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Authors: David Cyganski, Bill Page
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Description: Jupyter notebook Use_Sage_in_Jupyter_notebook_on_SageMathCloud.ipynb

From Poirier's Bohmian Mechanics without Wavefunctions to Hall's Many Interacting Worlds in More Than One Dimension###

Ref:

  1. Quantum Mechanics Without Wavefunctions Jeremy Schiff and Bill Poirier J. Chem. Phys. 136, 031102 (2012)

  2. Quantum Phenomena Modeled by Interactions between Many Classical Worlds Michael J. W. Hall Dirk-André Deckert and Howard M. Wiseman, PHYSICAL REVIEW X 4, 041013 (23 October 2014)

  3. Verlet integration (Wikipedia)



  4. Explicit, Time Reversible, Adaptive Step Size Control Ernst Hairer and Gustaf Söderlind SIAM Journal on Scientific Computing. 2005, vol. 26, no. 6, p. 1838-1851

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%display latex
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from numpy import array,concatenate,isnan from mpmath import erfinv hbar=var('hbar',latex_name='\\hbar') mu=var('mu',latex_name='\mu') hbar,mu
(,μ)\left({\hbar}, {\mu}\right)
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sin(x)
sin(x)\sin\left(x\right)
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1+1
22
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vars = ['x','y']; d = len(vars) def argscript(self, *args): return "%s_{%s}"%(self.name(),','.join(map(repr, args))) X = map(lambda nam:function(nam, print_latex_func=argscript),vars); x,y = X n,m = var('n,m'); ind=[n,m] # position of particle in world (n,m) Enm=map(lambda x:x(*ind),X);show(Enm)
[xn,m,yn,m]\left[x_{n,m}, y_{n,m}\right]
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def Dminus(x,i):return(x-x.subs(ind[i]==ind[i]-1)) def Dplus(x,i):return(x.subs(ind[i]==ind[i]+1)-x)
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Jnm = matrix(map(lambda e:[Dminus(e,i) for i in range(d)],Enm));show(Jnm)
(xn1,m+xn,mxn,m1+xn,myn1,m+yn,myn,m1+yn,m)\left(\begin{array}{rr} -x_{n - 1,m} + x_{n,m} & -x_{n,m - 1} + x_{n,m} \\ -y_{n - 1,m} + y_{n,m} & -y_{n,m - 1} + y_{n,m} \end{array}\right)
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# inverse Jacobian Knm = matrix(map(lambda x: map(lambda y:y.normalize(),x),Jnm^(-1))) show(Knm)
(yn,m1yn,mxn,m1yn1,mxn,myn1,mxn1,myn,m1+xn,myn,m1+xn1,myn,mxn,m1yn,mxn,m1xn,mxn,m1yn1,mxn,myn1,mxn1,myn,m1+xn,myn,m1+xn1,myn,mxn,m1yn,myn1,myn,mxn,m1yn1,mxn,myn1,mxn1,myn,m1+xn,myn,m1+xn1,myn,mxn,m1yn,mxn1,mxn,mxn,m1yn1,mxn,myn1,mxn1,myn,m1+xn,myn,m1+xn1,myn,mxn,m1yn,m)\left(\begin{array}{rr} \frac{y_{n,m - 1} - y_{n,m}}{x_{n,m - 1} y_{n - 1,m} - x_{n,m} y_{n - 1,m} - x_{n - 1,m} y_{n,m - 1} + x_{n,m} y_{n,m - 1} + x_{n - 1,m} y_{n,m} - x_{n,m - 1} y_{n,m}} & -\frac{x_{n,m - 1} - x_{n,m}}{x_{n,m - 1} y_{n - 1,m} - x_{n,m} y_{n - 1,m} - x_{n - 1,m} y_{n,m - 1} + x_{n,m} y_{n,m - 1} + x_{n - 1,m} y_{n,m} - x_{n,m - 1} y_{n,m}} \\ -\frac{y_{n - 1,m} - y_{n,m}}{x_{n,m - 1} y_{n - 1,m} - x_{n,m} y_{n - 1,m} - x_{n - 1,m} y_{n,m - 1} + x_{n,m} y_{n,m - 1} + x_{n - 1,m} y_{n,m} - x_{n,m - 1} y_{n,m}} & \frac{x_{n - 1,m} - x_{n,m}}{x_{n,m - 1} y_{n - 1,m} - x_{n,m} y_{n - 1,m} - x_{n - 1,m} y_{n,m - 1} + x_{n,m} y_{n,m - 1} + x_{n - 1,m} y_{n,m} - x_{n,m - 1} y_{n,m}} \end{array}\right)

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