Deriving the Schrödinger equation from Newtonian kinematics

Julio Alfonso Alonso, M. Paz Nieto, Emilio Santos

Noboru Kohiyama

The quantum mass (QM) is expressed using the expectation value of the kinetic energy derived from the Schrödinger equation (SE). The mass‐energy equation (ME) is expressed using the kinetic energy of a body in Newtonian mechanics. The SE is modified using the QM and ME. The Adams‐Williamson equation adeptly characterizes the density distribution within Earth's interior, highlighting the role of pressure in storing elastic energy. This stored energy forms the basis for deriving the Newtonian gravitational field. Intriguingly, the eigenvalue of electron's energy in the hydrogen-like atom derived from the modified SE exhibits variation in response to gravitational field strength.

Viraht Sahni

Jorge Krause

The formalism of non-Abelian quantum kinematics is applied to the Newtonian symmetry group of the harmonic oscillator. Within the regular ray representation of the group, the Schrödinger operator, as well as two other (new) invariant operators, are obtained as Casimir operators of the extended kinematic algebra. Superselection rules are then introduced, which permit the identification (and the explicit calculation) of the physical states of the system. Next, a complementary ray representation, attached to the space-time realization of the group, casts the Schrödinger operator into the familiar time-dependent space-time differential operator of the harmonic oscillator and thus, by means of the superselection rules, one obtains the time-dependent Schrödinger equation of the sytem. Finally, the evaluation of a Hurwitz invariant integral, over the group manifold, affords the well known Feynman space-time propagator 〈t′,x′‖t,x〉 of the simple harmonic oscillator. Everything comes out from the assumed symmetries of the system. The whole approach is group theoretic and ‘‘relativistic.’’ No classical analog is used in this ‘‘quantization’’ scheme.

Thomas B. Andrews

Aloysius F. Kracklauer

Nelson's derivation of Schrödinger's equation with a stochastic argument is shown to be compatible with demonstrations that quantum probabilities evolve deterministically.Received 16 April 1973DOI:https://doi.org/10.1103/PhysRevD.10.1358©1974 American Physical Society