10 Physical Systems10.1 Nature of Quantum Mechanics10.2 Introduction to Quantum Mechanics10.3 Stationary States10.4 Multi-State Model10.4.1 Energy Systems10.4.2 Information SystemsChapter 10Physical SystemsUntil now we have ignored most aspects of physical systems by dealing only with abstract ideas suchas information. 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 been that way, and it will not b e that way in the future. In past centuries, the physicalmanifestation of information was of great importance because of its gre at cost. To preserve or communicateinformation, books had to be written or even words cut into stone. For example, think of the processof creating a medieval manuscript during the middle ages. Pages were laboriously copied and illustrated.The results may be viewed today with great admiration for their artistry and cultural importance, in partbecause they were so expensive to create—society could only afford to deal with what it considered themost important information, and the cost of superb art work was not high compared with the other costs ofproduction.Advances over the years have improved the efficiency of information storage and transmission—think ofthe printing press, telegraph, telephone, radio, television, digital signal processing, semiconductors, and fiberoptics. These technologies have led to sophisticated systems such as computers and data networks, and haveshaped the methods used for the creation and distribution of information-intensive products by companiessuch as those in the entertainment business. As the cost of processing and distributing data drops, it isrelevant to consider the case where that cost is small compared to the cost of creating, maintaining, andusing the information. It is in this domain that the abstract ideas of information theory, bits, coding, andindeed all of computer science are dominant. All sectors of modern society are coping with the increasingamount of information that is available. Fundamental ideas of intellectual property, copyrights, patents, andtrade secrets are being rethought in light of the changing economics of information process ing. Welcome tothe information age.The model of information separated from its physical embodiment is, of course, an approximation ofreality. Eventually, as we make microelectronic systems more and more c omplicated, using smaller andsmaller components, we will need to face the fundamental limits imposed not by our ability to fabricatesmall structures, but by the laws of physics. All physical systems are governed by quantum mechanics.Quantum mechanics is often believed to be of importance 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 processing in physical systems, it is pertinent 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 includeAuthor: Paul Penfield, Jr.This document: http://www.mtl.mit.edu/Courses/6.050/2008/notes/chapter10.pdfVersion 1.5, April 10, 2008. Copyrightc 2008 Massachusetts Institute of TechnologyStart of notes · back · next | 6.050J/2.110J home page | Site map | Search | About this document | Comments and inquiries11310.1 Nature of Quantum Mechanics 114• The digital abstraction made practical by devices that can restore data with small perturbations• Use of probability to express our knowledge in the face of uncertainty• The Principle of Maximum Entropy as a technique to estimate probabilities without bias10.1 Nature of Quantum MechanicsQuantum mechanics is weird. There seems to be no way to make it appear otherwise. Many of itspredictions are not w hat one would expect from everyday experience with objects of the size normallyencountered.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 advance d 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 purposes.In light of these disadvantages, 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 exp erimentally 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 physic al 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 me chanicsjustifies 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