for your attention and may be see you in another post of this website upon complex numbers.
A couple of weeks back I already showed this video from Dr. Erica Carlson on the other website. I did select that video because in the second half of that video she talks about electron spin configurations that minimize the energy in stuff that is known as spin ice. Since all those energy problems that I have with viewing electrons as bipolar tiny magnets are always skipped, I decided to use this video as a short post on magnetism. In videos like this the pattern is always the same: at the surface it all looks logical like in this video the spin configuration in that stuff known as spin ice. But video after video I have seen over the last years, always when we need to look at crazyland they always skip that. When the energy stuff gets crazy, they just skip it. Now this is absolutely not some form of a conspiracy, these people like Erica simply believe the bipolar magnetic electron is true and as such they have a blind spot into the problems: They just don’t see the problems because of their blind spot.
In the year 2015 I started to doubt that electrons were tiny magnets with two magnetic poles. I started doubting that after I tried for myself to explain the results of the so called Stern-Gerlach experiment. In my view the results were only explainable if we use magnetic monopole electrons. A few days later reading all those official explanations I understood I had to be cautious. And at the begining back in 2015 I knew nothing about electron spin, all I knew was that people from physics thought they were tiny (bipolar) magnets. It’s been a long journey from there back in 2015 and it will also be a long long journey going from our present year 2024. After all the belief that electrons are tiny macroscopic magnets is deeply rooted in 100% of the physics community.
In this post, for the first time since 2015, I included a simple expression about how the professional physics professors view the potential energy of electrons related to magnetism. It is somewhere below and it is the same as we have for macroscopic magnets like say two bar magnets. If you hold two bar magnets south to north pole, that is the minimum potential energy because it costs energy to separate them. And if you hold two bar magnets say north pole to north pole, that is the situation of high potential energy.
The post itself is four pictures and two additional figures and of course the perfect video from Dr. Erica Carlson. Say for yourself, this video is a perfect 10 with all kinds of animations I can only dream of. Ok ok, there is just one tiny tiny error in it: electrons are not tiny magnets. But for the rest it’s a “PERFECT 10” kind of video.
Well bipolar physics freaks: what is your explanation in detail?
That was more or less the end of this post but I made one more picture depicting another big energy problem that the official version of electron spin has: The behavior of a single electron in an applied magnetic field.
After all if it were true that electrons are tiny magnets, if you apply a magnetic field to electrons shouldn’t they all perfectly align with that magnetic field and as such fall into their lowest potential energy state?
Yes in an ideal world they should, but we live in a world where we not only have a lot of professional physics professors but also television physics professors. And they never talk about the energy problems there are with the electron as being tiny magnets. So this is a strange strange world where physics just ingores simple problems like the last picture of this post:
Oh yes the stability problems we have if it were true that electrons are tiny magnets. As you see in the video it is always skipped and their brains never go down that route… It is what it is and here is the video:
Erica knows how to flip a spin…
Lets leave it with that, the next post will be about matrix representations of conjugtes of 2D complex numbers. They are weird and also lovely now I have my new method of understanding the process of conjugation. And as always thanks for your attention and not falling asleep before you read these last words of this post.
A bit like in the spirit of Sophus Lie lately I was thinking “Is there another way of finding those tangets at the number 1?”. To focus the mind, if you have an exponential circle or higher dimensional curve, the tangent at 1 is into the direction of the logarithm you want to find. In the case of 2D and 3D numbers I always want to know the logarithm of imaginary units. A bit more advanced as what all started a long time ago: e^it = cos t + i sin t. An important feature of those numbers tau that are the sought logs is that taking the conjugate always the negative returns. Just like the in the complex plane the conjugate of i is –i.
The idea is easy to understand: The proces of taking a conjugate of some number is also a linear transformation. These transformations have very simple matrices and there all you do is try to find the eigenvector that comes with eigenvalue -1. The idea basically is that tau must like in the direction of that eigenvector.
That is what we are going to do in this post, I will give six examples of the matrices that represent the conjugation of a number. And we’ll look at the eigenvectors associated with eigenvalue -1.
