How can I make probabilty calculation with the eigenwavefunction gathered by eigs() function in MATLAB?

  h = 0.0454973;
    n = 10;
    V = 545.0362942355468 - 1998.817146282548*x + 2980.608037604876*x^2 - 2314.110473743709*x^3 + 989.3591697175247*x^4 - 221.48586861701781*x^5 + 20.33254549765507*x^6;
    %% Eigenvalue computation:
    [xmin, xmax] = domain(V);                          % domain of problem
    % Create a CHEBOP with Dirichlet BCs
    L = chebop([xmin, xmax]);
    L.lbc = 0; L.rbc = 0;
    L.op = @(x,u) -h^2*diff(u,2) + V.*u;               % Schroedinger operator
    [U, D] = eigs(L, n, 'sr');                         % compute evals/efuns
    d = diag(D);                                       % vector of evals
    [d, ii] = sort(d);                                 % sort them
    U;
    U = U(:,ii);
    plot(U(:,8))
    

Here is the code that I am using for finding eigenfunction and eigenvalue. I can manage to graph the eigenfunction by using the last code line. However, I don't know how to convert the U into something that can be integrable to calculate probabilty of corresponding eigenfunction. Can someone help me?

[David Goodmanson]

Hi semih,

do you mean probabilty of the corresponding eigenvalue?

You have a set of orthonormal eigenfunctions U. For probabilities you would appear to need a wave function Y from somewhere else so that you can calculate overlaps with the eigenvectors. So if Y is a chebfun, then

abs(sum(conj(Y).*U(:,n)))^2

is the probability if being in quantum state n.