where x ja ¼ ðE
ð0Þ
j À E
ð0Þ
a Þ= h denotes the natural transition frequencies. ^
H
ð1Þ and
^
H
ð2Þ are first- and second-order perturbing Hamiltonians.
RSPT formally yields a rapidly convergent series for the interaction energy of a
molecule in the presence of external perturbing fields and an easy-to-handle set of
computational recipes for related response properties up to fourth-order, which are
evaluated by differentiation with respect to perturbation parameters [3].
Nonetheless, the analysis of predictions arrived at by k-th order wavefunctions
W
ðkÞ
a ¼ W
ðkÞ
a ðx 1 ; x 2 . . .x n Þ depending on 6n space-spin coordinates x i r i s i ; i ¼
1; . . .n for an n-electron system [3, 15, 16], may be quite hard, if not impossible, in
the majority of cases. Propagator methods [17–20] and coupled cluster theory [21–
23] offer appealing alternatives to RSPT for accurate calculations in small and
medium-size molecules with ground states characterized by a dominant single
electronic configuration [24–27], but, in general, do not make simpler the interpretation of results.
7.1.2 The Hydrodynamical Approach to Quantum
Mechanics
In 1926 Schrödinger proposed a definition of quantum-mechanical current density,
which satisfies a continuity equation formally identical to that of classical electrodynamics, in his fourth paper on quantization as an eigenvalue problem [28]. A few
months later, in the same year, Madelung put forward an alternative foundation of
quantum theory allowing for a hydrodynamical analogy [29]. Fundamental contributions were given by de Broglie in 1926 [30] and 1927 [31]. Later on, more
advanced formulations were elaborated by Landau [32] and by London [33]. Within
the hydrodynamical approach to quantum mechanics, the continuity condition and a
vector equation, with the same form as the Hamilton-Jacobi equation of motion of
classical mechanics, provide a substitute of the wave equation [34–40]. The equation
of motion can also be recast as the Newton second law, taking into account a
nonlocal quantum potential
1 [34]. Takabayasi investigated a relativistic hydrodynamics, which offers a hydrodynamical model of the Dirac matter [41].
The profound physical meaning [42], the capacity to gain an accurate and deep
understanding of the phenomenology and the philosophical implications [43, 44] of
the representation of quantum mechanics proposed by Madelung [29], Landau [32],
and London [33] (MLL) cannot be overstressed. Bohm showed that the hydrodynamical quantum mechanics is deterministic and provides an interpretation of
physical reality alternative to that of the Copenhagen School [34, 35].
1
See Eq. (8a) of Ref. [34]. By combining the first two addenda on the l.h.s. of Eq. (2) of Ref. [36],
a quantum-mechanical relationship of the same form as the Newton’s second law is obtained for a
particle acted upon by the Lorentz force and by the nonlocal Bohm potential, see Eq. (94) of Ref.
[15].
154
P. Lazzeretti
ð0Þ
j À E
ð0Þ
a Þ= h denotes the natural transition frequencies. ^
H
ð1Þ and
^
H
ð2Þ are first- and second-order perturbing Hamiltonians.
RSPT formally yields a rapidly convergent series for the interaction energy of a
molecule in the presence of external perturbing fields and an easy-to-handle set of
computational recipes for related response properties up to fourth-order, which are
evaluated by differentiation with respect to perturbation parameters [3].
Nonetheless, the analysis of predictions arrived at by k-th order wavefunctions
W
ðkÞ
a ¼ W
ðkÞ
a ðx 1 ; x 2 . . .x n Þ depending on 6n space-spin coordinates x i r i s i ; i ¼
1; . . .n for an n-electron system [3, 15, 16], may be quite hard, if not impossible, in
the majority of cases. Propagator methods [17–20] and coupled cluster theory [21–
23] offer appealing alternatives to RSPT for accurate calculations in small and
medium-size molecules with ground states characterized by a dominant single
electronic configuration [24–27], but, in general, do not make simpler the interpretation of results.
7.1.2 The Hydrodynamical Approach to Quantum
Mechanics
In 1926 Schrödinger proposed a definition of quantum-mechanical current density,
which satisfies a continuity equation formally identical to that of classical electrodynamics, in his fourth paper on quantization as an eigenvalue problem [28]. A few
months later, in the same year, Madelung put forward an alternative foundation of
quantum theory allowing for a hydrodynamical analogy [29]. Fundamental contributions were given by de Broglie in 1926 [30] and 1927 [31]. Later on, more
advanced formulations were elaborated by Landau [32] and by London [33]. Within
the hydrodynamical approach to quantum mechanics, the continuity condition and a
vector equation, with the same form as the Hamilton-Jacobi equation of motion of
classical mechanics, provide a substitute of the wave equation [34–40]. The equation
of motion can also be recast as the Newton second law, taking into account a
nonlocal quantum potential
1 [34]. Takabayasi investigated a relativistic hydrodynamics, which offers a hydrodynamical model of the Dirac matter [41].
The profound physical meaning [42], the capacity to gain an accurate and deep
understanding of the phenomenology and the philosophical implications [43, 44] of
the representation of quantum mechanics proposed by Madelung [29], Landau [32],
and London [33] (MLL) cannot be overstressed. Bohm showed that the hydrodynamical quantum mechanics is deterministic and provides an interpretation of
physical reality alternative to that of the Copenhagen School [34, 35].
1
See Eq. (8a) of Ref. [34]. By combining the first two addenda on the l.h.s. of Eq. (2) of Ref. [36],
a quantum-mechanical relationship of the same form as the Newton’s second law is obtained for a
particle acted upon by the Lorentz force and by the nonlocal Bohm potential, see Eq. (94) of Ref.
[15].
154
P. Lazzeretti