At the end I give two examples for 4D numbers and on the one hand you see it is getten a bit more difficult over there. You can get multiple eigenvectors each having the eigenvalue -1. Here this is the case with the complex 4D numbers while their ‘split complex’ version or the circular 4D numbers have not. Now all in all there are six examples in this post and each is a number set on it’s own. So you must understand them a little bit. The 2D numbers we look at will be the standard complex plane we all know and love, the elliptic and hyperbolic variants from lately. After that the two main systems for 3D numbers, the complex and circular versions. At last the two 4D multiplications and how to take the conjugate on those spaces.
The post itself is seven pictures long and there are two additional pictures that proudly carry the names “Figure 1” and “Figure 2”. What more do you want? Ok, lets hang in the pictures:
The purple line segment points into the direction of tau.That’s why 4D split complex numbers are just as boring as their 2D counter parts.
Years ago it dawned on me that the numbers tau in higher dimensional spaces always come in linear combinations of pairs of imaginary units. That clearly emerged from all those calculations I made as say the 7D circular numbers. At the time I never had a simple thing to explain why it always had to be this pair stuff. So that is one of the reasons to post this simple eigen vector problem: Now I have a very simple so called eigen value problem and if the dimensions grow the solution always come in pairs…
That was it for this post, likely the next post is upon so called ‘frustrated’ magnetism because the lady in the video explains the importance of understand energy when it comes to magnetism. After that may be a new math post on matrix representations of the actual conjugates, so that’s very different from this post that is about the matrices from the process of taking a conjugate… As always thanks for your attention.
This is a horrible simple post, after all for the complex numbers ruled by i^2 = -1 + 3i all you have to do is look where the determinant of the matrix representaitons equals zero. Well yes that is what we do, at present day there is that cute package or applet (in the past that was a so called computer program, why is all that kind of stuff an “applet” nowadays?) called DESMOS. With DESMOS the two lines that make up the set of non-invertibles is easy to graph. These two lines are interesting because they are the asymptotes of all those hyperboles in this space, anyway those hyperboles that can be written as det(z) = constant. Furthermore the two lines where det(z) = 0 separate the parts of this complex plane where det(z) > 0 from the parts where det(z) < 0.
I took the opportunity to introduce a more comprehensive notation to denote such spaces of 2D complex and split complex numbers. I wrote it much more as a set like in set theory and it includes the ‘rule’ for the imaginary component.
For readers who are new to this website and don’t have a clue what ‘hyperbolic’ or ‘elliptical’ 2D complex numbers are, it is all basically rather simple: These numbers are complex because i^2 = -1 + something, in the case of this post it is i^2 = -1 + 3i. These 2D complex numbers have matrix representations and the determinant of these matrices are constant along certain hyperboles. The case det(z) = 1 is very interesting for any of such a complex plane because that is a multiplicative group. Just like the unit circle in the standard or ordinary complex plane is a multiplicative group.
All in all this post has five pictures of size 550×1200 pixels and an additional two figures from graphs from the DESMOS applet.
For me it was funny to write some ‘high school math’ with just a few parabole kind of stuff in it. During the writing of this post I came across the idea of making a matrix representation of all that conjugating stuff, as such I found a beautiful but still extremely simple way to find the direction of the number tau in a particular space. The great thing is that this time it works in all dimensions so not only the 2D complex numbers but likely much much more. Lets try to upload my post to the internet and may I thank you for your attention.
Yes yes I know we already calculated the number tau for this space equipped with a hyperbolic multiplication. (That was a few posts back using matrix diagonalization.) But I had a few reasons to write this anyway, one reason was just curiosity. I wanted to know how those integrals looked and since we had calculated the number tau anyway we did not need to solve these integrals with pencil and paper. I also wanted you to show how you can write the product of such a complex number z against it’s conjugate. On the standard complex plane this defines a circle and on our hyperbolic space it is of course a hyperbole. At last I wanted to pen down the formula for finding the inverse on this particular hyperbolic complex number space. It looks just like the way this is done on the ordinary complex plane with the exception that if you calculate it the conjugate is a bit different. For me it is funny that we have exactly the same looking formula for the calculation of inverses. All in all it shows that the fixation the professional math professors have on all that “The norm of the product is the product of the norms” kind of stuff is only true because on the standard complex plane the determinant equals the square of the norm of a complex number z. In our present case of hyperbolic complex numbers we devide the conjugate by the determinant and those determinant define hyperboles and not circles. So nothing of that “The norm of the product is the product of the norms” kind of stuff. The deeper underlying mechanism is just always the determinant of the matrix representation.
