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European Journal of Physics

On the first Schröedinger’s derivation of his time-independent equation using statistical mechanics and extremizing the energy of the hydrogen atom

Mladen Abramovic

Published May 21, 2025• 0 citations
Publisher

Abstract

Abstract This paper intends to provide instruction for a single lesson in an introductory course on quantum mechanics by tying together probabilistic methods from statistical mechanics, the use of complex numbers, and a more founded explanation for the derivation of Schrödinger’s time-independent equation. Here presented derivation strongly supports the Copenhagen interpretation of quantum mechanics. Niels Bohr suggested a hydrogen atom model (similar to Kepler’s sun-planet model), where the negatively-charged electron moves in orbit around the heavy positively charged nucleus, this electron having some kinetic and potential energy. Starting from Hamilton’s differential equation for Kepler like atomic model, Schrödinger introduced a new unknown real function ψ in his derivation of time-independent equation (initially, he did not recognize it as the wave function). We show that hypothetical statistical spread of action on many (indefinite) levels is mandatory for the derivation of Schrödinger’s time-independent equation. The result of this hypothesis, the statistical probability constraint formula, is used in this derivation as we look for a minimum of probability for the electron energy distribution E ψ in the whole space. The Euler–Lagrange equation, with normalization of ψ as a constraint limitation, then gives Schrödinger’s time-independent equation for real numbers. A closer inspection of the normalized function ψ shows that it could be complex and is related to the de Broglie–Bohr model of the hydrogen atom. It can be postulated that the classical solution of Schrödinger’s time-independent equation should be a normalized complex function ψ .

DOI
10.1088/1361-6404/addb91

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Abstract This paper intends to provide instruction for a single lesson in an introductory course on quantum mechanics by tying together probabilistic methods from statistical mechanics, the use of complex numbers, and a more founded explanation for the derivation of Schrödinger’s time-independent equation. Here presented derivation strongly supports the Copenhagen interpretation of quantum mechanics. Niels Bohr suggested a hydrogen atom model (similar to Kepler’s sun-planet model), where the negatively-charged electron moves in orbit around the heavy positively charged nucleus, this electron having some kinetic and potential energy. Starting from Hamilton’s differential equation for Kepler like atomic model, Schrödinger introduced a new unknown real function ψ in his derivation of time-independent equation (initially, he did not recognize it as the wave function). We show that hypothetical statistical spread of action on many (indefinite) levels is mandatory for the derivation of Schrödinger’s time-independent equation. The result of this hypothesis, the statistical probability constraint formula, is used in this derivation as we look for a minimum of probability for the electron energy distribution E ψ in the whole space. The Euler–Lagrange equation, with normalization of ψ as a constraint limitation, then gives Schrödinger’s time-independent equation for real numbers. A closer inspection of the normalized function ψ shows that it could be complex and is related to the de Broglie–Bohr model of the hydrogen atom. It can be postulated that the classical solution of Schrödinger’s time-independent equation should be a normalized complex function ψ .
The proposed work presents the derivation of the wave function and the Schrödinger equation based on the dual tangential equation of time with an imaginary rate. The derivation uses de Broglie formulas. The wave and complex-conjugate functions are considered, which participate in the probabilistic description of the position of a particle in a potential field. Another interpretation of the product of these functions is presented. Based on it, the classical structure of the electron is derived. The derivation determines the quantum value of the imaginary incident vector. When deriving wave functions, simple solutions immediately arise for energies corresponding to radiation emitted by portions and a microscopic harmonic oscillator. In quantum mechanics, the first solution was used by Planck as a hypothesis when deriving his famous formula for describing equilibrium thermal radiation. The second energy solution was obtained from the Schrödinger equation by cumbersome calculations.
Even after nine decades of successful run of the Quantum Mechanics (QM), different viewpoints on foundational problems of Quantum Physics are still being actively debated. That is because mathematical logic of QM often defies the physical intuition which constitutes the main spirit of Physics. De Broglie’s hypothesis of matter waves implied that the dynamic characteristics of a micro particle in motion, can be ascribed to the wave characteristics of the wavelet accompanying the particle. The Schrödinger equation models the matter-wave interactions through wavefunction ψ and effectively serves as the foundation of QM. Even though mathematical structure of the Schrödinger equation is sound and elegant, here we show a conceptual mistake in the development of this equation wherein the physical situation has not been correctly modeled in the equation. The Coulomb potential energy of the proton electron pair in Hydrogen atom is essentially the negative interaction energy between their superposed electrostatic fields which is inversely proportional to their instantaneous separation distance. Assuming the proton to be relatively fixed at the origin of an appropriate coordinate system, the potential energy of the orbiting electron will be a function of instantaneous position coordinates of the electron. This has not been properly modeled in the Schrödinger equation. The resulting errors in the solution have been quantitatively demonstrated in this paper. We have stressed the necessity of incorporating a specific correction in the potential energy term of the Schrödinger equation, after which it may facilitate the adoption of Bohmian QM.
We found in the previous chapter that the molecular partition function is required to determine the thermodynamic properties of an ideal gas. To evaluate the partition function, specification of pertinent energy levels and degeneracies is necessary. Such knowledge demands that we investigate at least the rudiments of quantum mechanics, and especially those quantum concepts required for subsequent applications to statistical thermodynamics. For this reason, we concentrate in the next few chapters on the Schrödinger wave equation, whose various solutions provide the ε j 's and g j 's needed for the eventual calculation of thermodynamic properties. Depending on your academic background, you might thus consider reviewing classical mechanics (Appendix G) and operator theory (Appendix H) in preparation for your upcoming study of quantum mechanics. We begin this chapter with a historical review of the developments leading to the formulation of quantum mechanics, subsequently focusing on the Bohr model for atomic hydrogen and the de Broglie hypothesis for matter waves. We then introduce the Schrödinger wave equation, the basic postulates of quantum mechanics, and salient insights from these postulates germane to the development of statistical thermodynamics. We next apply the Schrödinger wave equation to the translation energy mode of an atom or molecule. This application conveniently explains both quantum states and quantum numbers, including their relation to our previous notions of microstate and macrostate. We end this chapter by discussing the Heisenberg uncertainty principle, including its utility in defining indistinguishability and symmetry conditions for multiparticle systems.
Abstract A heuristic derivation of the one-dimensional Schrödinger equation is presented based on an energy interpretation of the wave function. The spatial derivative naturally produces a term proportional to the kinetic energy, while the temporal derivative yields a term proportional to the total energy. By relating both contributions through the classical relation for total mechanical energy, the Schrödinger equation is obtained. The proposed approach provides a simple and intuitive framework to introduce this equation in modern physics or introductory-level quantum mechanics courses.
The Schrödinger equation was really postulated rather than derived. But we can see why it was postulated. It has been proven to be valid, in that it provides information about small, slow moving particles subject to a wide variety of potentials. The derivation of the time-dependent Schrödinger equation (TDSE) follows rather directly from assuming a free particle wave: Ψ = e i(kx-ωt) . If this wave function is plugged into the classical wave equation, and quantum principles are used, the two sides of the equation are not equal (this means this function does not "solve" the equation). The quantum principles required are an understanding of the photoelectric effect (by which E = ħω) and the de Broglie hypothesis (λ = h/p). Using these relationships, we show that one derivative of time is equal to a double derivative with respect to position, with a few proportionality factors mixed in. Thus, the TDSE for a free particle is just a statement that kinetic energy equals total energy . For a non-free particle (subject to a potential), potential must merely be added to the left-hand side of the equation, to yield kinetic energy plus potential energy equals total energy .
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.