Two Slits Interference Is Compatible with\\ Particles' Trajectories
Vladimir V. Kisil
We propose a simple numerical experiment of two slits interference of particles. It disproves the popular belief that such an interference is incompatible with a knowledge which slit each particle came through or, more generally, ``quantum particles could not have trajectories''. Our model is an illustration to the contextual interpretation of quantum probabilities. \keywords{double slit interference, contextual probability} \AMSMSC{81-01}{81-04}
interfr1.tex
%       $Id: article.tex,v 1.16 2001/10/17 11:12:36 kisilv Exp $    
\begin{filecontents*}{interfr1.mu}
/* We start from the initialisation of variables and constants */

first_slit_is_open := TRUE:    // We could separately open the first
second_slit_is_open := TRUE:   // or the second slits

grid := 40:   // The number of positions for the particle gun per degree
coverage := 40:   // The average number of gun shots per position
num_orbits := 3: // Number of atom's orbits

/* Array of spin variables for each orbit */
spin := array(0..num_orbits, (i)=1 $ i=0..num_orbits):

/* The array stores results of our experiment (initialised with zeros)*/
result := array(-90..90, 1..2, (i,1)=0 $ i=-90..90,  (i,2)=0 $ i=-90..90):

/* We need few random generators for:                                 */
random_position := random(2*180*grid)/180/grid: // position of a particle gun
random_dr := random(2):  // one out of two possible directions of scattering

/* We will need a fast solution of equation %%$x=y+\sin y$~\eqref{eq:displacement}%% many times*/
interference_grid :=500:
small_step := 2*1.1:
inversion:=array(0..interference_grid, (i)=-1 $ i=0..interference_grid):
for i from 0 to 2*interference_grid*small_step+1 step 1 do
  x := i/interference_grid/small_step*PI:
  inversion[floor((x+sin(x))*interference_grid/2/PI)] := float(x/2/PI):
end_for:

/* Now we start our numerical experiment */

for i from 0 to 180*grid*coverage do
  position := random_position(): // We choose a position of the gun randomly
  direction:= random_dr(): // We choose randomly a direction of scattering
  /* There is nothing random beyond this point---everything is deterministic! */
  slit := floor(position+1):    // then find which slit is used
  position := abs(position-1):   // and change the position relative to the centre.

  if ((slit = 1 and first_slit_is_open)           // We let a particle go
      or (slit = 2 and second_slit_is_open)) then // through an open slit only
    x := position*num_orbits: // The position relative to orbits scales
    orbit := round(x):   // Which orbit is closest?

    /* This is the interference calculations; it is used only if  */
    if (spin[orbit] = slit) then   // the spin of the orbit equal to slit
      position:=frac((floor(x)+inversion[round(frac(x)*interference_grid)])/num_orbits):
      spin[orbit] := 3- slit:  // And interference flips the spin of the orbit!
    end_if:

    /* Scattering calculation by the formula %%$\phi=\pm\arcsin p$ \eqref{eq:scattering}%%. */
    angle := round((-1)^direction*arcsin(position)/PI*180): // calculate the angle
    result[angle,slit] := result[angle,slit]+1: // and add it to the result 

  end_if:
end_for:

/* And finally plot the result */

res1:=plot::Polygon(plot::Point(i,0,result[i,1]/grid) $ i=-90..90, Color=[0,1,0]):
res2:=plot::Polygon(plot::Point(i,10,result[i,2]/grid) $ i=-90..90, Color=[1,0,0]):
res3:=plot::Polygon(plot::Point(i,20,(result[i,1]+result[i,2])/grid) $ i=-90..90, 
                    Color=[0,0,1]):
plot(res1,res2,res3);

/* The end $*/
\end{filecontents*}

\begin{filecontents}{interfr1.mp}
%prologues:=2;
beginfig(1);
  u=0.5cm;
  def gray = withcolor .8white enddef;

  orbits=3;
  % Draw the source of partcles
  a0=0.5u;  a20=5; a2=a20*u;
  for i=-orbits+1 upto orbits-1:
    drawarrow (a0,a2+i*u)--(a0+u,a2+i*u);
  endfor;

