This article presents a detailed derivation of the hydrogen atom solutions in spherical coordinates, introducing the role of ordinary differential equations (ODEs), Laguerre polynomials, Legendre polynomials and associated Legendre polynomials. The separation of variables method is applied to the time-independent Schrödinger equation for a particle in the Coulomb potential, leading to the standard forms for radial and angular wavefunctions. Central to this derivation is the Frobenius method, which serves as the primary tool for obtaining polynomial solutions that satisfy essential boundary regularity. All sequences used here can be reproduced through their respective recurrence relations, enabling the systematic construction of the polynomial solutions. Building upon this mathematical foundation, the article explores Bohmian mechanics within a specific superposition of two eigenfunctions with varying angular momentum quantum numbers at a fixed principal quantum number. This configuration generates an autonomous (time-independent) velocity field, allowing for a detailed analysis of the influence of nodal surfaces on particle motion. The investigation extends to time-dependent radial superpositions for and states, demonstrating how the centrifugal barrier in p states shifts trajectories away from the nucleus and induces distinct oscillatory dynamics compared to s state configurations. In a superposition of only two eigenstates sharing the same principal quantum number, the system maintains a time-independent probability density. Consequently, the dynamics remain largely quasi-periodic and reside at the threshold of instability, exhibiting only weakly chaotic behavior. This configuration serves as the immediate precursor to the fully developed chaos that, as Parmenter et al. [47] and Frisk [48] demonstrated, typically requires a minimum of three eigenstates to emerge within a two-dimensional configuration space. These results suggest a regime of weak chaos within the Bohmian framework, indicating a need for further research into the long-term stability and predictability of such systems. It is anticipated that a higher number of superposed states would lead to more pronounced chaotic behavior and stronger trajectory divergence, a prospect that remains a subject for future investigation.
s
p
Part 1 : Bohmian Trajectories and the Role of Nodal Surface Manifolds in Hydrogen Eigenstates–Mathematical Foundations and Wavefunction Derivation
In the first part of this series, the mathematical and symbolic foundations for the hydrogen atom are established. The analysis begins with the time-independent Schrödinger equation in a Coulomb potential, utilizing the Frobenius method (a power series solution applicable to equations with regular singular points) to derive the radial wavefunctions and associated Laguerre polynomials. Furthermore, the angular components are developed through the Associated Legendre differential equations and Spherical Harmonics. This part focuses on the rigorous construction of the stationary eigenstates, providing the essential wave-mechanical framework required to define the nodal surface manifolds that govern the subsequent Bohmian analysis.
Full list of posts: https://community.wolfram.com/web/kvbloh/
Note: see the detailed introduction to the de Broglie Bohm theory with Mathematica here:
https://community.wolfram.com/groups/-/m/t/3000266
Note: see the detailed introduction to the de Broglie Bohm theory with Mathematica here:
https://community.wolfram.com/groups/-/m/t/3000266
0. Introduction
0. Introduction
Sometimes it is necessary to end a Wolfram Language kernel session or to clear all values, definitions, attributes, defaults, options and messages for the symbols.
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ClearAll["Global`*"]
Use the following command, if you need to save/export graphs created by Mathematica into the current notebook directory:
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SetDirectory[NotebookDirectory[]]
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I:\Mathematica Notebooks
To make the equations more readable, the standard form of partial derivatives is used. The definition is taken from the Wolfram blog.
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pdConv[f_]:=TraditionalForm[f/.Derivative[inds__][g_][vars__]:>Apply[Defer[D[g[vars],##]]&,Transpose[{{vars},{inds}}]/.{{var_,0}:>Sequence[],{var_,1}:>{var}}]]
Cells that contain ‘dynamic content’ often lead to a system message. As a result, it is sometimes necessary to switch off dynamic evaluation for certain calculations.
PaletteNotebook@Column[{Button["Dynamic off",SetOptions[NotebookSelection[],DynamicUpdating->False]],Button["Dynamic on",SetOptions[NotebookSelection[],DynamicUpdating->True]]}]
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SetOptions[$FrontEnd,"DynamicEvaluationTimeout"->30]
Feel free to use, reproduce, and modify this code in its entirety or in part of it. A further note: Some descriptions might be formulated too simply and could possibly require a more detailed explanation.
