Calc eigenvalues, eigenvectors, and evolution of Hamiltonian

% Calc eigenvalues and eigenvectors, evolution of Hamiltonian
% define symbols
syms Omega1 Omega2 t

% Hamiltonian in matrix form

H = zeros(4,'sym');
H(1,1) = Omega1;
H(2,3) = Omega1;
H(3,2) = Omega1;
H(3,4) = Omega2;
H(4,3) = Omega2;

% 
disp('Hamiltonian in matrix form');
disp(H)

% solve eigenvalues and eigenstates
[V,D] = eig(H);

% 
for idx = 1:length(diag(D))
    disp("Eigenvalue:");
    disp(D(idx, idx));
    disp("Eigenstate:")
    disp(V(:,idx));
end

% evolution
% initial state
psi0 = [0;0;1;0];

% state at time t
psi_t = expm(-1i*H*t)*psi0;

% 
disp("State at time t")
disp(psi_t)