  % Draw slits
  as=4.5u;
  drawdblarrow (as,a2-orbits*u)..(as,a2-0.1*u);
  drawdblarrow (as,a2+orbits*u)..(as,a2+0.1*u);

  % Draw the atom
  a1=8u;
  z0=(a1,a2);
  for i=1 upto orbits:
    draw (a1+i*u,a2)..(a1,a2+i*u)..(a1-i*u,a2)..(a1,a2-i*u)..cycle;
  endfor;
  draw (a1,a2-(orbits+0.5)*u)--(a1,a2+(orbits+0.5)*u) dashed evenly;
  pickup pencircle scaled 4pt;
  draw z0;

  % Draw the screen
  pickup pencircle scaled 4pt;
  r=7; a21=1;
  a22=sqrt(r*r-(a20-a21)*(a20-a21));
  draw (a1+a22*u,a21*u)..(a1+r*u,a2)..(a1+a22*u,2*a2-a21*u) gray;

  % Labels

  label.lft(btex Source of particles etex rotated 90, (0.5u,a2));
  label.lft(btex Slit 1 etex rotated 90, (as,a2-0.5*orbits*u));
  label.lft(btex Slit 2 etex rotated 90, (as,a2+0.5*orbits*u));
  label.bot(btex Atom etex, (a1,0.5u));
  label.rt(btex Screen etex rotated 90, (a1+(r+0.1)*u,a2));

endfig;
beginfig(2);
    u:=0.5cm;
    a20:=5; a2:=a20*u; 
    % Draw graph for interference displacements 
    a10:=1; steps=10; pi=3.1415926; s:=pi;
    draw (a10-s,a20-s)*u--(a10+s,a20-s)*u--(a10+s,a20+s)*u
          --(a10-s,a20+s)*u--(a10-s,a20-s)*u dashed evenly;
    draw (a10-s,a20-s)*u--(a10+s,a20+s)*u dashed evenly;
    draw (a10-s,a20)*u--(a10+s,a20)*u;
    draw (a10-s,a20-s)*u
       for i=-steps+1 upto steps: 
         ..(a10+s*i/steps+sind(i/steps*180-180),a20+s*i/steps)*u
       endfor;

    label.lft(btex (a) etex, (a10-s-0.2,a20+s-0.1)*u);
    label.bot(btex $y+\sin y  =x$ etex, (a10,a20-s-0.1)*u);
    label.lrt(btex $0$ etex, (a10,a20)*u);
    label.llft(btex $-\pi$ etex, (a10-s,a20)*u);
    label.lrt(btex $\pi$ etex, (a10+s,a20)*u);

    % Draw scattering paths of particles
    a1:=13u;r:=3u;
    z0=(a1,a2);
    draw (a1+r,a2)..(a1,a2+r)..(a1-r,a2)..(a1,a2-r)..cycle;
    b1=cosd(20)*r; b2=sind(20)*r;
    c1=cosd(50)*r; c2=sind(50)*r;
    z0=1/3[(a1-b1,a2+b2),z1];
    (a1+c1,a2+c2)=1/3[z2,z0];    
    draw (a1-r-2u,a2)--(a1+r+2u,a2) dashed withdots;
    drawarrow (a1-r-2u,a2+b2)--(a1-b1,a2+b2)--z1;
    drawdblarrow (a1-r-1.8u,a2)--(a1-r-1.8u,a2+b2);
    drawarrow (a1-r-2u,a2+c2)--(a1+c1,a2+c2)--z2;
    draw z0--(a1+c1,a2+c2) dashed evenly;
    z3=1/2[z0,(a1+r,a2)];
    z4=1/2[z0,(a1+b1,a2-b2)];
    draw z3{down}..z4;

    pickup pencircle scaled 3pt;
    draw z0;

    label.lft(btex (b) etex, (a1-r-2.5u,a2+r));
    label.bot(btex $\phi=\pm\arcsin p$ etex, (a1,(a20-s-0.1)*u));
    label.lft(btex $p$ etex, (a1-r-1.8u,a2+0.5*b2));
    label.lrt(btex $\phi$ etex, 3/4[z0,(a1+r,a2)]);
endfig;
end
\end{filecontents}
\documentclass[12pt]{article}

