Download e-book for iPad: Computational Physics: Simulation of Classical and Quantum by Philipp Scherer

By Philipp Scherer

ISBN-10: 3642139892

ISBN-13: 9783642139895

ISBN-10: 3642139906

ISBN-13: 9783642139901

This e-book encapsulates the assurance for a two-semester path in computational physics. the 1st half introduces the fundamental numerical tools whereas omitting mathematical proofs yet demonstrating the algorithms when it comes to a number 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 application examples are learned as Java applets able to run on your browser and don't require any programming abilities.

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5 More Dimensions Consider polynomials of more than one variable. In two dimensions we use the Lagrange polynomials L i, j (x, y) = k=i (x − xk ) (xi − xk ) j =l (y − yl ) . 35) The interpolating polynomial is f i, j L i, j (x, y). 39) and the Laplace operator ∇ 2 f (x0 , y0 ) ≈ ∇ 2 p(x0 , y0 ) 1 = 2 ( f (x0 , y0 + h) + f (x0 , y0 − h) + f (x0 , y0 + h) + f (x0 , y0 − h) − 4 f (x0 , y0 )) . 1 Numerical Differentiation In this computer experiment we calculate the derivative of f (x) = sin(x) numerically with (a) the single-sided difference quotient f (x + h) − f (x) df ≈ dx h (b) the symmetrical difference quotient f (x + h) − f (x − h) df ≈ Dh f (x) = dx 2h (c) higher order approximations which can be derived using the extrapolation method 4 1 Dh f (x) + Dh/2 f (x) 3 3 4 64 1 Dh f (x) − Dh/2 f (x) + Dh/4 f (x) 45 9 45 − The error of the numerical approximation is shown on a log–log plot as a function of the step width h.

Xr −1 xr ] . 18) They are invariant against permutation of the arguments which can be seen from the explicit formula 18 2 Interpolation r f (xk ) . i =k (x k − x i ) [x1 x2 . . 21) · · · + [xn xn−1 . . x0 ](x − x0 )(x − x1 ) · · · (x − xn−1 ), and the function q(x) = [x xn · · · x0 ](x − x0 ) · · · (x − xn ). 22) Obviously q(xi ) = 0, i = 0 · · · n, hence p(x) is the interpolating polynomial. 3 Interpolation Error The error of the interpolation can be estimated with the following theorem: If f (x) is n +1 times differentiable then for each x there exists ξ within the smallest interval containing x as well as all of the xi with n q(x) = (x − xi ) i=0 f (n+1) (ξ ) .

P01 P12 .. 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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