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MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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61 7 Dimensions An error of 15 percent is often small compared to the other inaccuracies in an approximate computation, so this method of approximate minimization is a valuable time-saver. Now return to the original problem: determining the Bohr radius. The approximate min- imization predicts the correct value. Even if the method were not so charmed, there is no point in doing a proper, calculus minimization. The calculus method is too accurate given the inaccuracies in the rest of the derivation. Engineers understand this idea of not over-engineering a system. If a bicycle most often breaks at welds in the frame, there is little point replacing the metal between the welds with expensive, high-strength aerospace materials. The new materials might last 100 years instead of 50 years, but such a replacement would be overengineering. To improve a bicy- cle, put effort into improving or doing without the welds. In estimating the Bohr radius, the kinetic-energy estimate uses a crude form of the uncer- tainty principle, p x ~ , whereas the true statement is that p x ~ / 2 . The estimate also uses the approximation E Kinetic ( p ) 2 / m . This approximation contains m instead of 2 m in the denominator. It also assumes that p can be converted into an energy as though it were a true momentum rather than merely a crude estimate for the root-mean-square momentum. The potential- and kinetic-energy estimates use a crude definition of position uncertainty x : that x r . After making so many approximations, it is pointless to mini- mize the result using the elephant gun of differential calculus. The approximate method is as accurate as, or perhaps more accurate than the approximations in the energy. This method of equating competing terms is balancing . We balanced the kinetic energy against the potential energy by assuming that they are roughly the same size. The conse- quence is that ~ 2 . a 0 m e ( e 2 / 4 π± 0 ) Nature could have been unkind: The potential and kinetic energies could have differed by a factor of 10 or 100. But Nature is kind: The two energies are roughly equal, except for a constant that is nearly 1. ‘Nearly 1’ is also called of order unity . This rough equality occurs in many examples, and you often get a reasonable answer by pretending that two energies (or two quantities with the same units) are equal. When the quantities are potential and kinetic energy, as they often are, you get extra safety: The so-called virial theorem protects you against large errors (for more on the virial theorem, see any intermediate textbook on classical dynamics). 7.5 Bending of light by gravity
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This note was uploaded on 02/24/2012 for the course MECHANICAL 6.055J taught by Professor Sanjoymahajan during the Spring '08 term at MIT.

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apr09 - MIT OpenCourseWare http:/ocw.mit.edu 6.055J /...

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