\usepackage[procnames,noindent]{lgrind}

\usepackage{graphicx}

\PassOptionsToPackage{pdfauthor={Vladimir Kisil},%
    pdftitle={},%
    pdfsubject={},%
%    pdfpagescrop={53 400 300 700}%
    backref=page,%
  pdfkeywords={symmetries}
}{hyperref}

\usepackage[marked,standardeqnum,printedin%,modest
]{myart2k}
\ppnum{LEEDS--MATH--PURE--2001--35}{\href{http://arXiv.org/abs/quant-ph/0111094}{quant-ph/0111094}}
{2001}

\font\eightrm=cmr8

%\bibliographystyle{unsrt} %for BibTeX - sorted numerical labels by
                          %order of first citation.

\makeatletter
  \let\old@cite=\@cite
\ifundefined{footheight}
  \let\footheight=\footskip
\fi
\makeatother
%\usepackage{sprocl}
\makeatletter
  \let\@cite=\old@cite
\makeatother

\input{mydef}
%\titleshort{}
%\authorshort{Vladimir V. Kisil}

\newcommand{\MuPAD}{\textsf{MuPAD}}
\renewcommand{\thethm}{\arabic{thm}}

\begin{document}

\title[A Path Through Two Slits]
{Two Slits Interference Is Compatible with\\ Particles' Trajectories}

\author[Vladimir V. Kisil]
{\href{http://maths.leeds.ac.uk/~kisilv/}{Vladimir V. Kisil}}

\address{%            
%Institute of Mathematics\\
%Economics and Mechanics\\
%Odessa State University\\
%ul. Petra Velikogo, 2\\
%Odessa-57, 270057, UKRAINE
School of Mathematics\\
University of Leeds\\
Leeds LS2\,9JT\\
UK
}

\email{\href{mailto:kisilv@maths.leeds.ac.uk}{kisilv@maths.leeds.ac.uk}}

\urladdr{\href{http://maths.leeds.ac.uk/~kisilv/}%
{http://maths.leeds.ac.uk/\~{}kisilv/}}

\maketitle

\begin{abstract}
  We propose a simple numerical experiment of
   two slits interference of particles. It disproves the popular belief
   that such an interference is incompatible with a knowledge which
   slit each particle came through or, more generally, ``quantum particles
   could not have trajectories''. Our model is an illustration to the
   contextual interpretation of quantum probabilities.
   \keywords{double slit interference, contextual probability}
   \AMSMSC{81-01}{81-04}
\end{abstract}

\epigraph{It is all mysterious. And the more you look at it the more
  mysterious it seems.}{Richard Feynman, \cite[\S~1--5]{FeynmanIII}.}{}

\section{Introduction}
\label{sec:introduction}

There is a recent interest in revisions of foundations of quantum
mechanics and its relation to classic one.  An example of such
fundamentals is the double slits interference of electrons, which usually
opens quantum mechanical textbooks~\cite{FeynmanIII}. A consideration
of that thought experiment (an actual interference could not be
realised on just two slits---one need a crystal lattice instead) is a
common conclusion that ``electrons could not move along definite
paths''. Besides that vague statement the experiment is used for s
derivation of the uncertainty principle and a justification of quantum
rules for adding probabilities.

Recently those rules were analysed from the viewpoint of the
\emph{contextual probability},
see~\cite{Ballentine86a,Khrennikov99a,Khrennikov01a}, papers in this
volume and references herein. It allows to explain a difference between
classic and quantum rules for addition of probabilities given by
identities:
\begin{eqnarray}
  \label{eq:classic-add}
  P_{12}& = & P_1+P_2\qquad\qquad\qquad\quad \quad\textrm{ and } \\
  \label{eq:quantum-add}
  P_{12}& = & P_1+P_2 +2\sqrt{P_1P_2}\cos\theta
\end{eqnarray} correspondingly. The contextual suggestion is to consider a
probability $P$ of an outcome $A$ depending from the context $S$ of
experiment, the contextual probability is denoted by $P(A|S)$. For
example, in the two slits experiment one may assume up to three different
contexts: $S_{12}=\,$both slits are open, $S_1=\,$only the first slit is open,
and $S_2=\,$only the second slit is open. Then in the above
formulas~\eqref{eq:classic-add}--\eqref{eq:quantum-add} probabilities
should be understood as follows:
\begin{eqnordisp}
  P_{12}=P(A,S_{12}), \qquad P_{1}=P(A,S_1), \qquad P_{2}=P(A,S_2).
\end{eqnordisp} 

