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The standard normal distribution is the normal distribution with a mean of zero and a standard deviation of one Because the graph of its probability density resembles a bell, it is often called a bell curve Approximately normal distributions occur in many situations, as a result of the central limit theorem When there is reason to suspect the presence of a large number of small effects acting additively and independently, it is reasonable to assume that observations will be normal There are statistical methods to empirically test that assumption Two examples of the occurrence of the normal law in physics:



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*Robert Davis, Shortage of surgeons pinches US hospitals, USA Today, February 26, 2008, wwwusatodaycom/news/health/2008-0226-doctor-shortage_Nhtm





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Photon counts Light intensity from a single source varies with time, and is usually assumed to be normally distributed However, quantum mechanics interprets measurements of light intensity as photon counting Ordinary light sources that produce light by thermal emission should follow a Poisson distribution or Bose Einstein distribution on very short time scales On longer time scales (longer than the coherence time), the addition of independent variables yields an approximately normal distribution The intensity of laser light, which is a quantum phenomenon, has an exactly normal distribution Measurement errors Repeated measurements of the same quantity are expected to yield results which are clustered around a particular value If all major sources of errors have been taken into account, it is assumed that the remaining error must be the result of a large number of very small additive effects, and hence normal Deviations from normality are interpreted as indications of systematic errors which have not been taken into account Note that this is the central assumption of the mathematical theory of errors A13212 Cauchy Distribution The Cauchy distribution is an example of a continuous distribution that does not have an expected value or a variance In physics it is usually called a Lorentzian, and it is the distribution of the energy of an unstable state in quantum mechanics In particle physics, the extremely short lived particles associated with such unstable states are called resonances The Cauchy distribution has the probability density

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Quantization of the motion of a particle on the circle (like the quantization of polar coordinates in the plane) is an old question with so far mildly evasive answers A large body of literature exists concerning the subject, more speci cally devoted to the problem of angular localization and related Heisenberg inequalities; see, for instance, [202] Let us apply our scheme of coherent state quantization to this particular problem Let us rst recall the material presented in the last section of 5 The observation set X is the phase space of a particle moving on the circle, precisely the cylinder S 1 ~ R = {x = ( , J), |0 u < 2 , J R}, equipped with the measure 1 (dx) = 2 d J d The functions n (x) forming the orthonormal system needed to

Coherent States in Quantum Physics Jean-Pierre Gazeau Copyright 2009 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 978-3-527-40709-5

construct coherent states are suitably weighted Fourier exponentials: n (x) =

(Continued )

Some configuration problems pass the syntax tests performed by apachectl, but then result in the HTTP daemon exiting immediately after reloading its configuration. If this happens, use the tail command to check Apache s error log for useful information. On Debian systems, the error log is in /var/log/apache/error.log. On other systems, the location can be found by looking for the ErrorLog directive in your Apache configuration. An error message that you might encounter looks something like this:

e 2 ( J n) e in ,

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where > 0 is a regularization parameter that can be arbitrarily small The coherent states [62, 63, 67] read as |x = | J, = 1 N ( J)

e 2 ( J n) e in |e n ,

That was simple enough. Without the login parameter, the role is not allowed to log in to the PostgreSQL server, but it can be assigned privileges. This type of role is called a Group Role.

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