WARNING: These notes are incomplete and may contain errors in the implementation of the code and in the implementation of the equations. Although every step in the derivation of the Hydrogen atom has been meticulously checked, potential errors may still exist. The essay has been written to the best of my knowledge and with great care. Every equation has been checked. It cannot be ruled out that errors may still exist. Some of the formulations are not introduced with mathematical precision and may contain linguistic inaccuracies. To simplify calculations, SI units are not used throughout this work. The author hopes, that readability is not significantly affected by this. The essay has not undergone a detailed peer review. Attention was paid to the clarity and comprehensibility of the equations. Therefore, many equations were not combined into a single expression. I applied parallelization of the computational steps only in areas where a significant increase in processing speed could be achieved. This sometimes results in less than optimal computational efficiency. They rather serve didactic purposes and aim to contribute to an understanding of handling mathematics with Mathematica. Similarly, I have not indicated quotes from my own works. To the best of my knowledge, I have listed all individuals from whom I have adopted parts of the programs. If I have overlooked people who have contributed to Bohm’s theory or to the algorithms used, this is solely my responsibility.
CAUTION! Enter parameters as rational numbers if possible, so that numerics are as accurate as possible.
Note: To maximize didactic clarity and ensure every step is transparent to the reader, the author has explicitly detailed nearly all computational steps, avoiding reliance on conventional shortcuts such as “as is easily seen” or “following simple rearrangement.”
This approach aligns with a pedagogical principle often emphasized by John Archibald Wheeler. As recounted by Kip Thorne, Wheeler once gave the following advice to his student Richard Feynman regarding the presentation of complex ideas: “Little steps for little people.” While some might find the phrase provocative, in the context of Wheeler’s mentorship, it represented a profound commitment to clarity. It serves as a reminder that even the most complex quantum mechanical derivations should be broken down into accessible stages. In this notebook, “little steps” are not a reflection of the reader’s ability, but a tribute to the transparency of the derivation. The author has adopted this philosophy from personal experience, recognizing that breaking down brilliant insights into small, logically linked steps is precisely what makes the mathematical derivation of the hydrogen atom accessible. Furthermore, the author believes that seeing a fully worked-out example provides a necessary sense of security and confidence, allowing the reader to verify each stage of the logic. This ensures that the path of reasoning remains transparent for everyone.
Furthermore, this aligns with the principle articulated by mathematician David Hilbert (1862–1943), which the author has adapted for this context: “In physics, we must aim for clarity. It is often the detailed execution, and not the mere suggestion of a method, which proves decisive. A presentation should be judged by how little it presumes the reader already knows.”
WARNING: These notes are incomplete and may contain errors in the implementation of the code and in the implementation of the equations. Although every step in the derivation of the Hydrogen atom has been meticulously checked, potential errors may still exist. The essay has been written to the best of my knowledge and with great care. Every equation has been checked. It cannot be ruled out that errors may still exist. Some of the formulations are not introduced with mathematical precision and may contain linguistic inaccuracies. To simplify calculations, SI units are not used throughout this work. The author hopes, that readability is not significantly affected by this. The essay has not undergone a detailed peer review. Attention was paid to the clarity and comprehensibility of the equations. Therefore, many equations were not combined into a single expression. I applied parallelization of the computational steps only in areas where a significant increase in processing speed could be achieved. This sometimes results in less than optimal computational efficiency. They rather serve didactic purposes and aim to contribute to an understanding of handling mathematics with Mathematica. Similarly, I have not indicated quotes from my own works. To the best of my knowledge, I have listed all individuals from whom I have adopted parts of the programs. If I have overlooked people who have contributed to Bohm’s theory or to the algorithms used, this is solely my responsibility.
CAUTION! Enter parameters as rational numbers if possible, so that numerics are as accurate as possible.
Note: To maximize didactic clarity and ensure every step is transparent to the reader, the author has explicitly detailed nearly all computational steps, avoiding reliance on conventional shortcuts such as “as is easily seen” or “following simple rearrangement.”
