This is the first book ever written that presents non-Hermitian quantum mechanics (NHQM) as an alternative to the standard (Hermitian) formalism of quantum mechanics. Previous knowledge of the basic principles of quantum mechanics and its standard formalism is required.

The standard formalism is based on the requirement that all observable properties of a dynamic nature are associated with the real eigenvalues of a special class of operators, called Hermitian operators. All textbooks use Hermitian Hamiltonians in order to ensure conservation of the number of particles. See, for example, the monumental book of Dirac on The Principles of Quantum Mechanics.

The motivation for the derivation of the NHQM formalism is twofold.

The first is to be able to address questions that can be answered only within this formalism. For example:

• –in optics, where complex index of refraction are used;

• – in quantum field theory, where the parity–time (PT) symmetry properties of the Hamiltonian are investigated;

• – in cases where the language of quantum mechanics is used, even though the problems being addressed are within classical statistical mechanics or diffusion in biological systems;

• – in cases where complex potentials are introduced far away from the interaction region of the particles. This approach simplifies the numerical calculations and avoids artificial interference effects caused by reflection of the propagated wave packets from the edge of the grid.

The second is the desire to tackle problems that can, in principle, also be solved within the conventional Hermitian framework, but only with extreme difficulty, whereas the NHQM formalism enables a much simpler and more elegant solution.