Monday, August 1, 2011

Turning Operators into Matrices

Turning Operators into Matrices

Friday, July 22, 2011

Chaos Experiment Number 1

Hello. Today is July 23rd, 2011. I have moved into my new place and less some space the home is coming together nicely. I decided I would start my Quantum search at the very beginning with a book by David J. Griffiths titled, "Introduction to Quantum Mechanics." But this is not the subject of the blog today. I haven't actually started reading it except for the first couple chapters but from what I've read this gets straight into the meat of Quantum Mechanics with the tools that one uses to perform simple calculations. From these initial tools I may try and explore philosophies about meaning. I hope though that we do not extend ourselves too far and get lost in the translation between the language of mathematics and English as there is no official English to Math dictionary or even an unofficial one. My biggest concern is that reality is obscured in the formalism of mathematics and what may make sense mathematically is not a true representation of the world. One of my other goals is to use LaTeX to express the mathematical relationships for you the reader. This may take me a bit to hash out so this blog will be a snails pace until I develop the means to communicate with you effectively.

Instead, today I will try and talk about Chaos in a very limited sense and hopefully add an entry to your Mathematical Dictionary. I have been reading a book titled, "The Essence of Chaos," by Edward N. Lorenz. He was very concerned with predicting the weather and is inadvertently the father of the "Butterfly Effect," from a paper named, Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas? Lorenz is quick to point out however that the likelihood that butterfly would cause a Tornado is equal to the likelihood that a butterfly would prevent a tornado in Texas. This illustrates on of Lorenz's big ideas which is Sensitive Dependence. Sensitive Dependence is the idea that beginning from an initial state looking at the same system later it will look nothing alike. A state has sensitive dependence if it can change by orders of magnitude very quickly. If you are not looking closely then you might interpret this as chaos when really it is sensitive dependence.
Imagine standing on a balcony in Grand Central Station and taking a picture of the lobby then waiting one minute and taking another picture. Continue in this fashion until you have one hundred pictures. Examining one state to the next there may be no coherence or regularity in the positions of people. This could be seen as chaos but we're really seeing a system that changes rapidly in short periods of time. If you were to instead, take pictures ever ten seconds, you may be able to track the motions of people and rather than concluding that the system is chaotic, you may discover that the system is merely volatile, or sensitive.
Despite making this discovery about Grand Central Station, we still cannot predict what the system will look like in another ten seconds. There are way too many factors in this case to track the thoughts, plans and movements of everyone who is in or will be inside Grand Central Station. If we observe long enough, we can begin to see behavior that is periodic however; there are more people in the Station in the morning and evening when people go to work or leave the city to go home. Though the system is unpredictable we can look at it and infer patterns. What then should we label this system, is it inherently chaotic or can we track everyone who uses the Station.
This example seems to have a few consequences first and foremost; chaos seems to be defined by our capacity to track changes. Spotting the difference between a ruler that is 1000 millimeters (mm) long and one that is 1005 mm long would be challenging but we assume that every ruler we use is exact despite this inability. Our inability to perceive differences becomes an assurance that everything is the same. Lorenz illuminates this idea perfectly showing that our ability to predict is limited by our ability to measure. Imagine a meter stick making machine that used the last stick as a guide but had an error of 5%. After three iterations a meter stick could be (1.05*1.05*1.05)=1.15763 meters long. The inability of our machine to detect a 5% difference led to a false prediction of the size of 1 meter.
As this post is called Chaos Experiment Number 1 I have a fun and easy project that I pulled from the pages of "The Essence of Chaos." Leave the cell A1 empty. In A2 input the equation ((A1^2)-A1). That is the quantity in A1 squared minus A1. Coy this cell and paste it as far as your curiosity allows. I made my spread sheet as columns of 100 up to 19,400. This will iteratively square the quantity before it then subtract the squared quantity. This is our system.

Now experiment. Input any number you want into the initial box. Once you've tried a few things read further and see a few of the things I discovered.

My favorite thing to do is always start with 0 as 0 has a tendency to test the limits of any system. Obviously every entry will consequently be 0. Our system, from an initial state of 0, is perfectly stable. Nothing will perturb it from being 0 and we can predict to the nth value of the system for any value of n where n greater than or equal to 1. This is will be our rock as we will always know where 0 takes us.

Next let us plug in 1. This leads us to the the exact same result as 0. -1 however provides an infinite string of 2.

My favorite is plugging in 2 which returns 2 over and over again to infinity. -2 however explodes and we end up in the billions after 6 iterations.

Positive 3 interestingly enough brings us to the exact same numbers as -2. If we continue this method you will discover than n and -n+1 will produce the exact same results. We have found that a relationship exists between the integers.

The next thing that caught my attention is that anything greater than 2 is explosive and outstrips the capabilities of my computer easily. Putting in a number between -1 and 2 that is not the number 1 or 0 will ultimately lead to a pattern. The values will oscillate between positive and negative values continually getting closer to 0. This pattern is consistent for all values that I checked. If you find something different please share it in the comments =).

