By Mark Greenshields

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**Additional info for 40 best machine code routines for the 64**

**Sample text**

20), f (x1 , v1 ,t) = dx2 dv2 , . . , dxN dvN f N (x1 , v1 , . . , xN , vN ,t) =N m 2 π h¯ M ∑ α =1 × ψαN ∗ x1 + pα ds1 dx2 , . . , dxN exp i m v1 s1 h¯ s1 s1 , x2 , . . , xN ,t ψαN x1 − , x2 , . . , xN ,t . 63) To follow the mean field approximation, we are tempted to assume the factorized form ψαN (x1 , x2 , . . 4 The Schr¨odinger–Poisson System 29 for the N-body wavefunction, fully neglecting the correlations due to the interaction potential. 64), which does not respect the Pauli principle.

More complete reviews on Wigner function methods can be found, for example, in [8, 18, 26, 36]. For simplicity, let us start with a one-dimensional, one-particle pure state quantum system, represented by a wavefunction ψ (x,t). In this case, the Wigner function f = f (x, v,t) is defined [38] as f= m 2 π h¯ ds exp imvs h¯ s s ψ ∗ x + ,t ψ x − ,t , 2 2 F. 1) 15 16 2 The Wigner–Poisson System where x is the position, v the velocity, t the time, m the particle’s mass and h¯ = h/(2π ), where h is Planck’s constant.

XN ) = ∑ W (|xi − x j |) + ∑ Wext (xi ,t). 37). 43) 4πε0 |r − r | where −e is the electron charge and ε0 is the vacuum permittivity constant. In terms of the self-consistent Wsc and some external Wext potentials, it is convenient to define the total electrostatic potential φ (r,t) so that φ (r,t) = φsc (r,t) + φext (r,t), where Wsc (r,t) = −eφsc (r,t) , Wext (r,t) = −eφext (r,t). 37) that ∇2 φsc = − e ε0 e ε0 e = ε0 = Moreover, dr dv f (r , v,t) ∇2 1 4π |r − r | dr dv f (r , v,t) δ (r − r ) dv f (r, v,t).