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Revista Brasileira de Ensino de Física

Derivation of the Schrödinger equation I: the characteristic function

Olavo Leopoldino da Silva Filho, Marcello Ferreira

Published Jan 1, 2024• 0 citations
Publisher

Abstract

In this paper, we present a mathematical derivation of the Schrödinger equation departing from only two axioms. We also show that, using this formal derivation process, it is possible to directly derive the Schrödinger equation in generalized curvilinear coordinate systems. This derivation is also shown to be equivalent to Feynman’s path integral approach, but goes further, allowing us to mathematically derive the Bohr-Sommerfeld quantization rules. The use of a small parameter, both in the present derivation, where it is δr, and Feynman’s derivation, where it is ϵ = δt, is also clarified in terms of the Central Limit Theorem. Therefore, the article makes a didactic transposition of the topic of quantization, allowing it to be addressed in the context of teaching Quantum Mechanics. The epistemological importance of axiomatic approaches for the mathematical derivation and the interpretation of the symbols of the theory is also considered.

DOI
10.1590/1806-9126-rbef-2024-0183

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In this paper, we present a mathematical derivation of the Schrödinger equation departing from only two axioms. We also show that, using this formal derivation process, it is possible to directly derive the Schrödinger equation in generalized curvilinear coordinate systems. This derivation is also shown to be equivalent to Feynman’s path integral approach, but goes further, allowing us to mathematically derive the Bohr-Sommerfeld quantization rules. The use of a small parameter, both in the present derivation, where it is δr, and Feynman’s derivation, where it is ϵ = δt, is also clarified in terms of the Central Limit Theorem. Therefore, the article makes a didactic transposition of the topic of quantization, allowing it to be addressed in the context of teaching Quantum Mechanics. The epistemological importance of axiomatic approaches for the mathematical derivation and the interpretation of the symbols of the theory is also considered.
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.
Courses about Quantum Mechanics are generally developed at the level of the mathematical formalism, its syntactic level, seldom treating the interpretation of the theory, its semantic level. In the majority of cases, this first formal approach is not followed by any other course addressing the numerous interpretations of the theory. It has become apparent that this strategy leads to a deficit in the professional formation of the students, exceedingly driving them to technical areas of application of Quantum Mechanics. However, applications such as those on quantum computing and entanglement, as they become more and more encrusted in the deepest foundations of the theory, are asking more and more for the development of this kind of ability. In this paper we develop an approach to deal with second quantization in the realm of Quantum Mechanics. We show that the second quantization Schrodinger equation can be mathematically derived from a classical Hamiltonian written in the phase space (Q,P)(Q,P)(Q,P), obtained by a canonical transformation upon the original phase space (q,p)(q,p)(q,p), using an axiomatic quantization procedure developed elsewhere. We apply this quantization process to a bosonic system, modelled by the harmonic oscillator problem. We then make reverse engineering to show that Schwinger's second quantization approach to fermionic systems furnishes the path to derive a Schrodinger equation explicitly written in terms of the usual momenta and positions operators (q^,p^)(\hat{q},\hat{p})(q^​,p^​) for such systems. Finally, we use these mathematical developments to address the commonly accepted statement that 'the spin has no classical analog'.
In this paper, we present a derivation of the relativistic equations of Quantum Mechanics. We focus on the second order equations (both Klein Gordon’s and Dirac’s), since the linear Dirac equation can be derived from these. This derivation is an extension of the same method applied to the non-relativistic case in a in a previous series of articles by us. We show that, to extend the theoretical framework to the relativistic case, we have only to extend the axioms in a relativistically compatible way, as should be the case with any extension of an axiomatic method. We also show that this quantization method throws light in a different way to interpret some descriptors of the usual formalism, mainly those regarding the expression for the probability density function. This, on the other hand, gives the teacher following this approach in classes on the subject to assess the epistemological issue of the role of manuals in the learning of Quantum Mechanics.
Abstract The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.
Since the early twentieth century, quantum mechanics has sought an interpretation that offers a consistent worldview. In the course of that, many proposals were advanced, but all of them introduce, at some point, interpretation elements (semantics) that find no correlate in the formalism (syntactics). This distance from semantics and syntactics is one of the major reasons for finding so abstruse and diverse interpretations of the formalism. To overcome this issue, we propose an alternative stochastic interpretation, based exclusively on the formal structure of the Schrödinger equation, without resorting to external assumptions such as the collapse of the wave function or the role of the observer. We present four (mathematically equivalent) mathematical derivations of the Schrödinger equation based on four constructs: characteristic function, Boltzmann entropy, Central Limit Theorem (CLT), and Langevin equation. All of them resort to axioms already interpreted and offer complementary perspectives to the quantum formalism. The results show the possibility of deriving the Schrödinger equation from well-defined probabilistic principles and that the wave function represents a probability amplitude in the configuration space, with dispersions linked to the CLT. It is concluded that quantum mechanics has a stochastic support, originating from the separation between particle and field subsystems, allowing an objective description of quantum behavior as a mean-field theory, analogous, but not equal, to Brownian motion, without the need for arbitrary ontological entities.