The old rule “equat causa effectum” will be restored through constructing sets of
orbits with initial conditions on spheres. These constructions are guided by the
conserved quantities total energy, angular momentum, and Laplace–Runge–Lenz
vector. Moreover, I will present criteria to be posed on the initial conditions for
obtaining the trajectories of highest possible symmetry.
Analogous considerations will be performed for Bohr orbitals, ψ nlm . Here, the
squared wave functions, |ψ nlm |
2 , are considered to be analogs to the stationary-state
functions momentum (Newton 1999, Axioms) and velocity (Euler 1750, Ch. 7), see
(Enders 2006, 2008, 2009, 2013; Enders and Suisky 2005; Suisky 2009). Then,
Unsöld’s (1927) theorem paves the way to spherically symmetric sets of stationary
states. As in the classical case, the set of all stationary-state functions with given
energy and/or modulus of angular momentum forms a spherically symmetric figure.
Moreover, there is an accidental or hidden symmetry of the 1/r-potential as
represented by the Laplace–Runge–Lenz vector (Hermann 1710, 1732; Laplace
1799, Premiere Partie, Livre II, pp. 165ff.; Runge 1921; Lenz 1924, Eq. (2)
5 ) and
its quantum analog, the “Pauli vector” (Pauli 1926, Eq. (50)), which may be
exploited for constructing sets of spherically symmetric orbits, too.
All that suggests the teaching of physics to pay due attention not only to the
mathematical role, but also to the physical role of the initial conditions in the
description of phenomena. Old rules like “equat causa effectum” do not solve
physical problems, but provide another fruitful view on them.
4.2 On the Initial Conditions
A single trajectory of a body of constant mass, m, is specified by the initial position, r
(t ¼ 0), and velocity, v(t ¼ 0), or momentum, p(t ¼ 0), of the body. The developments of canonical, statistical and quantum mechanics have led to treating position
and velocity/momentum on more or less equal footing. For our purpose, it is more
appropriate to consider their different relationships to the external force, in order to
understand the different symmetry properties of the curves r(t) on the one hand and v
(t) respectively p(t) on the other hand.
In contrast to the curves v(t) and p(t), the curve r(t) (the trajectory) is not
immediately connected with the force. According to Newton’s Law 2, the immediate
“effect” of the force, F,—the “cause”—is the “momentum trajectory,” p(t), hence—
at constant mass—the hodograph,
6 v(t) (Hamilton 1847). Following Euler (1750,
para. 56, 70; 1752, para. 20), we have r ¼ vdt and dv ¼ Fdt/m (the mass, m, being
5 For more sources and newer developments, see https://en.wikipedia.org/wiki/Laplace–Runge–
Lenz_vector, and links therein.
6 The literal meaning of the word “hodograph” is path describer, where “path” means not a
trajectory, but the trace of the top of a parameter-dependent vector with fixed origin when the
parameter (here, the time, t) is changing. For developments till Today, see Cariñena et al. (2016).
40
P. Enders
orbits with initial conditions on spheres. These constructions are guided by the
conserved quantities total energy, angular momentum, and Laplace–Runge–Lenz
vector. Moreover, I will present criteria to be posed on the initial conditions for
obtaining the trajectories of highest possible symmetry.
Analogous considerations will be performed for Bohr orbitals, ψ nlm . Here, the
squared wave functions, |ψ nlm |
2 , are considered to be analogs to the stationary-state
functions momentum (Newton 1999, Axioms) and velocity (Euler 1750, Ch. 7), see
(Enders 2006, 2008, 2009, 2013; Enders and Suisky 2005; Suisky 2009). Then,
Unsöld’s (1927) theorem paves the way to spherically symmetric sets of stationary
states. As in the classical case, the set of all stationary-state functions with given
energy and/or modulus of angular momentum forms a spherically symmetric figure.
Moreover, there is an accidental or hidden symmetry of the 1/r-potential as
represented by the Laplace–Runge–Lenz vector (Hermann 1710, 1732; Laplace
1799, Premiere Partie, Livre II, pp. 165ff.; Runge 1921; Lenz 1924, Eq. (2)
5 ) and
its quantum analog, the “Pauli vector” (Pauli 1926, Eq. (50)), which may be
exploited for constructing sets of spherically symmetric orbits, too.
All that suggests the teaching of physics to pay due attention not only to the
mathematical role, but also to the physical role of the initial conditions in the
description of phenomena. Old rules like “equat causa effectum” do not solve
physical problems, but provide another fruitful view on them.
4.2 On the Initial Conditions
A single trajectory of a body of constant mass, m, is specified by the initial position, r
(t ¼ 0), and velocity, v(t ¼ 0), or momentum, p(t ¼ 0), of the body. The developments of canonical, statistical and quantum mechanics have led to treating position
and velocity/momentum on more or less equal footing. For our purpose, it is more
appropriate to consider their different relationships to the external force, in order to
understand the different symmetry properties of the curves r(t) on the one hand and v
(t) respectively p(t) on the other hand.
In contrast to the curves v(t) and p(t), the curve r(t) (the trajectory) is not
immediately connected with the force. According to Newton’s Law 2, the immediate
“effect” of the force, F,—the “cause”—is the “momentum trajectory,” p(t), hence—
at constant mass—the hodograph,
6 v(t) (Hamilton 1847). Following Euler (1750,
para. 56, 70; 1752, para. 20), we have r ¼ vdt and dv ¼ Fdt/m (the mass, m, being
5 For more sources and newer developments, see https://en.wikipedia.org/wiki/Laplace–Runge–
Lenz_vector, and links therein.
6 The literal meaning of the word “hodograph” is path describer, where “path” means not a
trajectory, but the trace of the top of a parameter-dependent vector with fixed origin when the
parameter (here, the time, t) is changing. For developments till Today, see Cariñena et al. (2016).
40
P. Enders