First part of this article:
Bohmian trajectories for the quantum harmonic oscillator
https://community.wolfram.com/groups/-/m/t/3561950
Bohmian trajectories for the quantum harmonic oscillator
https://community.wolfram.com/groups/-/m/t/3561950
Abstract: Following the detailed analysis of the one-dimensional harmonic oscillator and its precise trajectories in Part 1, this section extends the causal interpretation to the domain of quantum statistics. It offers a Bohmian perspective on classical Maxwell–Boltzmann, as well as quantum Bose–Einstein and Fermi–Dirac statistics. Unlike the Copenhagen interpretation, which treats particles as fundamentally indistinguishable and represented by symmetrized or antisymmetrized wavefunctions, the causal model of quantum mechanics provides a more intuitive picture. In this framework, ontological particles possess definite positions and dynamically determined velocities at all times, guided by a quantum potential or, equivalently, by the current density arising from the continuity equation of the many-particle system. This approach makes it possible to explore the statistical behavior of quantum particles while retaining the concept of individual trajectories. Within this framework, quantum statistics appear in a new light. The non-classical behavior of bosons and fermions does not originate from an intrinsic lack of individuality or from purely abstract symmetry postulates. Instead, it emerges from the correlations encoded in the many-particle wavefunction and its associated quantum potential. Entanglement is thereby revealed not as a mysterious action at a distance, but as a manifestation of the nonlocal correlations between particle trajectories encoded in the guiding wavefunction. This leads to an effective “repulsion” between fermions (the Pauli exclusion principle) and an effective “attraction” between bosons, reproducing Bose–Einstein and Fermi–Dirac statistics in a deterministic and visually coherent manner. The analysis presented here draws on the foundational insights of Peter Holland [7] and on illustrative models developed by Dewdney, Kyprianidis, and Vigier ([14a], [14b]).
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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Quit[]
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Use the following command, if you need to save/export graphs created by Mathematica into the current notebook directory:
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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.
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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 harmonic oscillator 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.
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 harmonic oscillator 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.
Figure
0
. Chris Dewdney, Jean-Pierre Vigier and David Bohm (images from [Pari Center, quanthomme.org and Pari Center]). 1. Wavefunctions for Two - Particle Systems in Different Quantum Statistics
1. Wavefunctions for Two - Particle Systems in Different Quantum Statistics
In conventional quantum mechanics, the symmetrization postulate (for bosons) or the antisymmetrization postulate (for fermions) is an axiom that enforces the indistinguishability of identical particles, leading directly to quantum statistics such as the Bose–Einstein or Fermi–Dirac distributions. In contrast, Bohmian mechanics does not require such postulates, since particles are treated as distinct entities with their own trajectories and are therefore not strictly indistinguishable in the classical sense ([8], [14a], [14b]). Nevertheless, the wavefunction still governs particle motion and correlations, giving rise to statistical behavior consistent with conventional quantum mechanics.For a system of two particles in a one-dimensional harmonic oscillator potential, quantum statistics in Bohmian mechanics emerge dynamically from the causal, nonlocal correlations mediated by the quantum potential ([7], [14a], [14b], [18]). These statistics are not a consequence of intrinsic indistinguishability, but rather of the dynamics encoded in the wavefunction and the resulting particle trajectories.To describe the two distinguishable particles in this system, we’ll use the following coordinate notation: The coordinate of the first particle. The coordinate of the second particle.This notation provides clarity when distinguishing between the positions of the two distinguishable particles in the subsequent wavefunction expressions.Maxwell-Boltzmann Statistics (MB):In the classical limit, the particles are distinguishable, and there is no non-local correlation. The total wavefunction is a product of individual wavefunctions, and the density is the product of the individual probability densities:Particle 1 and 2 without non-local correlation: (x,y,t)(x,t)·(y,t).The Particles move independently of each other and the wave density is the simple product ·.This describes independent particles without any quantum correlation.Bose-Einstein Statistics (BE):For bosons, the symmetric wavefunction generates non-local correlations between the particles, with an attractive quantum potential keeping the particles close to each other. The wavefunction is symmetrized: (x,y,t)[(x,t)(y,t)+(y,t)(x,t)].The quantum potential dynamically drives the particles together, and the normalization condition is typically written as: .Fermi-Dirac Statistics (FD):For fermions, the antisymmetric wavefunction generates non-local correlations with a repulsive quantum potential, which enforces the Pauli exclusion principle. The wavefunction is antisymmetrized: (x,y,t)[(x,t)(y,t)-(y,t)(x,t)].The repulsive interaction, arising from the antisymmetry of the wavefunction (the minus sign), prevents two fermions from occupying the same quantum state, dynamically enforcing the Pauli exclusion principle.