In the classical situation the context $S_{12}$ is the disjoint union
of two contexts $S_1\cup S_2$ and classic addition
formula~\eqref{eq:classic-add} could be rewritten as
$P(A|S_{12})=P(A|S_{1})+P(A|S_2)$.  In the quantum case the third term
in the formula~\eqref{eq:quantum-add} reflects the change of
context (see~\cite{Khrennikov01a} and
Section~\ref{sec:do-particles-have} for details) between different
experimental settings, e.g. if different slits 
are open. This allows to wipe out any mystery from quantum
formula~\eqref{eq:quantum-add} on the theoretical level.

On the other hand the generality of that construction obscures its
connection with concrete experiments. It is desirable from
psychological and pedagogical point of view to have a toy model based
on contextual principles which could imitate double slit interference,
for example. We describe such a model in this paper and arrive to the
conclusion that \emph{interference of particles is perfectly
  compatible with the precise knowledge of their trajectories}. It
could appear to be disputing with the dominant Copenhagen
Interpretation of quantum mechanics. But we will argue in
Remark~\ref{rem:copenhagen} that in facts our model is even more in
line with the Copenhagen philosophy than those typical explanations
referring to mysterious particles-without-a-path.

\label{sec:scheme-experiment}
\begin{figure}[thb]
  \begin{center}
    \includegraphics[]{interfr1.1}
    \caption{Graphical representation of the model.}
    \label{fig:scheme}
  \end{center}
\end{figure}

\section{Scheme of the Experiment}
Let us describe the scheme of our experiment. There is no claim that
common words like ``atom'', ``electrons'', and ``spin'' in the text bellow
exactly correspond to a physical reality, merely they are chosen for
illustration purposes only.  The scheme is graphically represented on
the Figure~\ref{fig:scheme}. There is a source of particles, which
emits them in the horizontal direction with a uniform distribution over
a vertical segment.  On their way particles meet two
slits, which are named ``1'' and ``2'' and could be open or
closed in an arbitrary combination. A particle coming to a closed
slit disappears without a trace. 

Particles coming through an open slit will interact with an atom. The
atom have equidistant orbits. The \emph{principal
ingredient} of the model is that each orbit have a spin, which
could take exactly two values named again as ``1'' and
``2''. An interference of the particle is presented by a
displacement in the vertical direction without an alteration of its
velocity.  It could occur \emph{if and only if} the name of spin for
the nearest orbit is the same as the name of slit the particle
came through. If an interference did happen than the spin of that orbit
\emph{will be changed} to the opposite. This is exactly the place their our
model shows a \emph{contextual behaviour}. Indeed if only one slit is
open then each orbit could produce at most one interferential
displacement. However if particles come through both open slits then a
rich interference appears due to randomness in their emission.

\begin{figure}[thb]
  \begin{center}
    \includegraphics[]{interfr1.2}
    \caption{The rules for interference (a) and scattering (b).}
    \label{fig:rules}
  \end{center}
\end{figure}

Besides an interference all particles also experience a scattering on
the atom as a whole. This manifests itself in changes of their
directions of motion and is observed on a semicircular screen behind
the atom. We chose the rules for interference and scattering to be
simple yet reproducing the popular pictures from the quantum
mechanical textbooks~\cite[Fig.~1--3]{FeynmanIII},
\cite[Fig.~5.2]{Ryder96}. 
\begin{conv}
\label{conv:assumptions}
Our assumptions are as follows (see
Figure~\ref{fig:rules}):
\begin{enumerate}
\item[(a)] An \emph{interference displacement} (if occurs at all) is directed
  out of the nearest orbit to the particle when it crossed the dashed
  line on the Figure~\ref{fig:scheme}. The amount of displacement comes
  from the formula:
  \begin{eqnordisp}[eq:displacement]
    y+\sin y =x,
  \end{eqnordisp} where $x$ and $y$ are distances to the nearest orbit
  before and after interference correspondingly both measured in units
  equal $\pi*$(the distance between orbits). See
  Figure~\ref{fig:interference}(a) for the graph of that function.
\item[(b)] A particle \emph{scatters} with probability $1/2$ either from the
  front or the back semicircular boundary of the atom. The direction of
  scattering forms a sharp angle with the original horizontal direction
  and belong to the radius of circle coming through the point of
  their contact, see Figure~\ref{fig:rules}(b). Analytical expression
  of that configuration is:
  \begin{eqnordisp}[eq:scattering]
    \phi=\pm\arcsin p.
  \end{eqnordisp}
\end{enumerate}
\end{conv}