This approach aligns with a pedagogical principle often emphasized by John Archibald Wheeler. As recounted by Kip Thorne, Wheeler once gave the following advice to his student Richard Feynman regarding the presentation of complex ideas: “Little steps for little people.” While some might find the phrase provocative, in the context of Wheeler’s mentorship, it represented a profound commitment to clarity. It serves as a reminder that even the most complex quantum mechanical derivations should be broken down into accessible stages. In this notebook, “little steps” are not a reflection of the reader’s ability, but a tribute to the transparency of the derivation. The author has adopted this philosophy from personal experience, recognizing that breaking down brilliant insights into small, logically linked steps is precisely what makes the mathematical derivation of the hydrogen atom accessible. Furthermore, the author believes that seeing a fully worked-out example provides a necessary sense of security and confidence, allowing the reader to verify each stage of the logic. This ensures that the path of reasoning remains transparent for everyone.
Furthermore, this aligns with the principle articulated by mathematician David Hilbert (1862–1943), which the author has adapted for this context: “In physics, we must aim for clarity. It is often the detailed execution, and not the mere suggestion of a method, which proves decisive. A presentation should be judged by how little it presumes the reader already knows.”
1. The Time - Independent Schrödinger Equation for the Coulomb Potential
1. The Time - Independent Schrödinger Equation for the Coulomb Potential
The analytical solution of the time-independent Schrödinger equation for the Coulomb potential occupies a central place in both the historical development and the modern understanding of quantum mechanics. The separation of variables in spherical coordinates, the emergence of angular momentum quantization, and the reduction of the radial equation to a second-order ordinary differential equation form the mathematical backbone of the hydrogen atom problem [4]. Early insights into the structure of linear differential equations and series solutions can be traced back to Frobenius’ foundational work on power-series methods [1], which later became a standard tool in mathematical physics [2]. Schrödinger’s original formulation of quantum mechanics and his treatment of bound-state eigenvalue problems established the physical interpretation of these mathematical structures ([3], [4]), building on earlier semiclassical insights into atomic structure provided by Bohr’s model of the hydrogen atom [5]. Building on this foundation, Sommerfeld extended the theory by introducing additional quantum numbers and relativistic corrections, thereby accounting for fine-structure effects in atomic spectra. These developments provided an important bridge between early atomic models and the fully quantum-mechanical treatment that followed. Subsequent refinements clarified the role of angular momentum and symmetry in the hydrogen spectrum [6], while systematic textbook treatments placed the Coulomb problem within a broader theoretical framework ([7], [8]). Bohm’s treatment of the hydrogen atom in his 1951 monograph [17] is remarkably clear and thorough, providing a systematic derivation of the radial and angular solutions while highlighting the underlying physical structure of the atom in a way that remains insightful to this day. In later chapters, the Frobenius method will be discussed in detail and applied explicitly to the radial Schrödinger equation, highlighting its role in deriving the hydrogenic energy spectrum and wave functions.
Experimental and historical studies have provided a visual insight into the spatial structure of hydrogenic states. Recent experiments have revealed the intricate pattern of radial and angular nodes, directly imaging the nodal structures of hydrogen-like wave functions [9]. Recent historical analysis traced the evolution of pictorial representations of electron probability densities, highlighting the transition from classical orbits to quantum orbitals [10]. As early as 1931, illustrative depictions of the “electron cloud” for hydrogen-like atoms allowed physicists to gain an intuitive understanding of nodal surfaces and spatial distributions [11]. These studies complement the analytical radial and angular solutions by providing a tangible view of where the electron is likely to be found in three-dimensional space.
Experimental and historical studies have provided a visual insight into the spatial structure of hydrogenic states. Recent experiments have revealed the intricate pattern of radial and angular nodes, directly imaging the nodal structures of hydrogen-like wave functions [9]. Recent historical analysis traced the evolution of pictorial representations of electron probability densities, highlighting the transition from classical orbits to quantum orbitals [10]. As early as 1931, illustrative depictions of the “electron cloud” for hydrogen-like atoms allowed physicists to gain an intuitive understanding of nodal surfaces and spatial distributions [11]. These studies complement the analytical radial and angular solutions by providing a tangible view of where the electron is likely to be found in three-dimensional space.
