Dreamy Logic

So I passed my exam on probability. I scored 13 out of 15, I just messed up the part about the so called testing hypothesis =(. I feel happy.

In this post I wanted to share a couple of experiences that have been going on on my life , and somehow, they all seem to share some crucial aspects even when they are very different by definition.

It's like when a smell reminds you of the taste of something. I mean, by definition, smell and taste are different senses, and they are detected by different mechanisms (you would say at first), and still, our brain can compare these two types of signals...of course, when you think in terms of molecules of substance being detected by the cell-receptors in your nose and tongue, smell and taste don't look so different. But you have to admit that unless you know something about this molecular behavior, there is no apparent way of reconciling these two things, and its just weird to remember the taste of something by smelling something else.

Well, something like that has been happening to me in a slightly different context. I tried weed for the first time. And the experience I had was comparable to a dream.

I felt the very same things, though about the same things, and the shocking part, I used  the same type of logical approach when trying to solve a problem (a theoretical one...practical, more handy things usually brought me back to reality, they kinda make me wake up...if that makes any sense)

So apart from the weird sensation of me vibrating like a  membrane when hit by the sound waves in the music, or the feeling that everybody is watching you and you can't stop laughing about that, something worth mentioning actually happen....

The moment I excuse myself from the table, and went to the bathroom for a little bit of coolness, I start having an irrepressible rush of brilliant, novel, worth spreading on TED, set of ideas about pretty much every unsolved problem in physics I could think off !...and I was astonished with each one of them..... dazzled upon its beauty....I couldn't believe all this have been lying on my insignificant brain all this time!..trust me, you would've cried...it was the best time Ive had in a bathroom so far....=)

Sudenly I knew the QM wave function governing every particle in the universe, I knew how to derive it, how to solve it for every single case, I could visualize it....it wasn't a simple function, but it existed, and I had it in front of me!...

I got a solution to the Navier-Stokes  equations for fluid mechanics, I knew the solution, you just had to consider and extra thing that everybody else have been missing...till now, till me.

GTR wasn't right. Einstein field equations were close, but they have been manipulated in the wrong order. I had it done correctly, in my mind, in a matter of seconds...the solution was perfect, I start making ''computer'' simulations right away in my head, they all fitted, not dark matter needed, no dark energy....If you don't cry at this point you are a robot!

And these kind of ideas just kept coming to me...like I said before: An irrepressible rush.....It was a great time there...there, in the dreamy world.

There were also sad things. These ideas are only the few I remember, I kept forgetting most of them 10 seconds after they magically appear in my mind...it was very frustrating. Fortunately enough, I was able to record myself, describing them and getting desperate whenever I forget them (...of course I didn't forget everything about them...otherwise I couldn't grieve about forgetting them, most of the time, I knew I just forgot a great idea, but didn't know exactly about what.) That way I got a few left to tell you about.

The thing here is, that when I woke up next morning, and played the recording I found something incredibly odd. All the postulates of my ideas were correct, I knew about what I wanted to solve/accomplish, but from there on my logic was totally poor and wicked. I end up having solutions with strawberry pies as variables, policemen rearranging Einstein's equations, super glue aligning all the electron's spins, and all sort of weird and nonsense things!...I was so disappointed =(.

But at this point I knew it was OK. I mean , you hear people having weird experiences, talking to car plates, or whispering to the moon. So I stopped thinking about that, and continued with my day as if nothing had happened...

A few days ago I start wondering about the experience I had. I just didn't seem right the fact that my brain, a scientific one I would say, let such a  faulty arguments impregnate the beautiful, ordered, objective, and clear paradigm that science is. How is it possible that strawberries and covariant tensors were indifferent to me?. Why would I trust policemen when talking about GTR stuff ?. Why did I make such a stupid logical  conjectures!?. The more I though about it, the more disappointed I start feeling at my education....lol.

Finally, a couple of days ago, I had a dream. The dream started in the middle of a city which I knew perfectly, I was talking to a person I knew very well, I felt like we had a strong relation, I could trust that person. Somehow I didn't need any context, I knew it all. We were discussing about how our computers were talking to us all the time, not through a speaker, no...the very screen had a huge mouth and it would distore the image as it spoke. the computer was inviting us to visit the Governor. We were laughing because everybody knew there are no Governors in Mars!.....OK?.....ok....