It has to be remarked however that the study of normed spaces is important in itself and also in practice: If you can find a good norm for some difficult problem like the successive aproximations in say differential equations and you can prove using that norm the stuff converges, that is BINGO of course. Yet a norm is only a tool and not all there is inside that strange space known as human math.
The post itself is 6 images long, in it I have two (pairs of) integrals going from 1 to i. The integrals are of course the inverse of a complex number because the derivative of the logarithm is the inverse and we want to know the log of i because by definition that is the number tau.
I included a so called Figure 1 that show the evaluation of these integrals by the Wolfram package for definite integrals, it’s a handy online tool in case you don’t want to evaluate your integrals with pencil and paper. The last image is from the other website where once more I want to insult the math professors just a tiny tiny bit by using the standard formula of finding the inverso on the standard complex plane. So all in all this post is 8 pictures long.
That was it more or less for this post, I hope you are a bit more confident by now that you can actually integrate spaces like this more or less just like you do in the standard complex plane. The last image is from the other website, it contains a female robot. Why there are female robots is unknow to me, after all with other tools you never have females like have you ever heard of a female screw driver or a female mobile phone?
Ok, that was it for this post. Thanks for your attention and just like the female robot look up into the light and start thinking about the wisdom behind “The norm of the product is the product of the norms”.
This is now year nine or may be the tenth year that I started doubting that electrons were tiny bipolar magnets because it makes much more sense that they are magnetic monopoles. Over the years I have found out that logic just does not work and given the fact that physics people get a salery from tax payer money, that is weird behavior. But physics professors behave just like math professors who after 33 years of doing just nothing will keep on doing that and never ever talk about the three dimensional complex numbers. What explains that kind of behavior, after all it’s all tax payer money so they should be a bit more humble don’t you think? The way I see it is that university people like math and physics professors are some elite. And I don’t mean an elite in the sense they are the very best at their science, no it’s just a collection of overpaid snobs. You must not think I am emotional or so by using the word snobs, no it’s a cold hearted classification of their behavior. It is now 102 years since the original Stern Gerlach experiment and there is boatloads and boatloads of theory of how electrons should behave in case such an experiment is repeated (that is a squence of those magnetic fields) and it is easy to understand the very first experimental physics human that would do such a sequential SG experiment would likely be rewarded a Nobel prize. And in the physics community the Noble prize is what they all dream of. So in a century of time without doubt on many occasions such an attempt must have been undertaken. But there is no trace of any such experiment in the literature, the only experiment that was done was by Frisch and Segrè where they tried to flip the electron spin and that all failed big time. But when building their experimental setup Frisch and Segre got advice from Albert Einstein and likely because of that they got their (non) results published and as such we can find it back in the present day literature.
Now why should a succesful sequential Stern Gerlach experiment lead to an almost 100% probability of getting a Nobel prize? That is easy to explain: It would validate in a deep manner that quantum states like spin states are probabilistic in nature and as such would be a fundamental thing in say all the present day attempts there are in building quantum computers.
Another way of understanding there are just no successful sequential Stern Gerlach experiments done in the last 100 years is simple to do: Go to Youtube and search for it, all you find is animations that explain how it “should work”. But none of those videos give a hint of an experiment actually done…
Is it true there are no Nobel prizes rewarded in the last 100 years related to a repeated or sequential SG experiment? Well in this year 2024 the Nobel prize committee has a website and guess what? They have a search applet for their very website. If you search for “Stern Gerlach” you get something like 12 results and if you serach for “Stern Gerlach experiment” you only get 6 results. None of those results says anything about experimental validation of all that spinor crap or anything that shows you can actually flip the magnetic spin of an electron. I made a picture for the other website as you can see below:
If you want you can go to the website of the Nobel prize committee and look for yourself if you can find such a prize rewarded. Here is the link: https://www.nobelprize.org/. It’s all a big bunch of crap: Electrons are not tiny magnets, they carry magnetic charge just like they carry electric charge.
