1. Quantized-Energy Equation for N-Level Atom in the Probability Representation of Quantum Mechanics
- Author
-
Vladimir I. Man'ko, Margarita A. Man’ko, and Vladimir N. Chernega
- Subjects
Physics ,Hamiltonian matrix ,02 engineering and technology ,01 natural sciences ,Hermitian matrix ,Atomic and Molecular Physics, and Optics ,Schrödinger equation ,010309 optics ,symbols.namesake ,Matrix (mathematics) ,020210 optoelectronics & photonics ,Quantum state ,Quantum mechanics ,0103 physical sciences ,Homogeneous space ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Hamiltonian (quantum mechanics) ,Engineering (miscellaneous) ,Eigenvalues and eigenvectors - Abstract
For Hermitian and non-Hermitian Hamiltonian matrices H, we present the Schr¨odinger equation for qudit (spin-j system, N-level atom) with the state vector |ψ〉 in a new form of the linear eigenvalue equation for the matrix ℋ = (H ⊗ 1N) and the probability eigenvector |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.
- Published
- 2020