After scattering the particle is registered by a semicircular screen
which is sufficiently far away from the atom to neglect the atom size.

\begin{rem}
  Note that the only non-deterministic elements enter the scheme are: (i)
  the initial choice of the position of the particle, and (ii) the
  choice of $+$ or $-$ in the formula~\eqref{eq:scattering}. Those two
  choices are independent each other and we could associate them both
  with the process of the particle emission. Apart from them the rest
  of algorithm of interference-scattering and registration of particle
  is completely deterministic. We also able to keep information
  about the slit the particle come through at the moment of its
  registration on the screen.
\end{rem}

It is easy to realise the above scheme on many computer languages. We
chose a programming of the {\MuPAD}~\cite{MuPAD} open computer algebra
system because it provides a good graphical presentation of the
obtained results and has a free license for a usage within
academia. Therefore to reproduce and verify our results anyone does
not have to own an expensive licence for the commercial software. The
complete listing of the used program together with some comments is
given in Appendix~\ref{se:appendix}.

\begin{figure}[htb]
  \begin{center}
    \includegraphics[scale=0.5]{interfrb.epsi}
    \caption{An absence of interference if only one slit is open.}
    \label{fig:no-interference}
  \end{center}
\end{figure}


\section{Obtained Results}
\label{sec:obtained-results}

The described model could be tested in two essentially different
contexts: with just one skit or both slits open. The results of these two
tests are shown on Figures~\ref{fig:no-interference}
and~\ref{fig:interference} and will be explained now. The output consists
of the three graphs, which show the number of particles for each degree
on the screen coming through first, second, and both slits
correspondingly. Because these graphs could be identical or very close
we put them in perspective on three parallel planes and coded them by
colours. The nearest (green) graph shows particles coming through the
first slit, the second (red) is drawn for the second slit, and the
third (blue) graph is the sum of the both previous graphs.

If just the first slit is open (see Figure~\ref{fig:no-interference})
then the first graph is essentially affected only by scattering and
did not shows any visible signs of interference; the second graph is
identically zero; and therefore the third graph is equal to the first
one. This is exactly that we know from textbooks~\cite[part (b) of
Figs.~1--1, 1--2, 1--3]{FeynmanIII} for a behaviour common to both
quantum and classic mechanics. The included program produces that
output if experimenter sets in its beginning the following values of
variables:
\begin{figure}[htb]
  \begin{center}
    \includegraphics[scale=0.5]{interfra.epsi}
    \caption{The presence of interference if both slits are open.}
    \label{fig:interference}
  \end{center}
\end{figure}
% This document was generated automagically by lgrind.  DO NOT EDIT.
\LGinlinefalse\LGbegin\lgrinde
\L{\LB{\V{first\_slit\_is\_open}_:=_\K{TRUE}:}\Tab{31}{\C{}\CE{}}}
\L{\LB{\V{second\_slit\_is\_open}_:=_\K{FALSE}:}\Tab{31}{\C{}\CE{}}}
\endlgrinde\LGend

On the other hand, if both slits are open and particles will come
through them in a random order then the context of the experiment will
be different and interference will affect particles coming through
both slits. Such an outcome is shown on Figure~\ref{fig:interference}:
the both 1st (green) and 2nd (red) graphs are almost identical and the
total sum is like a double of any of them. This is in a good agreement
with the quantum interference of particle illustrated for example
in~\cite[Fig.~1--3]{FeynmanIII}, \cite[Fig.~5.2]{Ryder96}.  The
included program produces that output if values of both variables are
set to be ``TRUE'' (the actual values in the listing given in
Appendix~\ref{se:appendix}):
% This document was generated automagically by lgrind.  DO NOT EDIT.
\LGinlinefalse\LGbegin\lgrinde
\L{\LB{\V{first\_slit\_is\_open}_:=_\K{TRUE}:}\Tab{31}{\C{}\CE{}}}
\L{\LB{\V{second\_slit\_is\_open}_:=_\K{TRUE}:}\Tab{31}{\C{}\CE{}}}
\endlgrinde\LGend