I am very well aware that logic does not work, but say to yourself about the crap of the electron pair they have over there in the physics community: The Pauli exclusion principle says that those electrons must have opposite spins so what does that mean if it is true that electrons are tiny magnets? Well if they have anti-parallel or opposite spins, doesn’t it look like this:
But again logic does not work so I do not expect that in this year 2024 the physics people will stop talking their usual bullshit. No way, after all as a social community they are just another bunch of overpaid snobs…
After having said that, after about only one century of time there is only recently an English translation made of the publication of the original Stern Gerlach experiment. The translation is done by Martin Bauer and here is a link to the pdf as you can find it on the preprint archive
With a new number tau (see previous post) there is always a log of an suitable imaginary unit found, but that does not mean you have a parametrization instantly. And with ‘suitable’ I mean the determinant of the imaginary unit must be one because I always want to make some complex exponential, in this case a complex exponential hyperbole. The post is relatively short, I always try to write short posts and I always fail fail and fail in that. But now it’s only four pictures long so this comes close to a tiktok version of my ususal math postings. By the way it has nothing to do with this post, but do you like tiktok? I don’t like it very much, it is more for people that have a bit different mindset compared to the way my old brain works… In this post I first neatly write down the parametrization using the sinh and cosh to express the two coordinate functions. And after that I more or less express it all as much as possible into the two eigenvalues of our beloved imaginary unit i that rules this plane via: i^2 = -1 + 3i. May be you have never thought about imaginary units as having eigen values themselves. In that case I invite you to calculate the eigenvalues of the ordinary numbers z from the complex plane. You know that plane that is ruled by an imaginary unit i via the rule: i^2 = -1. You will find a very interesting answer and of course after that you wonder WTF are the eigen vectors?
But let us not digress and enjoy the beauty of a complex exponential that is a hyperbole in this case. Here we go:
Figure1: Don’t mind the ‘female robot’ because all female robots are fake.
I am sorry for those bad looking accolades, it is some small fault in the Latex math package or some other old computer feature. You see it in much more texts written in Latex; the { and the } are just not presented properly. That was it for this post, as always thanks for your attention.
Some posts ago I showed you how you can calculate the number tau (always the logarithm of a suitable imaginary unit) using integrals for an elliptic multiplication. To be precise you can integrate the inverse of numbers along a path and that gives you the log. Just like on the real line if you start integrating in 1 and integrate 1/x you will get log(x). If you have read that post you know or remember those integrals look rather scary. And the method of using integrals is in it’s simplest on the 2D plane, in 3D real space those integrals are a lot harder to crack. And if the dimension is beyond 3 it gets worse and worse. That is why many years ago I developed a method that would always work in all dimensions and that is using matrix diagonalization. If you want the log of an imaginary unit, you can diagonalize it’s matrix representation. And ok ok that too becomes a bit more cumbersome when the dimensions rise. I once calculated the number tau for seven dimensional circular numbers or if you want for 7D split complex numbers. As you might have observed for yourself: For a human such calculations are a pain in the ass because just the tiniest of mistakes lead to the wrong answer. It is just like multiplying two large numbers by hand with paper and pencil, one digit wrong and the whole thing is wrong. Now we are going to calculate a log in a 2D space so wouldn’t it be handy if at least beforehand we know in what direction this log will go? After all a 2D real space is also known as a plane and in a plane we have vectors and stuff.
So for the very first tme after 12 years of not using it, I decided to include a very simple idea of a guy named Sophus Lie. When back in the year 2012 I decided to pick up my idea’s around higher complex numbers again of course I looked up if I could use anything from the past. And without doubt the math related to Sophus Lie was the most promising one because all other stuff was contaminated by those evil algebra people that at best use the square of an imaginary unit. But I decided not to do it because yes indeed those Lie groups were smooth so it was related to differentiation but it also had weird stuff like the Lie bracket that I had no use for. Beside that in Lie groups and Lie algebra’s there are no Cauchy-Riemann equations. As such I just could not use it and I decided to go my own way. Yet in this post I use a simple idea of Sophus Lie: If you differentiate the group at 1, that vector will point into the direction of the logarithm of the imaginary unit. It’s not a very deep math result but it is very helpful. Compare it to a screwdriver, a screwdriver is not a complicated machinery but it can be very useful in case you need to screw some screws…
Anyway for the mulitiplication in the complex plane ruled by i^2 = -1 + 3i I used the method of matrix diagonalization to get the log of the imaginary unit i. So all in all it is very simple but I needed 8 pictures to pen it all down and also one extra picture know as Figure 1.
Figure1.
That was it for this post, we now have a number tau that is the logarithm of the imaginary unit i that rules the multiplication on this complex plane. The next post is about finding the parametrization for the hyperbole that has a determinant of 1 using this number tau. As always thanks for your attention and see you in the next post.