Now, why didn't I just stoped the conversation there, asked about how can computer talk to us, how come we are on Marse...I don't know...the stuff any person would ask. Why wasn't I using my usual intellect there? Where was all the reality gone? How did I know that city or that person?...

.....Why my logic was wretched?....

That totally reminded me of the weed experience, I have come to the conclusion that logic wasn't as straightforward as it seems when you are awake...a whole new set of possibilities existed. Things could exist regardless of the logical possibility/impossibility status they have on the awake reality. If you check the different types of Omnipotence, you can come to the conclusion, that God's attribute of Omnipotence is logically impossible (as well as Omniscience) in this world, under this "logical paradigm". However, if logically incompatible phenomena could coexist in the same "reality", God, and all it's attributes would be non-sense. Such reality can be identified with the reality I just experienced, a reality where the rules of logic, the very cornerstone of our civilization, work in a different way. Check this out, and tell me:  how this contradiction has permeated our religions, our believes, our lives?

What should we do about this? What can we do?. Can we just rise a flag and claim this part of the Universe as a wrecked-logic-free-zone or something? Should we leave this wretched-logical things confined to our dreams and trips? Or the opposite!....Should we study it, I mean the fact that such pathological logical objects are available to us only in dreams and trips sets the bar really high from an experimentalist point of view, but still. Its a fascinating idea.

QM and Probability Theory

I've been preparing myself for over a week for my Statistics and Probability exam. I understand if you think that  I'm reaaaally stupid, but I feel great going to an exam and knowing that your head will not stop thinking about number and formulas, knowing that last night the only things in your dream were the formulas you  just learned...that sort of gives me confidence =).

I think at this point , my brain is thinking about numbers all the time, I have always been a big fan of Poker, specially the good old Texas Hold'em version. But, at this point I should not be invited to any game or I will stop the whole thing just to calculate all the probabilities Ive got to win the hand.....living hell. Proabilities are all over my head, however I'm not a the ''dreaming with formulas'' level.....that will take me another 3 days and a lot more caffeine. =).

Anyway, around two days ago I bumped into the know concept of complete set of events, defined over a probability space. You know...the sort of thing you find everyday while eating bread or not.

Right after this useful concept, there is the theorem that allows you to express any event, as the sum of conditional probabilities. I mean, if  $\{n_1, n_2, n_3,\dots , n_N \}$is a set of complete events in a probability space, then for any event A holds:

This is when I thought, this look just like the formula used in QM to decompose any vector to a given basis. That is:

Where, , is a basis for the space. (for the sake of simplicity, a finite one.) 

The good thing is that this similarity will make it easier to remember. However, I realized that the resemblance is only aesthetic. I mean, the vectors in QM do not represent probabilities themselves. you always need to multiply them by their complex conjugate, and only then you will obtain a probability (a probability density will be more precise). So  I can always use the fact that I remember one to construct the other, but I have to be aware that they are not the same formula written with a different notation. 

I have the vague notion that multiplying the second formula by its complex conjugate will give me the first formula...but I haven't worked it out.....yet

Update on the atomic dipoles

So I did my homework. Surfing the web, looking for some confirmation to the idea that the electron clouds may generate by themselves a kind of electric polarization. The results were not very satisfactory...a closer look to the spherical harmonics shows that they actually ARE symmetric on the polar angle. I guess i had the wrong idea in my head, I was probably thinking about the wave function itself, and forgetting to calculate the actual probability (I guess my QM ideas were in fact a little dusty after all).So , I guess electric dipoles will not appear by themselves in an atom when lowering the temperature as I thought. However, I did read about the so called Rydberg atoms, atoms with very high excitation energies, whose electron clouds resemble the hydrogen structure, and they behave as described by the Bohr model....quite interesting. It turns out that the higher their energies are, the better response they have to external electric fields. It make sense, since the high energies will separate the electrons away from the nucleus and making them more ''volatile'' when an external force acts upon them.

But enough of the self-pitifulness. I also found an interesting fact. Even if the atomic orbital are symmetrical in the polar angle, they do have a dependence on it. This dependence crates closed loops where the electron cloud is more dense than else. Sometimes it can create several rings around the atom creating a magnetic dipole moment.

