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The_microscopic_world - How the microscopic world works...

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© M. S. Shell 2008 1/22 last modified 9/21/2009 How the microscopic world works ChE210A In order to understand the molecular origins of thermodynamic equilibrium, it is important to first understand the fundamental principles by which molecules interact. Moreover, as science and technology increasingly elucidate and rely upon nanoscale phenomena, molecular driving forces are a critical piece of any technical education. Quantum theory So, how does the world really work? What are the most fundamental, basic principles that form the basis of reality as we know it? Currently our understanding of reality rests upon two principle fields in physics: quantum theory and relativity. Both of these theories have been the subject of stringent experimental tests over the past century. Perhaps surprisingly, there are some incompatibilities between quantum theory and relativity, which are driving the develop- ment of new ideas like string theory. Luckily differences between the two manifest only under particular circumstances, as in black holes; moreover, it turns out that relativistic effects are not usually relevant to most everyday problems and to the kinds of problems studied in chemical thermodynamics. So we can safely start our discussion at the level of quantum theory. Quantum mechanics describes the complete time-evolution of any system. It is most easily described by imagining a system of fundamental particles, like electrons or protons. Each particle has associated with it a position variable, g1818 g2869 ,g1818 g2870 ,… , etc., where each g1818 is a vector g4666g1876,g1877,g1878g4667 . Then, the complete state of the system at any point in time is specified by a function Ψg4666g1818 g2869 ,g1818 g2870 ,…,g1872g4667 called the wavefunction . The wavefunction takes on complex values, of the form g1853g3397g1854g1861 . The physical significance of the wavefunction is that Ψ g1499 g4666g1818 g2869 ,g1818 g2870 ,…,g1872g4667Ψg4666g1818 g2869 ,g1818 g2870 ,…,g1872g4667 gives the joint probability that particle one is at g1818 g2869 , particle 2 is at g1818 g2870 , etc, at the time g1872 . Here g1499 denotes the complex conjugate, i.e., g1853g3398g1854g1861 . The quantity Ψ g1499 Ψ therefore is always real and positive, g1853 g2870 g3397g1854 g2870 . The central focus of quantum mechanics is the complex-valued wavefunction Ψ , whose interpretation is that Ψ g1499 Ψ gives the probability the system is at a specific microscopic state at a given time. The wavefunction describes the evolution of probabilities . This is very different from Newto- nian mechanics, in which each particle has an exact position at time g1872 , and not a probability distribution of positions. Quantum mechanics says that this distribution is the most we can possibly know about the system; we cannot predict the position of the particles to more
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© M. S. Shell 2008 2/22 last modified 9/21/2009 accuracy. There is some inherent randomness in nature, and the best that we can do is predict the probabilities of different possible outcomes. This may sound a bit strange, because we are not used to this kind of behavior at the macroscopic scale. Indeed, for large objects, these
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