Certain one- and two-dimensional reductions of the three-dimensional Schrödinger equation of the hydrogen atom are considered. These reductions are carried out from the point of view of the two common sets of space coordinates: Cartesian and spherical. The resulting systems have features that relate more readily to the old quantum theory models of Bohr and Sommerfeld than the general three-dimensional hydrogen atom. Furthermore, the considerations yield interesting insights into the quantum mechanics of the hydrogen atom and may serve as helpful intermediary preparation, in an introductory presentation of the subject, for the unreduced three-dimensional case.
We discuss the introduction of constraints in a system of n noninteracting particles which obey the laws of nonrelativistic quantum mechanics. In this paper the particles are first thought of as being unconstrained (described by the
3n Cartesian coordinates of a flat space
R), but subject to an external potential
V which, in a certian suitable limit, forces the system to remain in a curved subspace
V of
R. This idea was already employed in a previous work where we have discussed the motion of one constrained particle. It was then shown that in order to obtain a meaningful result the particle wave function should be "uniformly compressed" into a surface (or curve), avoiding, in this way, the tangential forces which correspond to the dissipative constraints of classical mechanics. The resulting Schr\"odinger equation could then be separated in such a way that the part which contained the surface (or curve) variables was independent of the potential
V employed in the constraining process. In the present paper we show that this procedure cannot be carried out for all subspaces
V of
R in the case of a many-particle system, unless
V satisfies a certain geometrical condition. The many-particle Schr\"odinger equation (as was already the case of the one-particle systems) contains, besides the kinetic energy term, potentials which are not bending invariants, that is, cannot be obtained from the metric tensor
gij of
V or its derivatives. This gives rise to different Schr\"odinger equations for isometric subspaces, in striking contrast with the usual quantization procedures which start from the classical Lagrangian of the constrained system.
An axiomatic approach to quantum mechanics that has, as a theorem, the Schrödinger equation may be of enormous value to cope with interpretation issues, since all the interpretation constructs must be present in the axioms, or directly derived by them from their mathematical unfolding. Thus, it is critical to show that this axiomatic derivation is reliable beyond any possible doubt. To show this, it is possible to make generalizations and extensions of the axioms to derive the Schrödinger equation in the underlying generalized or extended formats. In previous papers, we have shown that the axiomatic approach we propose can be used to derive the Schrödinger equation as a direct axiom. Since then, we have also shown that it was possible to generalize that derivation to coordinate systems other than the Cartesian, as well as its relativistic extensions that lead to the relativistic wave equations. An extension to dissipative systems was also performed, allowing us to mathematically derive the Caldirola–Kanai equation from first principles. All these derivations were performed using pure states and in the absence of the electromagnetic field. This means that we can further generalize the approach to embrace these two possibilities. Being an axiomatic approach, we show that we need only to slightly modify the axioms to derive the Schrödinger equation for these two contexts. Despite being quite direct, the algebraic complexity of these derivations should give the reader the desired confidence in the proposed axioms.
When quantum mechanical motion of a particle constrained to a curved surface in the Euclidean space is considered, three different approaches may be adopted to describe the system. One is the confining approach which enforces a particle to stay in the surface by a physical confining potential. Both of the others follow Dirac's general prescription to treat a constrained system, but employ different constraints, namely the usual and the conservative constraints, respectively. The three approaches lead to results which do not in general agree with each other, though they give the same classical equation of motion. It is emphasized that the arbitrariness in the ordering of operators cannot be avoided when the latter two approaches are adopted. In the approach with the conservative constraint, however, there is a physical ordering, which leads to the result in agreement with that of the confining approach. We interpret these discrepancy and agreement in simple physical terms.