QUANTIZED-ENERGY EQUATION FOR N-LEVEL ATOM IN THE PROBABILITY REPRESENTATION OF QUANTUM MECHANICS

For Hermitian and non-Hermitian Hamiltonian matrices H, we present the Schr center dot odinger equation for qudit (spin-j system, N-level atom) with the state vector vertical bar psi > in a new form of the linear eigenvalue equation for the matrix Script capital H = (H circle times 1(N)) and the probability eigenvector vertical bar p > identified with quantum states in the probability representation of quantum mechanics. We discuss the possibility to experimentally detect the difference between the system states described by the solutions, corresponding to the Schrodinger equation with Hermitian and non-Hermitian Hamiltonians, by measuring the probabilities of artificial spin-1/2 projections m = +/- 1/2, sets of which are identified with qudit states. We show that different symmetries of systems, including PT-symmetry and broken PT-symmetry, are determined by a set of N complex eigenvalues of the Hamiltonian matrix H.