By Philipp Scherer
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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Extra resources for Computational Physics: Simulation of Classical and Quantum Systems
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  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 .
Computational Physics: Simulation of Classical and Quantum Systems by Philipp Scherer