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Edit detail for SandBoxMIWHall revision 4 of 14

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Editor: Bill Page Time: 2015/07/17 18:23:33 GMT+0
Note:

changed:
-Gausssian
Gaussian

added:

Cauchy
\begin{axiom}
PdfCauchy(x)==1/%pi/(1+x^2)
eval(RHall,[P(q)=PdfCauchy(q),D(P(q),q)=D(PdfCauchy(q),q),D(P(q),[q,q])=D(PdfCauchy(q),[q,q])])
eval(Rforce,[P(q)=PdfCauchy(q),D(P(q),q)=D(PdfCauchy(q),q),D(P(q),[q,q])=D(PdfCauchy(q),[q,q]),D(P(q),[q,q,q])=D(PdfCauchy(q),[q,q,q])])
\end{axiom}

Many Interacting Worlds - Hall, et al.

fricas

P:=operator('P);

Type: BasicOperator?

fricas

Q:=operator('Q);

Type: BasicOperator?

fricas

g:=operator('g);

Type: BasicOperator?

fricas

U:=operator('U);

Type: BasicOperator?

fricas

r:=operator('r);

Type: BasicOperator?

fricas

ℏ:=h;

Type: Variable(h)

Hall equations 15,19,20

fricas

eq15:= r(q) = -D(U(q),q)

$\label{eq1}{r \left({q}\right)}= -{{U_{\ }^{,}}\left({q}\right)}$

(1)

Type: Equation(Expression(Integer))

fricas

eq19:= U(q) = 1/(2*m)*g(q)^2

$\label{eq2}{U \left({q}\right)}={{{g \left({q}\right)}^{2}}\over{2 \ m}}$

(2)

Type: Equation(Expression(Integer))

fricas

eq20:= g(q) = ℏ/2 * 1/P(q)*D(P(q),q)

$\label{eq3}{g \left({q}\right)}={{h \ {{P_{\ }^{,}}\left({q}\right)}}\over{2 \ {P \left({q}\right)}}}$

(3)

Type: Equation(Expression(Integer))

fricas

eval(eq19,eq20)

$\label{eq4}{U \left({q}\right)}={{{{h}^{2}}\ {{{P_{\ }^{,}}\left({q}\right)}^{2}}}\over{8 \ m \ {{P \left({q}\right)}^{2}}}}$

(4)

Type: Equation(Expression(Integer))

fricas

RHall:=eval(eq15,D(lhs %,q)=D(rhs %,q))

$\label{eq5}{r \left({q}\right)}={{-{{{h}^{2}}\ {P \left({q}\right)}\ {{P_{\ }^{,}}\left({q}\right)}\ {{P_{\ }^{, ,}}\left({q}\right)}}+{{{h}^{2}}\ {{{P_{\ }^{,}}\left({q}\right)}^{3}}}}\over{4 \ m \ {{P \left({q}\right)}^{3}}}}$

(5)

Type: Equation(Expression(Integer))

Hall equations 6, 7, A3

fricas

eq6:= r(q) = -D(Q(q),q)

$\label{eq6}{r \left({q}\right)}= -{{Q_{\ }^{,}}\left({q}\right)}$

(6)

Type: Equation(Expression(Integer))

fricas

eq7:= Q(q) = 1/sqrt(P(q))*-ℏ^2/2/m*D(sqrt(P(q)),[q,q])

$\label{eq7}{Q \left({q}\right)}={{{2 \ {{h}^{2}}\ {P \left({q}\right)}\ {{P_{\ }^{, ,}}\left({q}\right)}}-{{{h}^{2}}\ {{{P_{\ }^{,}}\left({q}\right)}^{2}}}}\over{8 \ m \ {{P \left({q}\right)}^{2}}}}$

(7)

Type: Equation(Expression(Integer))

fricas

Rforce:=eval(eq6,D(lhs eq7,q)=D(rhs eq7,q))

$\label{eq8}\begin{array}{@{}l} \displaystyle {r \left({q}\right)}={{\left( \begin{array}{@{}l} \displaystyle -{{{h}^{2}}\ {{P \left({q}\right)}^{2}}\ {{P_{\ }^{, , ,}}\left({q}\right)}}+ \ \ \displaystyle {2 \ {{h}^{2}}\ {P \left({q}\right)}\ {{P_{\ }^{,}}\left({q}\right)}\ {{P_{\ }^{, ,}}\left({q}\right)}}-{{{h}^{2}}\ {{{P_{\ }^{,}}\left({q}\right)}^{3}}}$

(8)

Type: Equation(Expression(Integer))

Gaussian

fricas

PdfNorm(x)==1/2*sqrt(2)*exp(-1/2*x^2)/sqrt(%pi)

Type: Void

fricas

eval(RHall,[P(q)=PdfNorm(q),D(P(q),q)=D(PdfNorm(q),q),D(P(q),[q,q])=D(PdfNorm(q),[q,q])])

fricas

Compiling function PdfNorm with type Variable(q) -> Expression(
      Integer)

$\label{eq9}{r \left({q}\right)}= -{{{{h}^{2}}\ q}\over{4 \ m}}$

(9)

Type: Equation(Expression(Integer))

fricas

eval(Rforce,[P(q)=PdfNorm(q),D(P(q),q)=D(PdfNorm(q),q),D(P(q),[q,q])=D(PdfNorm(q),[q,q]),D(P(q),[q,q,q])=D(PdfNorm(q),[q,q,q])])

$\label{eq10}{r \left({q}\right)}= -{{{{h}^{2}}\ q}\over{4 \ m}}$

(10)

Type: Equation(Expression(Integer))

Cauchy

fricas

PdfCauchy(x)==1/%pi/(1+x^2)

Type: Void

fricas

eval(RHall,[P(q)=PdfCauchy(q),D(P(q),q)=D(PdfCauchy(q),q),D(P(q),[q,q])=D(PdfCauchy(q),[q,q])])

fricas

Compiling function PdfCauchy with type Variable(q) -> Expression(
      Integer)

$\label{eq11}{r \left({q}\right)}={{{{{h}^{2}}\ {{q}^{3}}}-{{{h}^{2}}\ q}}\over{{m \ {{q}^{6}}}+{3 \ m \ {{q}^{4}}}+{3 \ m \ {{q}^{2}}}+ m}}$

(11)

Type: Equation(Expression(Integer))

fricas

eval(Rforce,[P(q)=PdfCauchy(q),D(P(q),q)=D(PdfCauchy(q),q),D(P(q),[q,q])=D(PdfCauchy(q),[q,q]),D(P(q),[q,q,q])=D(PdfCauchy(q),[q,q,q])])

$\label{eq12}{r \left({q}\right)}={{{2 \ {{h}^{2}}\ {{q}^{3}}}-{4 \ {{h}^{2}}\ q}}\over{{m \ {{q}^{6}}}+{3 \ m \ {{q}^{4}}}+{3 \ m \ {{q}^{2}}}+ m}}$

(12)

Type: Equation(Expression(Integer))