Here are some drawins of an example of such case (n,l,m=7,6,0)

The cool part is that this magnetic dipole moment will appear without any external field!. Like I though it could happen with the electric counterpart.

So far so good.  Now, another thrilling (on my standards) idea came to my mind. When you plot the same atom with a change in the magnetic number m, form =0 to =6 or =-6, you get, what seems to me, circular coils around the toroidas direction, just like in a tokamak.(n,l,m=7,6,6)

Will this resemblance with toroidal coils will behave anything like a tokamak? will create a toroidal magnetic field? what would the magnetic dipole moment be?....just wondering.

Atomic Dipoles

So here I am, writing again, trying to think what to say in order to meet the high expectancies I build in the first post (Big mistake). I kinda explicitly said that I fin all of the ideas interesting, and that is true, the thing is, I kinda wanted to imply, that they may not seem so cool to you.....but WTF, since nobody read this, I will give it a shot...

Last week, during the visit of the Colombian ambassador in Prague, about 30 Colombian fellows gathered after the meeting and went together for a beer in Strahovsky Klaster (very nice place btw). During the conversation, somebody mention one of Colombian typical 'sports', Tejo. We were discussing how there are world championships of such discipline (If it is a discipline at all...) and teams from different countries get drunk while trying to explode gunpowder bags located about 20 mts away swinging through the air metallic discs. But that is the OK part of it. After a while, people stared confessing how bad they are at Tejo, how they have blown up the roof of the place, how professionals attach their own disk to their belts using a powerful magnet, or how foreigns thought somebody called ''mecha'' has died of a gunshot (my favorite..).

Between laughs I kept thinking about that little magnet. And apparently the players not only use it as an easy way to store their precious discs while not playing (Or a way to wake up hangover and still have it with them), but apparently they put the magnet so somehow the metallic disc will gain an extra symmetrical property (placing the magnet concentric with the disc) thus allowing better performances. As the only guy who liked physics in there, it was my responsibility to clarify things, so I explained how most of the day to day metals, interact with magnetic fields in the atomic scale aligning their little magnetic dipoles as a response to the external field, creating then, their own magnetic field. I tried to make clear that as soon as the external field will disappear, the magnetization of the disc will also disappear, and therefore the expected extra symmetrical property. In the best of the cases, regions with different magnetization will appear, but still, the symmetry will disappear....It felt nice.

During the next day or two, I kept thinking about the argument I used, about how the little dipoles tend to align in an external magnetic field, and so on. I start remembering the dusty ideas from QM about the exited states of the atoms, Spherical Harmonics, the Bose-Einstein condensate....and then the idea strike me.

Imagine a molecule at room temperature. The molecule will be jumping from one exited state to another, and the different energies will loose importance when they are averaged over time or over the ensemble. However as we lower the temperature the states will start ''jumping down'' the energy scale.  Low enough to appreciate the effect of the polarization created by the electron cloud itself (since the Spherical Harmonics are not symmetrical in the polar coordinate). I mean, there will be no need of an external field to create such polarization.

It may seem very obvious just by looking at the first spherical harmonics. But I guess I have never realized that. I wonder if there is any way to measure that little polarization...I mean, is not like I cant think about one, but rather,have anybody done it? I know you can calculate it doing integrals over the wave function and so on. But..like always...just wondering......just wondering.

Pilot

It was almost midnight, I still had a little bit more than 8 hours of work, the night looked long, the chair seemed uncomfortable and my mind was starting to die out of boredom. My future seemed grey, my dreams unreachable and even if If reach them, they would seem worthless. I realize my spirit was dying and I had to do something about it. I spend the rest of my night-shift thinking what to do, how to change, how to power my thoughts, how to canalize all the random ideas that sometimes hit me. By 5 am I had found a solution.

Two days later, another night-shift was about to start, another 9 hours of mindless work were ahead of me. but this time I had with me a small pad full of random thoughts on the broadest spectrum of things, from going into the Vacuum Industry , to a different and equivalent approach to Asimov's three laws of robotics. I was armed and ready to fire. I felt a nice warmth in my veins, and my spirit felt satisfied.

From now on I have been keeping track of all the little ideas that hit me from time to time during the day. I certainly find all of them interesting, but of course, I might be a little biased while affirming so. I would like then to star sharing some of them here.