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1.1 Visualization in the (x,y) Configuration Space
1.1 Visualization in the () Configuration Space
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Visualizing trajectories in the () configuration space provides an especially clear way to illustrate the dynamical consequences of quantum statistics in the Bohmian framework. In this space, each axis represents the position of one particle, so the full state of a two-particle system is captured at every instant.
For Maxwell-Boltzmann statistics, the trajectories can cross freely, reflecting the independence of the particles. For Bose-Einstein statistics, trajectories tend to cluster along the diagonal (time axis), showing the constructive interference and the effective “attraction” induced by symmetrization. In contrast, Fermi-Dirac trajectories are repelled from the diagonal, highlighting the Pauli exclusion principle through destructive interference.
This type of visualization is particularly effective because it simultaneously shows the non-crossing property of bosons and fermions, the relative separation of maxima, and the strength of interference patterns. It provides an intuitive, geometric picture of how the symmetry or antisymmetry of the wavefunction is dynamically expressed in particle motion, making the abstract concept of quantum statistics tangible and accessible. Consequently, plotting in the () space complements numerical computations, allowing one to directly observe the statistical differences in Bohmian trajectories.
x,y
For Maxwell-Boltzmann statistics, the trajectories can cross freely, reflecting the independence of the particles. For Bose-Einstein statistics, trajectories tend to cluster along the diagonal
x(t)≈y(t)
This type of visualization is particularly effective because it simultaneously shows the non-crossing property of bosons and fermions, the relative separation of maxima, and the strength of interference patterns. It provides an intuitive, geometric picture of how the symmetry or antisymmetry of the wavefunction is dynamically expressed in particle motion, making the abstract concept of quantum statistics tangible and accessible. Consequently, plotting in the (
x,y
2. Maxwell–Boltzmann Statistics (MB): Independent Trajectories
2. Maxwell–Boltzmann Statistics (MB): Independent Trajectories
In the Bohmian picture, Maxwell-Boltzmann particles are treated as distinguishable and the total wavefunction is simply a product of single-particle states. Each particle follows its own trajectory independently, guided by the phase of the corresponding single-particle wavefunction. As a result, trajectories can cross freely in configuration space. There are no additional interference effects from particle exchange.
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para={ωx->1/2,ωy->1/2};(*Frequency*)xL=-4;(*Forthegraphics*)xR=4;yL=-4;yR=4;psi1[x_,t_,x0_]=/.m->1/.ℏ->1psi2[y_,t_,y0_]=/.m->1/.ℏ->1Print["Psi"]Psi[x_,y_,t_]=psi1[x,t,2]psi2[y,t,-2]//PowerExpand//SimplifyPrint["Complex Conjugate Psi"]PsiC[x_,y_,t_]=Psi[x,y,t]/.Complex[0,n_]->-Complex[0,n]Print["Wave density: "]Prob[x_,y_,t_]=Simplify[ComplexExpand[(Psi[x,y,t]PsiC[x,y,t]),TargetFunctions{Re,Im}],Assumptions{Element[_,Reals],ωx>0,ωy>0,t>0}]/.m->1/.ℏ->1Print["Initial position for t=0: ",Prob[x,y,0]/.para]Print["Total phasefunction: "]S[x_,y_,t_]=ArcTan[ComplexExpand[Im[Psi[x,y,t]]]/ComplexExpand[Re[Psi[x,y,t]]]//Simplify]Print["Velocity Sx: "](Sx[x_,y_,t_]=D[S[x,y,t],x]//Factor//TrigReduce//Simplify)//TimingPrint["Velocity Sx: "](Sy[x_,y_,t_]=D[S[x,y,t],y]//Factor//TrigReduce//Simplify)//Timing
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initialMBplot=DensityPlot[Chop[Prob[x,y,0]/.para],{x,xL,xR},{y,yL,yR},PlotPoints->60,Mesh->False,Frame->True,FrameLabel->{"x","y"},ColorFunction->"BlueGreenYellow",PlotLabel->Style["Initial wave density",Bold,14]];SetDirectory[NotebookDirectory[]];directory="AnimationsFiles";filename="initialMBplot_5_10_2025.png";filePath=NotebookDirectory[]<>FileNameJoin[{directory,filename}];Export[filePath,initialMBplot]Import[filePath]
A concise notation for determining the velocity vector can be given by the following equation, but the calculation time increases significantly for large terms.
