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more quantum mechanical}. For a book that puts a greater emphasis on technological applications oflow-noise lasers, see [469]. Specifically on the quantum theory of the laser, I would recommend the following books. Haken's twovolume work [260] is an excellent introduction. A more specialized treatment is given in [261], [341], and [533]. For more exotic lasers such as the correlated emission laser (CEL), see Scully and Zubairy's book [552]. The history of the laser is nicely told by one of its creators, Charles Townes, in [602]. See also the review paper [375]. In 1900 a student of Jules Henri Poincare, Louis Bachelier, anticipated many of Einstein's ideas on Brownian motion by studying the apparently completely different problem of the changes in the price of shares in the Parisian stock market [25]. Bachelier assumed that share prices obeyed a Gaussian distribution. This means that in his model, unlike in reality, share prices could become negative. 32 It was Sprenkle [574] who, in the 1960s, first corrected Bachelier's assumption, eliminating negative prices with the adoption of a lognormal distribution instead of a Gaussian. This corresponds to a geometric Brownian motion, where the step size is proportional to the distance position rather than the ordinary arithmetic Brownian motion seen in this chapter. For more on these ideas, see Osborne's excellent book [473]. These developments eventually lead to the Black and Scholes theory of option pricing [59], which practically began the field of quantitative finance. For more on quantitative finance, see the following excellent books [304, 476, 653]. To know more about stochastic calculus from a physicist's perspective, see [224, 232, 580, 612-614]. If you prefer a rigorous approach in the axiomstheorem-proof style, have a look at [340, 371]. Many of the important historical references can be found in the collection [623]. The bible of numerical methods to solve stochastic differential equations is Kloeden and Platen's book [358]. An important milestone before the realization of the one-and-the-same atom laser was Kyangwon An and Michael S. Feld's single-atom laser [16]. This device is similar to the micro maser except that it operates in the optical regime and the Rabi frequency is comparable to the decay rates. As in the micro maser, the cavity field is built by a succession of atoms rather than the same single atom as in the Caltech experiment [436]. For more on An and Feld's single-atom laser, see [lfrI9, 192,405,423]. The combination of strong coupling in the optical regime with advanced trapping technology also allows the generation of single-photon states on demand (Le., deterministically). See [356, 372, 373] and the recent realization of a deterministic source of single photons [438].
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32Japanese interest rates. however. seem to fit a Gaussian distribution better than a lognormal one.
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8.1 When a child on a swing lowers and raises her center of gravity, the effective length of the pendulum string corresponding to the swing (see the model presented in 2) changes. Considering that this change in the effective length is small in comparison with the length itself, assuming that the swing only swings by small angles and that the child lowers and raises her center of gravity sinusoidally, derive the following equation for the pumped-up swing:
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+ 'Y dt + {I + E cos(wt)} woO =
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where 0 is the angle of the swing with the vertical, 'Y is the coefficient of viscous friction, E 1, Wo is the ordinary resonance frequency of the swing, and w is the pumping frequency due to the lowering and raising of the center of gravity. Assuming that'Y < < wo, 0(0) = 0, and 0(0) = v, solve this equation by Laplace transform. Show that the oscillations are amplified when w = 2wo. 8.2 Solve (8.20) assuming that the particle starts the diffusion at x = 0 at t = 0
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[i.e.,p(x,O) = <5(x)].
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Hint: Apply a Fourier spatial transform to get an ordinary first-order differential equation from (8.20). You can easily solve this equation for the transform of p(x). To invert the transform, complete the squares in the exponential to obtain a Gaussian.
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Using (8.31) and (8.35) in (8.34), obtain (8.36).
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For a function Z == f(Y1 , , Yn ) of the n random variables Y1 , , Yn , the probability distribution Q( z) of Z is given in terms of that of Y1 , , Yn , P(YI, ... , Yn), by [232]
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dYn P(YI,. . ,Yn) <5 (z - f(YI. ,Yn))'
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Using this, show that if Z = in + .jV Y, where in, v are deterministic variables and Y is a normal variable of mean zero and variance 1 whose probability distribution is given by
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P(y)= ~'
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