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 variety of desktop experiments. the second one half focuses on simulation of classical and quantum structures with instructive examples spanning many fields in physics, from a classical rotor to a quantum bit. All software examples are discovered as Java applets able to run on your browser and don't require any programming talents.

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5) The result has the form ⎛ A(1) Now subtract ai2 a22 a11 ⎜ 0 ⎜ ⎜ 0 =⎜ ⎜ ⎜ ⎝ 0 0 a12 . . a1n−1 (1) (1) a22 . . a2n−1 (1) a32 . . . .. ⎞ a1n (1) a2n ⎟ ⎟ (1) ⎟ a3n ⎟ . ⎟ ⎟ . 6) (1) (1) an2 . . . ann times the second row from rows 3 . . n. 7) ⎛ ⎞ 1 ⎜0 1 ⎟ ⎜ ⎟ ⎜ 0 −l32 1 ⎟ L2 = ⎜ ⎟ ⎜ .. . ⎟ ⎝. ⎠ 1 0 −ln2 with (1) li2 = ai2 (1) a22 . 8) The result is ⎛ A(2) (2) (2) a11 a12 ⎜ (2) ⎜ 0 a22 ⎜ 0 0 =⎜ ⎜ .. ⎜ .. ⎝ . 0 0 ⎞ (2) (2) a13 . . a1n (2) (2) ⎟ a23 . . a2n ⎟ (2) (2) ⎟ a33 . . a3n ⎟ .

40) These are known as Legendre polynomials. Consider now a polynomial p(x) of order 2n − 1. It can be interpolated at the n sample points xi using the Lagrange method by a polynomial p(x) of order n − 1: n L j (x) p(x j ). 41) 44 4 Numerical Integration Then p(x) can be written as p(x) = p(x) + (x − x 1 )(x − x2 ) · · · (x − xn )q(x). 42) Obviously q(x) is a polynomial of order (2n − 1) − n = n − 1. Now choose the positions xi as the roots of the nth order Legendre polynomial: (x − x1 )(x − x2 ) · · · (x − xn ) = Pn (x).

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 .