of the computer revolution, the field of quantum chemistry prospered, and it is
viewed today as a fundamental area becoming more or less synonymous with the
fields of theoretical chemistry and chemical physics.
Despite these successes there remain many inconsistencies and conundrums
plaguing the fundamental physical formulation. Primas [17] in his thought provoking evaluation of the myth of universal laws brings up ten profound problems
connected with the incapabilities of traditional quantum chemistry promoting the
so-called structural approach. Although many of the puzzles have a metaphysical
flavour, they aim at the deeper meaning of chemistry and a worldview, unus
mundus, which should also incorporate the dimensions of a semiotic analysis. The
present author has reformulated some of the most serious ones in Brändas [6], in
particular the well-known issues of the uni-directedness of time and the associated
irreversibility of the macro-world (e.g. molecular chirality), as well as the elimination of the law of causality in the microscopic domain. Closely related is also the
inquiry of the absolute nature of the second law, Sklar [18]. These problems have
been reconsidered in the light of recent non-Hermitian quantum mechanics, see
Nicolaides and Brändas [19] and Moiseyev [20] suggesting potential solutions to
the paradox, see also Ref. [6].
Much could be said about the necessity to realize and accomplish a quantum
mechanical extension,
4 however it should be enough to briefly explain what we
mean when saying that the Schrödinger equation is extended or continued beyond
Hermitian territory. In the portrayal we refer to the popular analytic dilation technique for a simple case of a proper potential, V, exhibiting a point spectrum below a
positive continuum.
The method simply amounts to multiplying the space coordinate x with a
complex scale factor g ¼ e
ih , with h ¼ arg(gÞ for some 0 h\h 0 , where h 0
depends on the potential V, leading to a simple but nontrivial analytic extension of
the spectrum of the differential operator that derives from a standard molecular
Schrödinger equation. A comparison between the spectrum of the Schrödinger
differential operator, before and after scaling, in the simple case where the point
spectrum r P accumulates at a given point, defining the onset of the absolutely
continuous spectrum r C , yields a rotation of the absolutely continuous spectrum
around the accumulation point (threshold) with the angle À2arg(gÞ, originating
from the (dominating) kinetic part of the Hamiltonian being related the second order
differential operator D ¼ r
2 . The mathematical theorem that provides the bound
states, the continuum and the so-called resonance states, in the sector between the
real axis and its rotated analogue, is due to Balslev and Combes [21]. For rigorous
treatments including also the electromagnetic field, see e.g. resonance studies in
4 This extension rests on a rigorous mathematical theory, i.e. the Balslev-Combes theorem [21],
see also Simon [22], and it is vital to understand and appreciate non-Hermitian quantum mechanics
and its consequences for the dynamics of resonance states embedded in the continuum and their
properties for higher order dynamics.
260
E.J. Brändas
viewed today as a fundamental area becoming more or less synonymous with the
fields of theoretical chemistry and chemical physics.
Despite these successes there remain many inconsistencies and conundrums
plaguing the fundamental physical formulation. Primas [17] in his thought provoking evaluation of the myth of universal laws brings up ten profound problems
connected with the incapabilities of traditional quantum chemistry promoting the
so-called structural approach. Although many of the puzzles have a metaphysical
flavour, they aim at the deeper meaning of chemistry and a worldview, unus
mundus, which should also incorporate the dimensions of a semiotic analysis. The
present author has reformulated some of the most serious ones in Brändas [6], in
particular the well-known issues of the uni-directedness of time and the associated
irreversibility of the macro-world (e.g. molecular chirality), as well as the elimination of the law of causality in the microscopic domain. Closely related is also the
inquiry of the absolute nature of the second law, Sklar [18]. These problems have
been reconsidered in the light of recent non-Hermitian quantum mechanics, see
Nicolaides and Brändas [19] and Moiseyev [20] suggesting potential solutions to
the paradox, see also Ref. [6].
Much could be said about the necessity to realize and accomplish a quantum
mechanical extension,
4 however it should be enough to briefly explain what we
mean when saying that the Schrödinger equation is extended or continued beyond
Hermitian territory. In the portrayal we refer to the popular analytic dilation technique for a simple case of a proper potential, V, exhibiting a point spectrum below a
positive continuum.
The method simply amounts to multiplying the space coordinate x with a
complex scale factor g ¼ e
ih , with h ¼ arg(gÞ for some 0 h\h 0 , where h 0
depends on the potential V, leading to a simple but nontrivial analytic extension of
the spectrum of the differential operator that derives from a standard molecular
Schrödinger equation. A comparison between the spectrum of the Schrödinger
differential operator, before and after scaling, in the simple case where the point
spectrum r P accumulates at a given point, defining the onset of the absolutely
continuous spectrum r C , yields a rotation of the absolutely continuous spectrum
around the accumulation point (threshold) with the angle À2arg(gÞ, originating
from the (dominating) kinetic part of the Hamiltonian being related the second order
differential operator D ¼ r
2 . The mathematical theorem that provides the bound
states, the continuum and the so-called resonance states, in the sector between the
real axis and its rotated analogue, is due to Balslev and Combes [21]. For rigorous
treatments including also the electromagnetic field, see e.g. resonance studies in
4 This extension rests on a rigorous mathematical theory, i.e. the Balslev-Combes theorem [21],
see also Simon [22], and it is vital to understand and appreciate non-Hermitian quantum mechanics
and its consequences for the dynamics of resonance states embedded in the continuum and their
properties for higher order dynamics.
260
E.J. Brändas