The video shows the density of the wavefunction and the possible trajectories depending on the starting points and current positions.
Each particle follows its own trajectory independently, guided by the phase of the corresponding single-particle wavefunction. As a result, trajectories can cross freely in configuration space. There are no additional interference effects from particle exchange.
3. Bose–Einstein Symmetrization (BE) and Trajectory Clustering
3. Bose–Einstein Symmetrization (BE) and Trajectory Clustering
For bosons, the wavefunction is symmetrized with respect to particle exchange, leading to constructive interference whenever particles occupy similar positions. In Bohmian mechanics, this symmetry produces trajectories that never cross and tend to cluster along regions where the wavefunction amplitude is large. Interference effects are strongest in this case, producing pronounced trajectory bundling and sharp density maxima along the diagonal in configuration space.
The video shows the density of the wavefunction and the possible trajectories depending on the starting points and current positions. The animation clearly shows that interference is much stronger in this case, causing the particles to move closer together and even cluster. This is in sharp contrast to the previous example (MB) and visually highlights the effect of bosonic symmetry.
The displayed figure illustrates the trajectories of the bosons in configuration space. The pronounced bundling of the trajectories is a direct consequence of the constructive interference characteristic of symmetric wavefunctions. This clustering behavior, which forms distinct interference patterns or fringes, visually underscores the fundamental difference between bosonic and fermionic systems.
4. Fermi–Dirac (FD) Antisymmetrization and Trajectory Avoidance
4. Fermi–Dirac (FD) Antisymmetrization and Trajectory Avoidance
Fermionic wavefunctions are antisymmetric under particle exchange, leading to nodes along the diagonal where two particles coincide. In Bohmian terms, this antisymmetry enforces non-crossing trajectories and a mutual avoidance of particles in configuration space. The interference effects are weaker than in the bosonic case, resulting in smoother trajectories, density maxima that are further apart, and less pronounced modulation in the configuration space. This makes the suppression of overlap visually evident and highlights the dynamic expression of the Pauli exclusion principle.
The video shows the density of the wavefunction and the possible trajectories depending on the starting points and current positions. The animation clearly shows the strong influence of antisymmetry, forcing the particles to move away from each other. This is in sharp contrast to the previous example and visually highlights the effect of the Pauli exclusion principle, which prevents particles from occupying the same position and results in a mutual avoidance.
The graphic illustrates how the antisymmetric wavefunction prevents the particles from approaching each other. This antisymmetry gives rise to destructive interference at regions where the particle coordinates coincide. The resulting nodes effectively prohibit the particle trajectories from crossing, thereby visually enforcing the Pauli exclusion principle in the system’s configuration space.
5. Summary of Bohmian Trajectories and the Dynamical Origins of Quantum Statistics
5. Summary of Bohmian Trajectories and the Dynamical Origins of Quantum Statistics
In the Bohmian interpretation, the statistical differences between Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac particles are vividly expressed through their trajectories, since the symmetry of the many-particle wavefunction directly shapes their motion. Maxwell-Boltzmann particles, described by unsymmetrized product states, have trajectories that cross freely. By contrast, the symmetric wavefunction of bosons enhances constructive interference, causing clustering, while the antisymmetric wavefunction of fermions enforces nodes in configuration space, forcing trajectories apart and thus dynamically realizing the Pauli exclusion principle.