Note that if an experimenter changes the context again and will firstly
send a half of particles (still randomly distributed) only
to the first slit and afterwards another half to the second slit then
interference will not occur again---just like in the case of one open
slit. This is similar to the thought experiment with ``watching
electron'', which destroys the interference
completely~\cite[\S~1--6]{FeynmanIII}. In that way our model reproduces
one more feature associated with mysterious quantum behaviour.


\section{Do Particles Have Trajectories?}
\label{sec:do-particles-have}

\renewcommand{\thethm}{\Alph{thm}}

Let us recall the ``Proposition A'' from  of the
popular textbook~\cite[\S~1--5]{FeynmanIII}:

\begin{prop}
  \label{prop:A}
\textup{Each electron \emph{either} goes through slit $1$ \emph{or} it
  goes through slit $2$.
}\end{prop}

Unfortunately the another
important assumption was not highlighted by the authors
of~\cite{FeynmanIII} so explicitly:

\begin{prop}
  \label{prop:B}
\textup{The distribution of electrons coming through slit $1$ is the same
  \emph{regardless} either slit $2$ is open or not.}
\end{prop}

This two assumptions together imply the addition rule
$P_{12}=P_1+P_2$~\eqref{eq:classic-add}, which is wrong in the quantum
case: we could not get the third joint (blue) distribution on the
Figure~\ref{fig:interference} adding together two identical (green)
distributions from the first slit on
Figure~\ref{fig:no-interference}. Because Proposition~\ref{prop:B}
was hidden in~\cite{FeynmanIII} the only responsible for that failure
seems to be Proposition~\ref{prop:A}:

\begin{quote}
  \ldots undoubtedly we should conclude that \emph{Proposition~\ref{prop:A}
    is false}. It is \emph{not} true that the electrons go \emph{either}
  through hole $1$ or $2$. (\cite[\S~1--5]{FeynmanIII}, emphasis of
  the original.)
\end{quote}

But as we saw in the previous section Proposition~\ref{prop:A} is
perfectly compatible with the interference of electron. Instead we
would blame Proposition~\ref{prop:B} for the failure of an accurate
prediction. This proposition is explicitly \emph{non}-contextual, it
supposes that an outcome of a particular event is independent from the
whole context. In contextual framework we drop the
Proposition~\ref{prop:B} and get in the terms of contextual probabilities:
\begin{eqnordisp}
  P(E_1|S_1)\neq P(E_1|S_{12}), \qquad \textrm{ and } \qquad
  P(E_2|S_2)\neq P(E_2|S_{12}),
\end{eqnordisp}
where $E_i$ are the events that an electron come through slit $i$,
$i=1, 2$, and $S_1$, $S_2$, and $S_{12}$ are contexts where only slit 1,
only slit 2 is, and both slits are open. Then instead the definitely
wrong statement $P(E_{12}|S_{12})= P(E_{1}|S_{1}) + P(E_{2}|S_{2})$
the Figure~\ref{fig:interference} represent a true contextual addition
of probabilities:
\begin{eqnordisp}[eq:quant-add-2]
  P(E_{12}|S_{12})= P(E_{1}|S_{12}) + P(E_{2}|S_{12}).
\end{eqnordisp}
\renewcommand{\thethm}{\arabic{thm}}
\setcounter{thm}{2}
\begin{rem}
  \label{rem:copenhagen}
  It is interesting to note that our model for interference should
  satisfy a most orthodox follower of the Copenhagen
  Interpretation. Indeed in that interpretation the measuring
  apparatus is a part of the measured system and the \emph{uncertainty
    principle} is a consequence of impossibility to make its
  backreaction to the system arbitrary small. From that point of view
  the two slits are also parts of the system and their influence
  on an outcome of experiment could not be neglected. In our model slits
  preserve some information about electrons coming through them. This
  provides a device for an indirect interaction between electrons even
  if any direct interaction is excluded.
\end{rem}

It could be interesting to construct more advanced and realistic
models based on the same principle as our one. For example, we can study
an interference on a regular lattice of simple atoms without a
complicated internal structure. In that way it is reasonably to expect
that a collective behaviour of atoms in the lattice will allow to get
consequences similar to our Convention~\ref{conv:assumptions} starting
from simpler and less artificial assumptions.

