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
In [2]:
%load_ext sage
In [3]:
from numpy import array,concatenate,isnan
from mpmath import erfinv
hbar=var('hbar',latex_name='\\hbar')
mu=var('mu',latex_name='\mu')
In [4]:
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)

[x(n, m), y(n, m)]
In [5]:
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)
In [6]:
Jnm = matrix(map(lambda e:[Dminus(e,i) for i in range(d)],Enm));show(Jnm)

[-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)]
In [8]:
# inverse Jacobian
Knm = matrix(map(lambda x: map(lambda y:y.normalize(),x),Jnm^(-1)))
show(Knm)
Out[8]:
\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)
In [ ]: