Free fallers

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In a recent post, we have discussed how external forces are described alongside with quantum forces in the local-realistic model of quantum mechanics that we defend in this blog.

In order to simulate force field scenarios, we use the accelerated code presented in this post (the one simulating the expected motion) and we modify it as shown below. In this post we discuss a constant force scenario (suggestively denoted here as "the free faller"), characterized by a constant parameter f.



%%% Simulate an ensemble of Np particles emitted at intervals Ti 
%%% from either of Ns distinct sources xs(1,...,Ns) having
%%% probability Ps(1,...,Ns), in the presence of a constant force %%% field f. Evaluate the frequency of arrivals at a 'screen' 
%%% after Nt iterations.
%%%
%%% Parameters: Np,Ns,Nt,xs,Ps,f.
%%%
%%% Evaluate the number of possible bosons for this scenario 

B = Ns*(Ns-1)/2; 

%%% Evaluate the probability of each Quantum Reset 

iB = 0; 
for i = 1:Ns-1 
   for j = i+1:Ns 
      iB = iB+1; 
      Pb(iB) = 2*Ps(i)*Ps(j); 
      dd(iB) = abs(xs(i)-xs(j)); 
   end
end 

%%% Start simulating 

for i = 1:Np 
   
   %%% Attribute a random source momentum 

   v0 = -1+2*rand; 
   q = 0; 
  
   %%% Choose the source node according to probability vector 
  
   x0 = xs(find(rand<cumsum(Ps),1,'first'));

   %%% Position the particle to be emitted at the source node x0

   x = x0;

   %%% Simulate Nt time steps 

    vF = 0;

   for t = 1:Nt

      %%% Evaluate momentum and energy propensity

      w = sqrt(Pb/2).*sin(pi*dd*q-pi*ee)./(pi*dd);
      wsum = sum(w);
      vQ = v0-2*wsum;
      vF = vF+f;
      V = vQ+vF;
      e = (1+V^2)/2;

      %%% Average motion only

       q = q*(lag-1)/lag+(vQ)*(1/lag);
      v = V;

      %%% Increase the particle's position x according to momentum

      x = x+v;

   end

   %%% Record arrival node for further analysis

   X(i) = x;

end

 Note that the momentum propensity, as calculated by this code, can exceed the speed of light limits and thus needs a correction according to special relativity that will be discussed in a future post. 

 Some example results are shown below. The first figure is obtained for one single source (no quantum forces, only external forces), with Nt = 200, Np = 5000, and f = 0.002. The frequency of arrivals tends to be constant as expected around the average-motion position of fNt2/2 (that is, 40 lattice nodes).

 The second figure is for a Gaussian wave preparation (inducing quantum forces, see this post) with Ns = 101 (variance of 12.5) vm = 0.1. Clearly, the obtained frequency of arrivals matches the theoretical distribution (calculated using the propagator for a free faller as explained in this paper).




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