Apple once sent me their theorem, they wrote: "Physical systems are divided into types according to their unchanging (or ‘state-independent’) properties, and the state of a system at a time consists of a complete specification of those of its properties that change with time (its ‘state-dependent’ properties). To give a complete description of a system, then, we need to say what type of system it is and what its state is at each moment in its history. A physical quantity is a mutually exclusive and jointly exhaustive family of physical properties (for those who know this way of talking, it is a family of properties with the structure of the cells in a partition). Knowing what kinds of values a quantity takes can tell us a great deal about the relations among the properties of which it is composed. The values of a bivalent quantity, for instance, form a set with two members; the values of a real-valued quantity form a set with the structure of the real numbers. This is a special case of something we will see again and again, viz., that knowing what kind of mathematical objects represent the elements in some set (here, the values of a physical quantity; later, the states that a system can assume, or the quantities pertaining to it) tells us a very great deal (indeed, arguably, all there is to know) about the relations among them.
In quantum mechanical contexts, the term ‘observable’ is used interchangeably with ‘physical quantity’, and should be treated as a technical term with the same meaning. It is no accident that the early developers of the theory chose the term, but the choice was made for reasons that are not, nowadays, generally accepted. The state-space of a system is the space formed by the set of its possible states, i.e., the physically possible ways of combining the values of quantities that characterize it internally. In classical theories, a set of quantities which forms a supervenience basis for the rest is typically designated as ‘basic’ or ‘fundamental’, and, since any mathematically possible way of combining their values is a physical possibility, the state-space can be obtained by simply taking these as coordinates. So, for instance, the state-space of a classical mechanical system composed of n particles, obtained by specifying the values of 6n real-valued quantities — three components of position, and three of momentum for each particle in the system — is a 6n-dimensional coordinate space. Each possible state of such a system corresponds to a point in the space, and each point in the space corresponds to a possible state of such a system. The situation is a little different in quantum mechanics, where there are mathematically describable ways of combining the values of the quantities that don't represent physically possible states. As we will see, the state-spaces of quantum mechanics are special kinds of vector spaces, known as Hilbert spaces, and they have more internal structure than their classical counterparts.
A structure is a set of elements on which certain operations and relations are defined, a mathematical structure is just a structure in which the elements are mathematical objects (numbers, sets, vectors) and the operations mathematical ones, and a model is a mathematical structure used to represent some physically significant structure in the world.
The heart and soul of quantum mechanics is contained in the Hilbert spaces that represent the state-spaces of quantum mechanical systems. The internal relations among states and quantities, and everything this entails about the ways quantum mechanical systems behave, are all woven into the structure of these spaces, embodied in the relations among the mathematical objects which represent them. This means that understanding what a system is like according to quantum mechanics is inseparable from familiarity with the internal structure of those spaces. Know your way around Hilbert space, and become familiar with the dynamical laws that describe the paths that vectors travel through it, and you know everything there is to know, in the terms provided by the theory, about the systems that it describes.
By ‘know your way around’ Hilbert space, I mean something more than possess a description or a map of it; anybody who has a quantum mechanics textbook on their shelf has that. I mean know your way around it in the way you know your way around the city in which you live. This is a practical kind of knowledge that comes in degrees and it is best acquired by learning to solve problems of the form: How do I get from A to B? Can I get there without passing through C? And what is the shortest route? Graduate students in physics spend long years gaining familiarity with the nooks and crannies of Hilbert space, locating familiar landmarks, treading its beaten paths, learning where secret passages and dead ends lie, and developing a sense of the overall lay of the land. They learn how to navigate Hilbert space in the way a cab driver learns to navigate his city.
How much of this kind of knowledge is needed to approach the philosophical problems associated with the theory? In the beginning, not very much: just the most general facts about the geometry of the landscape (which is, in any case, unlike that of most cities, beautifully organized), and the paths that (the vectors representing the states of) systems travel through them. That is what will be introduced here: first a bit of easy math, and then, in a nutshell, the theory."
Yeah, I concur with Donnie
I really don't care, I just need this address for a project at school
We are sorry you were not at Las Vegas CES!! Our family just enjoys Apple and trying to convince Dad to join the Apple Group. Have a Great New Year!!
amen to that r man! what the heck does all that technical nonsense mean? i just want the adress for a school project!
Dude. YES, the adress is all I needed hahah school clearly gives projects by the trillions on this stuff
i hate school project
Hi Steve Jobs,
If you are still out there facing your terminal cancer which I read in today's news, will you want to try my natural healing. All I ask is to give me a chance.
"Insanity: doing the same thing over and over again and expecting different results." Albert Einstein. Steve, your doctors are "doing the same thing (Western Medicine) over and over again and expecting different results." It's about time you seek to do it differently.
Also, Steve, "Everything should be as simple as it is, but not simpler." Albert Einstein. The solution to your health issues is a simple one and therefore you need a simpler solution.
Human issues can only be healed by human hands, not by machines; human and machines are not related in the slightest way, genetically or otherwise. The best machines are programmed by human's ability to transfer his knowledge into it, but not all his knowledge; so, why get a 'thing' which is more inferior than human to address human issues?. I am sure you should have worked it out by now.
Before I sign off, please start massaging (squeeze your ears between your thumb and fore finger) your external ears until I meet you.
I will hear from you soon.
My family owns Apple stock and we are thrilled to see your company doing so well. I would love to see Apple show the USA that Apple cares about our national debt. I think it would be huge if Apple shared some of it's wealth by giving back. I think this would make a terrific commercial-support Apple because Apple supports you. Apple could bring in new customers while donating money for every Apple product bought to our nations debt.
Just an idea.
I am an educational consultant and writer. I would like to speak to someone from your company in regard to educational books, toys and products that I have developed for children and their parents. I have written two parenting books Learning Togerher With Children is on how children learn language. The second book, Thinking Together With Children focuses on how children learn to think. Both of these books are aimed at parents as primary educators of their children. I also have written and am developing books more for children. My first book Hurry Up Joey, It's Time To Go! is for preschool children being introduced to language. The second book for upper elementary children studying geography, history and social sciences, Once Upon A Rock is into the content of our earth and its inhabitants. I have a registered trademark that I have created and own called "The Marsupians." You can check me out on my website: iOriginsBooks.com. Also on the web under my full name Jeanette Kroese Thomson. At present I am taking your One to One course to become more efficient in using two new Apple computers--the 27" I Mac 12.2, OSX NX, VERSION 10.8 ns the 11" MacBook Air. I live in Laval, Quebec, Canada and go to the Laval Carrafour Shopping Center to your store there. At present it is in a temporary location and will soon be in a new two story setting early next year. I have had wonderful service and connections with all the team of Apple employers in this store. In fact, it is from being introduce to this process that it came to me that you might like my line of educational material. I have also purchased the Wacom Bamboo Capture Pen & Touch software to help me become my efficient in writing my next book, Once Upon A Rock In Doggerland. I have just returned from England and Holland after researching for this book that deals with the sea level rising at the end of the Mesolithic period 8000 to 10,000 years ago. This book will also be aimed at upper elementary students. My husband Gary A. Thomson and I are both American citizens born and raised in Nebraska-Lancaster and Otoe counties. We have degrees from the University of Nebraska and other advanced degrees as well. We have just purchased a Winnabego RV and will be traveling in the U.S. especially in the Southwest and Califorinia where we have family. I have a sister in Burbank, CA. So, it would be possible for us to journey up to Cupertino in the near future. My intention for all my efforts is to help parents to understand how important it is that they communicate with their children to help them learn. When parents do this, the child has the foundation and vehicle to move through the educational process more successfully. My telephone number is: 450 689 0993, email: email@example.com; Website: iOriginsBooks.com Jeanette Kroese Thomson (Wishing to enhance your wonderful products that I have used since the late 80s & early 90s. Your computers are the means by which we have been able to create and publish our books independently.)