Another interesting direction for a research is as follows. We
confirmed a correct form~\eqref{eq:quant-add-2} for addition of
quantum probabilities which is similar to the classic
one~\eqref{eq:classic-add}, both of them are just different
realisations the same general contextual formula. Similarly it was
proposed recently~\cite{Kisil00a} to obtain both quantum and classic
brackets from the same common source---the $p$-mechanical brackets. It
may be possible and promising to combine both approaches (contextual
probability and $p$-mechanical brackets) in order to wipe out an
unnatural opposition of quantum and classic worlds.

\setcounter{section}{0}

\renewcommand{\thesection}{\Alph{section}}
\section{Appendix: the Listing of the {\MuPAD} Program} 
\label{se:appendix}

Here is the listing of the program for the {\MuPAD}~\cite{MuPAD}
software to create Figures~\ref{fig:no-interference}
and~\ref{fig:interference}. The listing is typeset with the help of
the \emph{free software} \texttt{Lgrind}~\cite{Lgrind}.

The code is short but you do not need even to
retype it to use for our own numerical experiments. If you get the
source of this article~\cite{Kisil01c} form the
\href{http://arxiv.org}{\texttt{arhiv.org}} and type the command
\begin{eqnordisp}
  \texttt{latex 0111094.tex}
\end{eqnordisp} then the file \texttt{interfr1.mu}, which contains
this code, will be created in
current directory. 

Because it could be regarded as a piece of software
I have to include a license to conform with the present legal
climate. I choose the \emph{GNU General Public License}~\cite{GNUGPL},
please read it before use the code.

\lgrindfile{interfr1.lt}

%\renewcommand{\thesection}{}
\section*{Acknowledgments}
\label{sec:acknowledgments}
I am grateful to SciFace~\circledR\ for granting me and other
scientists a free license to use \MuPAD~\cite{MuPAD}. I acknowledge
also the usage of free software \textsf{Lgrind}~\cite{Lgrind} and
\textsf{MetaPost}~\cite{MetaPost}. And should we say once more our thanks
to Donald E.~Knuth for the wonderful \TeX?

\small
%\bibliographystyle{plain}
%\bibliography{abbrevmr,akisil,aphysics,analyse,arare}
\newcommand{\noopsort}[1]{} \newcommand{\printfirst}[2]{#1}
  \newcommand{\singleletter}[1]{#1} \newcommand{\switchargs}[2]{#2#1}
  \newcommand{\irm}{\textup{I}} \newcommand{\iirm}{\textup{II}}
  \newcommand{\vrm}{\textup{V}}
  \providecommand{\cprime}{'}\providecommand{\arXiv}[1]{\eprint{http://arXiv.o%
rg/abs/#1}{arXiv: #1}}
\begin{thebibliography}{10}

\bibitem{Lgrind}
\textsf{Lgrind}: Produce beautiful listings of source code with {\LaTeX}.
\newblock
  \url{http://www.ctan.org/tex-archive/help/Catalogue/entries/lgrind.html}.

\bibitem{Ballentine86a}
L.~E. Ballentine.
\newblock Probability theory in quantum mechanics.
\newblock {\em Amer. J. Phys.}, 54(10):883--889, 1986.
\newblock \MR{87m:81050}.

\bibitem{MuPAD}
SciFace \circledR.
\newblock \textsf{MuPAD}: the open computer algebra system.
\newblock \url{http://www.mupad.de/}.

\bibitem{FeynmanIII}
Richard~P. Feynman, Robert~B. Leighton, and Matthew Sands.
\newblock {\em The {F}eynman lectures on physics. {V}ol. 3: {Q}uantum
  mechanics}.
\newblock Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965.
\newblock \MR{MR35:3944}.

