4.6 Summary and Conclusions
The metaphysical postulate “equat causa effectum” does not solve physical problems, but it helps to explore them more profoundly and hence should be exploited in
teaching physics. Here, it leads to the examination of the role of the initial conditions
as another “cause” of a trajectory, additionally to the force. If the initial conditions
“equal” the cause, the single effect does so as well.
Because the force acts directly upon the momentum and velocity, respectively,
but only indirectly upon the position, the hodograph is more symmetric than the
trajectory. For a central force field, both are plane. This has been shown by means of
qualitative and intuitive arguments, which are much more easily accessible than
angular momentum conservation.
In both the classical and quantum cases, two different initial conditions lead to
equivalent trajectories/wave functions, if the one can be obtained from the other one
through a symmetry transformation of the force field/potential. The set of all such
trajectories respectively wave functions squared forms a figure that exhibits the
symmetry of the force field/potential.
For Kepler orbits, there are no boundary conditions; for Bohr orbitals, the
symmetry of the boundary conditions equals that of the force field. For this, they
play no role for the symmetry of the motion. If their symmetry is lower than that of
the force field, their influence on the symmetry of the motion can be treated
analogously to that of the initial conditions.
Acknowledgment I would like to thank Michael Erdmann, Joseph Rosen, Dieter Suisky, and a
former colleague not wishing to be named for numerous enlightening discussions. I have benefited
from numerous excitations in science as well in arts during my stay at Orihuela/Oriola and thus feel
highly indebted to the organizers of FFP15.
References
Bode JE (1772) Deutliche Anleitung zur Kentniss des Gestirnten Himmels. Harmsen, Hamburg
Cariñena JF, Rañada MF, Santander M (2016) A macroscopic test of the Aharonov-Bohm effect.
arXiv:0708.2428v1
Chandrasekhar S (1946) On a new theory of Weizsäcker on the origin of the solar system. Rev Mod
Phys 18:94–102
Conway JH, Burgiel H, Goodman-Strauss C (2008) The symmetries of things. A K Peters/CRC
Press, New York
Curie P (1894) Sur la symétrie dans les phénomènes physiques, symétrie d’un champ électrique et
d’un champ magnétique. J Phys (Paris) 3(3):393–415. https://hal.archives-ouvertes.fr/jpa00239814. Engl.: On symmetry in physical phenomena, symmetry of an electric field and of a
magnetic field, In: Rosen (1982, pp 17–25)
Einstein A (1923) Bietet die Feldtheorie Möglichkeiten für die Lösung des Quantenproblems?
Sitzungsber Preuss Akad Wiss phys-math Kl 13:XXXIII
Enders P (2006) Von der klassischen Physik zur Quantenphysik. Eine historisch-kritische
deduktive Ableitung mit Anwendungsbeispielen aus der Festkörperphysik. Springer, Berlin
4 “Equat Causa Effectum”
45
The metaphysical postulate “equat causa effectum” does not solve physical problems, but it helps to explore them more profoundly and hence should be exploited in
teaching physics. Here, it leads to the examination of the role of the initial conditions
as another “cause” of a trajectory, additionally to the force. If the initial conditions
“equal” the cause, the single effect does so as well.
Because the force acts directly upon the momentum and velocity, respectively,
but only indirectly upon the position, the hodograph is more symmetric than the
trajectory. For a central force field, both are plane. This has been shown by means of
qualitative and intuitive arguments, which are much more easily accessible than
angular momentum conservation.
In both the classical and quantum cases, two different initial conditions lead to
equivalent trajectories/wave functions, if the one can be obtained from the other one
through a symmetry transformation of the force field/potential. The set of all such
trajectories respectively wave functions squared forms a figure that exhibits the
symmetry of the force field/potential.
For Kepler orbits, there are no boundary conditions; for Bohr orbitals, the
symmetry of the boundary conditions equals that of the force field. For this, they
play no role for the symmetry of the motion. If their symmetry is lower than that of
the force field, their influence on the symmetry of the motion can be treated
analogously to that of the initial conditions.
Acknowledgment I would like to thank Michael Erdmann, Joseph Rosen, Dieter Suisky, and a
former colleague not wishing to be named for numerous enlightening discussions. I have benefited
from numerous excitations in science as well in arts during my stay at Orihuela/Oriola and thus feel
highly indebted to the organizers of FFP15.
References
Bode JE (1772) Deutliche Anleitung zur Kentniss des Gestirnten Himmels. Harmsen, Hamburg
Cariñena JF, Rañada MF, Santander M (2016) A macroscopic test of the Aharonov-Bohm effect.
arXiv:0708.2428v1
Chandrasekhar S (1946) On a new theory of Weizsäcker on the origin of the solar system. Rev Mod
Phys 18:94–102
Conway JH, Burgiel H, Goodman-Strauss C (2008) The symmetries of things. A K Peters/CRC
Press, New York
Curie P (1894) Sur la symétrie dans les phénomènes physiques, symétrie d’un champ électrique et
d’un champ magnétique. J Phys (Paris) 3(3):393–415. https://hal.archives-ouvertes.fr/jpa00239814. Engl.: On symmetry in physical phenomena, symmetry of an electric field and of a
magnetic field, In: Rosen (1982, pp 17–25)
Einstein A (1923) Bietet die Feldtheorie Möglichkeiten für die Lösung des Quantenproblems?
Sitzungsber Preuss Akad Wiss phys-math Kl 13:XXXIII
Enders P (2006) Von der klassischen Physik zur Quantenphysik. Eine historisch-kritische
deduktive Ableitung mit Anwendungsbeispielen aus der Festkörperphysik. Springer, Berlin
4 “Equat Causa Effectum”
45