The time-independent Schrödinger equation for a particle of mass in the attractive Coulomb potential (with ) is a second-order ordinary differential equation [41] (ODE):In this context, represents the radial distance from the nucleus, defined as the magnitude of the position vector, , in spherical coordinates.In the following main chapters, atomic units are adopted, so that the Coulomb potential is written as , and in later sections the hydrogen case is considered, for which .We use the standard separation ansatz in spherical coordinates (e.g. [4], [7], [8], [17]):,where denotes the angular eigenfunctions and . The Laplacian [39] in spherical coordinates has the form+sinθ+θ.Inserting , dividing by , and multiplying by separates the equation into radial and angular parts.The separation constant is crucial for the mathematical separation of the Schrödinger equation in spherical coordinates. Its value is determined by the angular part of the equation, specifically, by requiring that the solutions (θ,ϕ) (the spherical harmonics ([34], [44]), which describe the angular momentum and orbital shape) are well-behaved and single-valued. By extracting this constant from the angular equation and introducing it into the radial equation , we connect the orbital shape (angular momentum) to the electron’s distance from the nucleus (radius and energy). This procedure allows the angular part, which is independent of the potential , to be solved first, enabling the entire equation to be broken down into two solvable, independent parts. Physically, the term manifests in the radial equation as the centrifugal potential that prevents the electron from falling into the nucleus when . For orbitals (s-states), the centrifugal barrier vanishes, and these electrons possess their maximum of the configuration-space density at the nucleus (but not for ). In the Bohmian view, the particle is prevented from falling into the nucleus by the repulsive energy generated by the Quantum Potential , which becomes infinite as , effectively creating a dynamical barrier that balances the infinite attraction of the Coulomb force. Note: In this context, the terms hydrogen-like, hydrogenic, and Coulomb potential are used synonymously to refer to atomic systems whose structure, interactions, and energy levels resemble those of the hydrogen atom.
m
V(r)-
Z
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1.1 Radial Equation and Laguerre Polynomials
1.1 Radial Equation and Laguerre Polynomials
The radial dynamics are governed by a one-dimensional Schrödinger equation, where the use of a reduced wave function simplifies the kinetic energy term and leads to solutions defined by the quantization of energy and angular momentum, where the three-dimensional Laplace operator [39] decomposes into a purely radial part and an angular part, -, where is the squared orbital angular momentum operator acting only on the angular variables . The replacement of the operator with the term ℓ(ℓ+1) arises from the fact that the spherical harmonics (θ,ϕ) are eigenfunctions of the squared angular momentum operator, satisfying the eigenvalue equation ℓ(ℓ+1).The radial ODE reads:with eliminating the first derivative term.In atomic units ():or equivalentlyDefine for bound states (). The asymptotic solution for is . Substitutingreduces the equation to a confluent hypergeometric ODE whose polynomial solutions are the associated Laguerre polynomials [32]: (2κr),yielding the hydrogen-like radial functions: (r)∝(2κr).
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1.2 Angular Equation and Legendre Polynomials
1.2 Angular Equation and Legendre Polynomials
The angular dependence of the wave function is determined by the eigenfunctions of the squared angular momentum operator, which separates into two independent differential equations governing the polar and azimuthal orientations (e.g. [7], [8], [17]).The angular part satisfies Y(θ,ϕ)ℓ(ℓ+1)Y(θ,ϕ),where the squared angular momentum operator is-sinθ+θ.The polar function satisfiessinθ+ℓ(ℓ+1)-θΘ0.Introducing converts this into the associated Legendre ODE [43]:(1-)+ℓ(ℓ+1)-Θ0.The regular solutions on are the associated Legendre polynomials [43](x), and for these reduce to the Legendre polynomials (x). The full angular eigenfunctions (spherical harmonics) are(θ,ϕ)(cosθ).The azimuthal function satisfies the ODEΦ+Φ0, Φ(ϕ).
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2. Short Introduction to the Radial Equation
2. Short Introduction to the Radial Equation
4. Symbolic Implementation of the Frobenius Method for the Radial Hydrogen Equation
4. Symbolic Implementation of the Frobenius Method for the Radial Hydrogen Equation
6.0 Physical and Mathematical Justification of the Associated Legendre Factorization
6.0 Physical and Mathematical Justification of the Associated Legendre Factorization
6.4 Spherical Harmonics
6.4 Spherical Harmonics
To be continued in part 2.....
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Bohmian trajectories and the role of nodal surface manifolds in hydrogen eigenstates: Part 1
by Klaus von Bloh
Wolfram Community, STAFF PICKS, February 26, 2026
https://community.wolfram.com/groups/-/m/t/3645941
by Klaus von Bloh
Wolfram Community, STAFF PICKS, February 26, 2026
https://community.wolfram.com/groups/-/m/t/3645941