Non-Hermitian Quantum Mechanics

Описание

2 Different formulations of quantum mechanics
energy term in quantum mechanics. The propagation axis Z serves as time in quantum
mechanics. The imaginary part of the complex index of refraction indicates the amount
of absorption loss of the propagated light when it passes through the wave guide. The
complex potential renders the Hamiltonian non-Hermitian and therefore such systems
can be only studied within the framework of the non-Hermitian formalism discussed
here.
(3) Simplification of numerical propagation of wave packets in time.
The propagation of matter waves by the Schrodinger equation and the propagation of ¨
light in waveguides in the paraxial approximation are associated with two different
physical phenomena but they obey the same mathematical equation. The numerical
propagation of wave packets is much more simple when taken within the framework
of the non-Hermitian formalism of quantum mechanics rather than in the standard
(Hermitian) formalism. This is due to the inclusion of a reflection-free complex absorbing potential (RF-CAP) in the Hamiltonian which attains non-zero values only in the
non-interacting region in the coordinate space where the physical potentials vanish.
This approach enables one to avoid the artificial reflections from the edge of the numerical grid when a finite number of grid points (or a finite number of basis functions)
are used to describe a propagated wave-packet. By adding the complex non-Hermitian
potential to the Hamiltonian one can carry out numerical calculations using a finite
number of grid points or a finite number of basis functions (after all, our computers
are finite) and have a numerically exact propagated wave-packet in the region where
the RF-CAPs vanish. By numerically exact, we mean that the wave-packet which is
obtained by introducing a RF-CAP into the calculations, is exactly as the wave-packet
which would be calculated (if it were possible) by computers which are infinite (i.e.
infinite capacity, memory and computational power). The derivation of RF-CAPs by
carrying out a smooth exterior scaling transformation of the spatial coordinates is
presented in Chapter 5.
(4) Another numerical example for the advantage of the use of the non-Hermitian formalism
over the standard one is when the dynamics of a given system can be described by a
small number of resonance states.
Often it is enough to describe the dynamical process and to calculate all possible measurable quantities just from a single resonance state. See, for example, in Chapter 8, the
calculations of the high-harmonic-generation (HHG) spectra (i.e. the emitted high frequency radiation) and the calculations of the above-threshold-ionization (ATI) spectra
from a single quasi-energy photo-induced resonance state when atoms or molecules
interact with strong laser fields.
(5) Within the framework of the non-Hermitian formalism of quantum mechanics, one can
get a better understanding of different methods and theories developed in the standard
(Hermitian) formalism of quantum mechanics.
The first example is the Rayleigh–Schrodinger perturbation theory where the full ¨
Hamiltonian is defined as Hˆ = Hˆ0 + λVˆ , where λ is the perturbation strength parameter. The interesting non-trivial cases occur when Hˆ0 and Vˆ do not commute. The radius

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