Schrödinger Equation for Energy Levels as a Linear Equation for Probability Distributions Identified with Quantum States

In this paper, we show that the energy spectrum of a qudit system (spin- j , N -level atom) with Hamiltonian N × N matrix H is determined by the equation for eigenvalues of the N 2 × N 2 matrix ℋ and its eigenvectors | p ⟩ with components, which are probability distributions identified with quantum states in the probability representation of quantum mechanics. We derive this equation based on the conventional Schrödinger equation for the state vector | ψ ⟩. We discuss the example of qubit in detail and study new entropic characteristics of the energy levels.