New PDF release: Computational Physics: Simulation of Classical and Quantum

By Philipp Scherer

ISBN-10: 3642139892

ISBN-13: 9783642139895

ISBN-10: 3642139906

ISBN-13: 9783642139901

This booklet encapsulates the insurance for a two-semester direction in computational physics. the 1st half introduces the elemental numerical tools whereas omitting mathematical proofs yet demonstrating the algorithms in terms of a number of computing device experiments. the second one half focuses on simulation of classical and quantum platforms with instructive examples spanning many fields in physics, from a classical rotor to a quantum bit. All software examples are learned as Java applets able to run on your browser and don't require any programming talents.

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Example text

3 Interpolation error. 25). Its roots xi are given by the x values of the sample points (circles). Inside the interval x 0 . . 26) whereas the error increases rapidly outside the interval 0 < x < 2π (Fig. 3). Algorithm The divided differences are arranged in the following way: 20 2 Interpolation f0 f1 .. [x0 x1 ] .. .. . 4 Neville Method The Neville method [5] is advantageous if the polynomial is not needed explicitly and has to be evaluated only at one point. Consider the interpolating polynomial for the points x0 .

In this computer experiment we determine the smallest and largest integer numbers. Beginning with I = 1 we add repeatedly 1 until the condition I + 1 > I becomes invalid or subtract repeatedly 1 until I − 1 < I becomes invalid. For the 64-bit long integer format this takes too long. Here we multiply alternatively I by 2 until I − 1 < I becomes invalid. For the character format the corresponding ordinal number is shown which is obtained by casting the character to an integer. 3 Truncation Error This computer experiment approximates the cosine function by a truncated Taylor series n max cos(x) ≈ mycos(x, n max ) = (−)n n=0 x 2n x2 x4 x6 =1− + − + ··· (2n)!

P012 .. .. 32) . Pn Pn−1,n Pn−2,n−1,n · · · P01···n The first column contains the function values Pi (x) = f i . 3 Spline Interpolation Polynomials are not well suited for interpolation over a larger range. Often spline functions are superior which are piecewise defined polynomials [6, 7]. The simplest case is a linear spline which just connects the sampling points by straight lines: yi+1 − yi (x − xi ), xi+1 − xi s(x) = pi (x) where xi ≤ x < xi+1 . 34) The most important case is the cubic spline which is given in the interval xi ≤ x < xi+1 by pi (x) = αi + βi (x − xi ) + γi (x − xi )2 + δi (x − xi )3 .

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Computational Physics: Simulation of Classical and Quantum Systems by Philipp Scherer

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