#not OK to post homework
#Nick Belov, Feb 9nd 2025, Assignment 4

read "C5.txt":
#PM(): the three famous Pauli matrices sigma_x,sigma_y, sigma_z
#RHM(n,K): a random n by n Hermitian matrix with entriries from -K to K
#EV(L): inputs a Hermitian matrix (say n by n) and outputs the LIST of eigenvalues
#EVC1(L,g): inputs a matrix L and an eigenvalue g finds the NORMALIZED eigenvector
#EVC(L): inputs a Hermitian matrix and outputs the pair ListOfEigenvalues, ListOfEigenvectors
#ProbDist(L,A): Given an obervalble represented by the Hermitian matrix L and a state vector A outputs first all the eigenvectors (pure states) and the prob. of winding in the respective pure states followed by the expected value of A (i.e. if you make the observation L and the state is A the average in the long run


#RandomState(n,K)
#that inputs a pos. integer n and a pos. integer K and outputs a random state vector with n complex components whose real and imaginary parts are from -K to K, then normalize it.

RandomState:=proc(n,K)
    local i,res,ra,m;
    ra:=rand(-K..K);
    res:=[seq(ra()+ra()*I,i=1..n)];
    m:=add(res[i]^2,i=1..n);
    m:=sqrt(m);
    res:=[seq(res[i]/m,i=1..n)];
    return res;
end:

#inputs a NORMALIZED vector n=[nx,ny,nz]
#that constructs the 2 by 2 matrix. at the bottm of p. 349 of the Susskind-Frieman Quantum Mechanics book book.
GeneralPauli := proc(v)
    local nx, ny, nz;
    nx := v[1];
    ny := v[2];
    nz := v[3];
    # Return the 2x2 matrix as a list of lists:
    return [ [ (nz), nx - I*ny ],
             [ nx + I*ny, -nz ] ];
end:
#eigen values seem to be 1 and -1 !