\bibitem{GNUGPL}
Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA
  02111-1307, USA.
\newblock {\em GNU General Public License}, second edition, 1991.
\newblock \url{http://www.gnu.org/licenses/gpl.html}.

\bibitem{MetaPost}
John~D. Hobby.
\newblock \textsf{MetaPost}: A \textsf{MetaFont} like system with postscript
  output.
\newblock \url{http://www.tug.org/metapost.html}.

\bibitem{Khrennikov99a}
Andrei Khrennikov.
\newblock {\em Interpretations of probability}.
\newblock VSP, Utrecht, 1999.
\newblock \MR{MR2001j:81004}.

\bibitem{Khrennikov01a}
Andrei Khrennikov.
\newblock `{Quantum} probabilities' as context depending probabilities.
\newblock 2001.
\newblock \arXiv{quant-ph/0106073}.

\bibitem{Kisil01c}
Vladimir~V. Kisil.
\newblock Two slits interference is compatible with particles' trajectories.
\newblock 2001.
\newblock \arXiv{quant-ph/0111094}.

\bibitem{Kisil00a}
Vladimir~V. Kisil.
\newblock Quantum and classic brackets.
\newblock {\em Int. J. Theor. Phys.}, 41(1):63--77, 2002.
\newblock \arXiv{math-ph/0007030}.

\bibitem{Ryder96}
Lewis~H. Ryder.
\newblock {\em Quantum Field Theory}.
\newblock Cambridge University Press, Cambridge, 2nd edition,
  {\noopsort{1985}}1996.

\end{thebibliography}

\end{document}

mydef.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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quant-ph0111094.extracted_bib
\begin{thebibliography}{10}

\bibitem{Lgrind}
\textsf{Lgrind}: Produce beautiful listings of source code with {\LaTeX}.
\newblock
  \url{http://www.ctan.org/tex-archive/help/Catalogue/entries/lgrind.html}.

\bibitem{Ballentine86a}
L.~E. Ballentine.
\newblock Probability theory in quantum mechanics.
\newblock {\em Amer. J. Phys.}, 54(10):883--889, 1986.
\newblock \MR{87m:81050}.

\bibitem{MuPAD}
SciFace \circledR.
\newblock \textsf{MuPAD}: the open computer algebra system.
\newblock \url{http://www.mupad.de/}.

\bibitem{FeynmanIII}
Richard~P. Feynman, Robert~B. Leighton, and Matthew Sands.
\newblock {\em The {F}eynman lectures on physics. {V}ol. 3: {Q}uantum
  mechanics}.
\newblock Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965.
\newblock \MR{MR35:3944}.

\bibitem{GNUGPL}
Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA
  02111-1307, USA.
\newblock {\em GNU General Public License}, second edition, 1991.
\newblock \url{http://www.gnu.org/licenses/gpl.html}.

\bibitem{MetaPost}
John~D. Hobby.
\newblock \textsf{MetaPost}: A \textsf{MetaFont} like system with postscript
  output.
\newblock \url{http://www.tug.org/metapost.html}.

\bibitem{Khrennikov99a}
Andrei Khrennikov.
\newblock {\em Interpretations of probability}.
\newblock VSP, Utrecht, 1999.
\newblock \MR{MR2001j:81004}.

\bibitem{Khrennikov01a}
Andrei Khrennikov.
\newblock `{Quantum} probabilities' as context depending probabilities.
\newblock 2001.
\newblock \arXiv{quant-ph/0106073}.

\bibitem{Kisil01c}
Vladimir~V. Kisil.
\newblock Two slits interference is compatible with particles' trajectories.
\newblock 2001.
\newblock \arXiv{quant-ph/0111094}.

\bibitem{Kisil00a}
Vladimir~V. Kisil.
\newblock Quantum and classic brackets.
\newblock {\em Int. J. Theor. Phys.}, 41(1):63--77, 2002.
\newblock \arXiv{math-ph/0007030}.

\bibitem{Ryder96}
Lewis~H. Ryder.
\newblock {\em Quantum Field Theory}.
\newblock Cambridge University Press, Cambridge, 2nd edition,
  {\noopsort{1985}}1996.

\end{thebibliography}