FriCAS worksheet example
From Poirier's Bohmian Mechanics without Wavefunctions to Hall's Many Interacting Worlds
Ref:
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)
Quantum Mechanics Without Wavefunctions
Jeremy Schiff and Bill Poirier
[quant-ph] 11 Jan 2012
Verlet integration
set
for n=1,2,... iterate
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1-D
Hall's MIW quantum force eqs. (24,25)
Schiff and Poirier eq. (18)
In 1-D this is equivalent to eq. (10).
Now replace derivatives with differences
Compiling function diff1 with type (BasicOperator,Variable(n)) -> Expression(Integer)
Compiling function diff1 with type (BasicOperator,Polynomial(Integer)) -> Expression(Integer)
Compiling function diff2 with type (BasicOperator,Variable(n)) -> Expression(Integer)
Compiling function diff2 with type (BasicOperator,Polynomial(Integer)) -> Expression(Integer)
Compiling function diff3 with type (BasicOperator,Variable(n)) -> Expression(Integer)
Compiling function diff3 with type (BasicOperator,Polynomial(Integer)) -> Expression(Integer)
Compiling function diff4 with type (BasicOperator,Variable(n)) -> Expression(Integer)
Compiling function sif with type Polynomial(Integer) -> Fraction(Polynomial(Integer))
Compiling function sigma with type Polynomial(Integer) -> Fraction(Polynomial(Integer))
Compiling function sif with type Variable(n) -> Fraction(Polynomial(Integer))
Compiling function sigma with type Variable(n) -> Fraction(Polynomial(Integer))
Compiling function r with type Variable(n) -> Fraction(Polynomial(Integer))
$${{\left( -{x _{{n+2}}}+{x _{{n+1}}}+{x _{{n -1}}} -{x _{{n -2}}} \right)} \ {{{x _{n}}} ^{4}}}+{{\left( {{\left( {x _{{n+1}}}+{9 \ {x _{{n -1}}}} -{6 \ {x _{{n -2}}}} \right)} \ {x _{{n+2}}}} -{{{x _{{n+1}}}} ^{2}}+{{\left( -{{12} \ {x _{{n -1}}}}+{9 \ {x _{{n -2}}}} \right)} \ {x _{{n+1}}}} -{{{x _{{n -1}}}} ^{2}}+{{x _{{n -2}}} \ {x _{{n -1}}}} \right)} \ {{{x _{n}}} ^{3}}}+{{\left( {{\left( {{\left( -{{12} \ {x _{{n -1}}}}+{9 \ {x _{{n -2}}}} \right)} \ {x _{{n+1}}}} -{{12} \ {{{x _{{n -1}}}} ^{2}}}+{9 \ {x _{{n -2}}} \ {x _{{n -1}}}} \right)} \ {x _{{n+2}}}}+{{\left( {{15} \ {x _{{n -1}}}} -{{12} \ {x _{{n -2}}}} \right)} \ {{{x _{{n+1}}}} ^{2}}}+{{\left( {{15} \ {{{x _{{n -1}}}} ^{2}}} -{{12} \ {x _{{n -2}}} \ {x _{{n -1}}}} \right)} \ {x _{{n+1}}}} \right)} \ {{{x _{n}}} ^{2}}}+{{\left( {{\left( {{\left( {5 \ {x _{{n -1}}}} -{5 \ {x _{{n -2}}}} \right)} \ {{{x _{{n+1}}}} ^{2}}}+{{\left( {{11} \ {{{x _{{n -1}}}} ^{2}}} -{8 \ {x _{{n -2}}} \ {x _{{n -1}}}} \right)} \ {x _{{n+1}}}}+{6 \ {{{x _{{n -1}}}} ^{3}}} -{5 \ {x _{{n -2}}} \ {{{x _{{n -1}}}} ^{2}}} \right)} \ {x _{{n+2}}}}+{{\left( -{6 \ {x _{{n -1}}}}+{6 \ {x _{{n -2}}}} \right)} \ {{{x _{{n+1}}}} ^{3}}}+{{\left( -{{14} \ {{{x _{{n -1}}}} ^{2}}}+{{11} \ {x _{{n -2}}} \ {x _{{n -1}}}} \right)} \ {{{x _{{n+1}}}} ^{2}}}+{{\left( -{6 \ {{{x _{{n -1}}}} ^{3}}}+{5 \ {x _{{n -2}}} \ {{{x _{{n -1}}}} ^{2}}} \right)} \ {x _{{n+1}}}} \right)} \ {x _{n}}}+{{\left( {{\left( -{x _{{n -1}}}+{x _{{n -2}}} \right)} \ {{{x _{{n+1}}}} ^{3}}}+{{\left( -{2 \ {{{x _{{n -1}}}} ^{2}}}+{2 \ {x _{{n -2}}} \ {x _{{n -1}}}} \right)} \ {{{x _{{n+1}}}} ^{2}}}+{{\left( -{3 \ {{{x _{{n -1}}}} ^{3}}}+{2 \ {x _{{n -2}}} \ {{{x _{{n -1}}}} ^{2}}} \right)} \ {x _{{n+1}}}} -{{{x _{{n -1}}}} ^{4}}+{{x _{{n -2}}} \ {{{x _{{n -1}}}} ^{3}}} \right)} \ {x _{{n+2}}}}+{{\left( {x _{{n -1}}} -{x _{{n -2}}} \right)} \ {{{x _{{n+1}}}} ^{4}}}+{{\left( {3 \ {{{x _{{n -1}}}} ^{2}}} -{3 \ {x _{{n -2}}} \ {x _{{n -1}}}} \right)} \ {{{x _{{n+1}}}} ^{3}}}+{{\left( {3 \ {{{x _{{n -1}}}} ^{3}}} -{2 \ {x _{{n -2}}} \ {{{x _{{n -1}}}} ^{2}}} \right)} \ {{{x _{{n+1}}}} ^{2}}}+{{\left( {{{x _{{n -1}}}} ^{4}} -{{x _{{n -2}}} \ {{{x _{{n -1}}}} ^{3}}} \right)} \ {x _{{n+1}}}}$$$$4 \ {\left( {x _{{n -1}}} -{x _{{n -2}}} \right)} \ {\left( {x _{{n+2}}} -{x _{{n+1}}} \right)} \ {\left( {x _{n}} -{x _{{n+1}}} \right)} \ {\left( {x _{n}} -{x _{{n -1}}} \right)}$$2-D
1 particle moving in 2 dimensions (x,y) in many worlds, each indexed by 2 uniform parameters (n,m)
Note that there are only 28 unique derivatives since:
etc.
Difference operators
Compiled code for D- has been cleared.
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Compiling function D- with type (Expression(Integer),PositiveInteger) -> Expression(Integer)
Compiling function D+ with type (Expression(Integer),PositiveInteger) -> Expression(Integer)