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MIT 6 050J - Physical Systems

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11 Physical Systems11.1 Nature of Quantum Mechanics11.2 Introduction to Quantum Mechanics11.3 Stationary States11.4 Multi-State Model11.5 Energy Systems11.6 Information SystemsChapter 11Physical SystemsUntil now we have ignored most aspects of physical systems by dealing only with abstract ideas such asinformation. Although we assumed that each bit stored or transmitted was manifested in some physicalobject, we focused on the abstract bits, and ignored any limitations caused by the laws of physics. This isthe fundamental mantra of the information age.It has not always b e en that way, and it will not be that way in the future. In past centuries, the physicalmanifestation of information was of great importance because of its great cost. To preserve or communicateinformation, b ooks had to be written or even words cut into stone. For example, think of the process ofcreating a medieval manuscript during the middle ages. Pages were laboriously copied and illustrated. Theresults may be viewed today with great admiration for their artistry and cultural importance, in part becausethey were so expensive to create – society could only afford to deal with the most important information,and the cost of superb art work was not high compared with the other costs of production.Advances over the years have improved the efficiency of information storage and transmission – thinkof the printing press, telegraph, telephone, radio, television, digital signal processing, semiconductors, fiberoptics. These have enabled complicated systems such as computers, data networks, and even economicsystems for the creation and distribution of entertainment. As the cost of processing data drops, it isrelevant to consider the case where the cost is negligible compared to the cost of creating, maintaining,and making use of information. It is in this domain that the abstract ideas of information theory, bits,coding, and indeed all of computer science are dominant. All aspects of modern society are coping withthe increasing amount of information that is available. Even the ideas of intellectual property, copyrights,patents, and trade secrets are being redefined in light of the changing economics of information processing.This is the information age.This model of information separated from its physical embodiment is, of course, an approximation ofreality. Eventually, as we make microelectronic systems more and more complicated, using smaller andsmaller components, we will need to face the fundamental limits imposed not so much by our ability tofabricate sm all structures, but by the basic laws of physics. The basic law which governs all physical systemsis quantum mechanics.Quantum mechanics is often believed to be of im portance only for small structures, say the size of anatom. Although it is unavoidable at that length scale, it also governs everyday objects. When dealing withinformation proc es sing in physical systems, it is p ertinent to consider both very small systems with a smallnumber of bits of information, and large systems with large amounts of information.The key ideas we have used thus far that need to be re-interpreted in the regime where quantum mechanicsis important include• The digital abstraction made practical by devices that can restore data with small perturbationsAuthor: Paul Penfield, Jr.Version 1.0.2, April 7, 2003. Copyrightc 20 03 Massachusetts Institute of TechnologyURL: http://www-mtl.mit.edu/Courses/6.050/notes/chapter11.pdfTOC: http://www-mtl.mit.edu/Courses/6.050/notes/index.html9311.1 Nature of Quantum Mechanics 94• Use of probability to express our knowledge in the face of uncertainty• The Principle of Maximum Entropy as a technique to estimate probabilities without bias11.1 Nature of Quantum MechanicsQuantum mechanics is weird. There seems to be no way to make it appear otherwise. Many of its predictionsare not consistent with expectations that arise from everyday experience.Quantum mechanics is mysterious, even to very good physicists. The underlying philosophy and inter-pretation of its equations and techniques are controversial.Quantum mechanics is difficult to use. Relatively advanced mathematical skills are needed. The basicequation, although linear, is a partial differential equation that cannot be solved analytically except in avery few simple situations. Usually numerical solutions are necessary.Quantum mechanics, like other physical theories, requires skill and judgement both in modelling and inmathematics. It is not generally taught in any depth before the graduate or advanced undergraduate level.Quantum mechanics comes in different forms. It has many alternate formulations. Generally these areequivalent in the sense that they predict the same results of experiments, but are not equally easy to learnor to use for particular purpose s.In light of these attributes, why is quantum mechanics important? Because it works. It is the ONLYfundamental physical theory that works over such a wide range of situations. Its predictions have beenverified experimentally time after time. It applies to everyday size objects, and to astronomical objects(although it is usually not necessary for them). It applies to atomic-size objects, to electromagnetic waves,and to sub-atomic objects. There is a version that is compatible with the theory of special relativity. Aboutthe only physical phenomenon not handled well at this time is gravity; quantum mechanics has not yet beenextended to be compatible with the theory of general relativity.In these notes we cannot cover quantum mechanics in much depth. For the purpose of examininginformation processing in physical systems, we only need to understand a few of the general features suchsystems must have. In particular, we need a model of physical systems in which there are many possible states,each with its own probability of being the one the system is actually in (i.e., the state “occupied”). Thesestates all have physical properties associated with them, and energy is one of these. Quantum mechanicsjustifies this model.We will use this model in two situations. The first (below) is one with many states, where the objective isto understand how the information associated with the occupancy of these states affects the flow of energy.The second (in a later chapter of these notes) is one with a very small number of states, where informationis represented using the occupancy of these states, and the objective is to understand both the


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MIT 6 050J - Physical Systems

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