Chaos. Based on a few examples it seems as if we defined some rules of this system. We can predict what will happen based on the number we put in but can we easily predict what those numbers will be? If we didn't know the algorithm before hand how long would it take for this system to cease being chaotic to us? Chaos is matter of what we are perceiving. Ultimately it seems as if Chaos is a necessary generalization that can qualify a great deal of information quickly.
I will say that Chaos is indeed something real. It exists just outside our perception and is manifest when we can see how systems change. We can be aware of our uncertainty but that does not eliminate it. This brings up something to ponder: If we could somehow track ever single variable, every single motion and every single fluctuation in existence, have we eliminated Chaos? Can we ever eliminate uncertainty in measurement?

Thanks much. Don't hesitate to point out mistakes in my logic or errors in my grammar and feel free to share your thoughts or ideas.

Monday, June 13, 2011

2011 Back and Back and Again

I first made this blog when I was leaving New Orleans I think. Maybe I made it while I was still living there but it has been a while. I've decided to reestablish my online presence and contribute the to the interweb along with everyone else. My interests have changed a lot through the last few years however and as of 4-6 weeks after June 11th, 2011, I will be in possession of a Bachelor of Arts in Chemistry. I'm looking to try and share some of my knowledge to those who will listen and add a piece of myself to other people's lives.
Right now I'm in limbo, wrapping up the pedestrian tasks involved in moving and reestablishing oneself in a new location. This may damper my new enthusiasm for blogging, but I can still let my non-existent readers know that I hope to find my way through various texts on quantum mechanics and regurgitate them in a more friendly manner to you the reader. The more I study, the more I'm convinced anyone can understand this material if they try. Hopefully I can be a bridge between you and curiosities that may not be in your disciplines but still occupy a space in your mind. I already have a few books available and once I get out of B-Ham I will start on chapter one.

So cheers to new things, old things and you.

Tuesday, July 22, 2008


"Beginning to think is to be undermined. Society has but little connection with such beginnings. The worm is in man's heart. That is where it must be sought. One must follow and understand this fatal game that leads from lucidity in the face of existence to flight from light."

"It often happens that those who commit suicide were assured of the meaning of life."

I'm trying to reoncile what I know of Sartre's idea's on perception and Camus's idea of the Absurd. They each seem all inclusive in themselves, inclusive of their own idea's as well as the rest of the world. But both deal with perception in a different way. Sartre is talking almost about the gears of a watch to Camus who is trying to tell the time. The only way I reconcile what I think of both is to think that, while Camus deny's it, his philosophy is metaphilosophy. That understanding the world through that perception of the Absurd applies only to those who understand. And it seems as if Camus unknowingly admits this, as anyone who cannot understand the Absurd is dead is dead in his mind. Literal death, suicide, or metaphorical death, hope. I'm not implying that Absurdism is a a means to hope, I do beleive that it stands on it's own. Imagine having a photograph, a memory, and being at the Grand Canyon. All three change your impressions of the others but by having a memory of the three phenomenon you know your perception has changed. Camus philosophy applies whenever you wake up from life and examine it for it is when you wake up that you deal with the absurd. Then one must either return to the pattern, kill oneself or deal with the Absurd and create meaning in the face of meaninglessness. But only when the idea of the Absurd is implanted in ones head can the cycle of Absurdity begin. It' like another vicious cycle, or mind game to play that take advantage of cathartic life moments. I quite liken it to the Allegory of the Cave. When one discovers the Absurdity of life they are not only exposed to the bright sun but ultimately doomed to toil in the shackles from where they had just broken free until such a day comes when the darkness sets in and Catharsis is possible once more. I imagine Camus's philosophy as implying a wisdom that is constantly and at one moment, infinite and freeing. And indeed, his use of the Myth of Sisyphus complete's his idea's whereas Sisyphus is always pushing the rock, it breaking, walking back, and then struggling once more. It seems to imply that no catharsis is ever infinite, but can in the future prevent you from experienceing the same cathartic path. Thus Sisyphus, the wisest and most daring man was sentenced to a life of complete and utter ignorance.

Thursday, July 3, 2008

Still Life Pastel Colors

weiss/ phthalogrun dunkel
white/ dark pthalo green

kadmiumgelb zitron/ chromoxydgrun feurig
cadmium yellow lemon/ viridian

kadmiumgelb/ permanentgrun
cadmium yellow/ permanent green

lasurorange/ grungold
orange glaze/ gold green

purpurrosa hell/ umbra gebrannt
light purple pink/ burnt umber

alizarinkarmesin/ rotel
alizarin crimson/ sanguine

magenta/ lichter ocker
magenta/ light yellow ochre

violett/ strohgelb
violet/ cream

blauviolett/ zimtbraun
blue violet/ cinamon

ultramarin/ kaltgrau II
ultramarine/ cool grey II

kobaltblau/ kaltgrau VI
cobalt bue/ cool grey VI

helioturkis/ schwarz
helio turquoise/ black