MATLAB: How to normalize a function solved numerically

normalizepdfquantumwavefunction

I'm trying to make the pdf of a quantum harmonic oscillator, once the equation is solved numerically, the pdf area should be 1 but instead it is 2.2604e-28. Thanks for advanced
function test(n,y0,dy0)
syms y(x)
m=9.1093821545*1e-31; k=m*(4.57*1e14)^2;  w=sqrt(k/m); hb=(6.6260695729*1e-34)/(2*pi);  En=hb*w*(n+.5); c1=-2*m/(hb^2); c_i0=sqrt(hb/(m*w));
[V] = odeToVectorField(diff(y,x,2) == c1*(En-.5*k*x.^2).*y);
x1=0;
x2=2e-9;
M = matlabFunction(V,'vars',{'x','Y'});
[p,q]= ode45(M,[x1 x2],[y0 dy0]);
dens=(q(:,1).*q(:,1));
% A plotear
plot([-p p],En+dens,'-k') %psi quadrat
%plot(-p,En+q(:,1),'-k') psi
hold on
%plot(p,En+q(:,1),'-k') psi
potencial=0.5*k*p.^2;
plot([-p p],potencial,'-r') %potencial
plot([-p p], ones(1,length(p))*En,'-b') %En
% natural frequency of the oscillator w = 4.57e14 Hz
hold off
grid on
u=zeros(1,length(dens));
2*trapz(p,dens) %area of the pdf
end

Best Answer

Hi david,
A normalization integral that small shows that the scaling could be improved upon, but as a workaround is this what you are looking for?
densnorm = dens/(2*trapz(p,dens));
2*trapz(p,densnorm)                     %area of the pdf
ans = 1
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