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 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 variety of laptop experiments. the second one half makes a speciality of 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 learned as Java applets able to run on your browser and don't require any programming abilities.

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22) . 23) with the recursion formula Ti, j = Ti+1, j−1 + Ti, j−1 − Ti+1, j . 4 Higher Derivatives Difference quotients for higher derivatives can be obtained systematically using polynomial interpolation. Consider equidistant points xn = x0 + nh = . . , x0 − 2h, x0 − h, x0 , x0 + h, x0 + 2h, . . 28) which are evaluated at x0 : 1 1 f (x0 + h) − f (x0 − h) y−1 + y1 = , 2h 2h 2h f (x0 − h) − 2 f (x0 ) + f (x0 + h) f (x0 ) ≈ p (x0 ) = . 30) Higher order polynomials can be evaluated with an algebra program.

5! + 4 3 f (x) + h 20 h 40 f (x) + f (5) (x) + · · · 4 · 3! 16 · 5! = f (x) − 1 h 40 (5) f (x) + · · · 4 5! 15) 32 3 Numerical Differentiation 100 10–2 h–1 absolute error 10–4 10–6 (a) 10–8 (b) 10–10 h 10–12 (c) (d) h2 10–14 h4 6 h 10–4 10–2 10–16 10–16 10–14 10–12 10–10 10–8 10–6 step size h 100 d Fig. 2 Numerical differentiation. 17(d)). For very small step sizes the error increases as h −1 due to rounding errors that the error order is O(h 40 ). For three step widths h 0 = 2h 1 = 4h 2 we obtain the polynomial of second order (in h 2 ) (Fig.

J. 1007/978-3-642-13990-1_5, C Springer-Verlag Berlin Heidelberg 2010 47 48 5 Systems of Inhomogeneous Linear Equations ⎛ ⎞ 1 ⎜ −l21 1 ⎟ ⎜ ⎟ ⎜ −l31 1 ⎟ L1 = ⎜ ⎟ ⎜ .. ⎟ ⎝ . ⎠ 1 −ln1 li1 = ai1 . 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 = ⎜ ⎟ ⎜ .. . ⎟ ⎝.

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


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