As emphasized by Holland [7] and Dewdney, Kyprianidis, and Vigier ([14a], [14b]), these patterns are not abstract statistical postulates but the direct outcome of quantum statistics and nonlocal dynamics. The trajectory plots also reveal a further nuance not highlighted in earlier analyses: in the fermionic case, antisymmetry suppresses interference amplitudes relative to the bosonic case. As a result, fermionic patterns appear smoother and less corrugated, with broader maxima and weaker interference fringes, whereas bosonic symmetrization amplifies constructive interference and produces more pronounced clustering. This shows that exchange statistics affect not only the topology of trajectories but also the degree to which interference structures are dynamically expressed.
A further conceptual lesson emerges from these trajectories. In Bohmian mechanics, particles are always distinct entities with well-defined positions and paths — or, as Sheldon Goldstein memorably put it, “Particles are particles” (personal communication, Foundations and Farewell to Detlef Dürr, LMU Munich, November 2016). Exchange effects are therefore not a fundamental property of the particles themselves but arise dynamically from the symmetry of the guiding wavefunction.
Finally, these distinctive patterns for Bose-Einstein and Fermi-Dirac systems are not due to classical forces acting between the particles. They result from the nonlocal structure of the quantum potential, through which the motion of each particle is instantaneously correlated with that of all others. The symmetrized or antisymmetrized form of the wavefunction therefore governs the dynamics of the entire system at once. For Maxwell-Boltzmann particles, by contrast, the absence of exchange symmetry means that such exchange-induced nonlocal correlations do not arise. These three distinct causal pictures were first dynamically illustrated by Dewdney, Kyprianidis, and Vigier in [14a] and [14b], arguing that quantum statistics arise from the causal, nonlocal influence of the wave function rather than from axiomatic postulates of indistinguishability.
As emphasized by Holland [7] and Dewdney, Kyprianidis, and Vigier ([14a], [14b]), these patterns are not abstract statistical postulates but the direct outcome of quantum statistics and nonlocal dynamics. The trajectory plots also reveal a further nuance not highlighted in earlier analyses: in the fermionic case, antisymmetry suppresses interference amplitudes relative to the bosonic case. As a result, fermionic patterns appear smoother and less corrugated, with broader maxima and weaker interference fringes, whereas bosonic symmetrization amplifies constructive interference and produces more pronounced clustering. This shows that exchange statistics affect not only the topology of trajectories but also the degree to which interference structures are dynamically expressed.
A further conceptual lesson emerges from these trajectories. In Bohmian mechanics, particles are always distinct entities with well-defined positions and paths — or, as Sheldon Goldstein memorably put it, “Particles are particles” (personal communication, Foundations and Farewell to Detlef Dürr, LMU Munich, November 2016). Exchange effects are therefore not a fundamental property of the particles themselves but arise dynamically from the symmetry of the guiding wavefunction.
Finally, these distinctive patterns for Bose-Einstein and Fermi-Dirac systems are not due to classical forces acting between the particles. They result from the nonlocal structure of the quantum potential, through which the motion of each particle is instantaneously correlated with that of all others. The symmetrized or antisymmetrized form of the wavefunction therefore governs the dynamics of the entire system at once. For Maxwell-Boltzmann particles, by contrast, the absence of exchange symmetry means that such exchange-induced nonlocal correlations do not arise. These three distinct causal pictures were first dynamically illustrated by Dewdney, Kyprianidis, and Vigier in [14a] and [14b], arguing that quantum statistics arise from the causal, nonlocal influence of the wave function rather than from axiomatic postulates of indistinguishability.
References
References
A heartfelt thank you to the Wolfram Team for their invaluable support in developing my programs with Mathematica. I am especially grateful to Vitaliy Kaurov and Ahmed Elbanna for their expert guidance and patience throughout the entire project.
A selection of animations can be seen in some of the author’s videos (Other Internet Sources: [1-6]).
A selection of animations can be seen in some of the author’s videos (Other Internet Sources: [1-6]).
Wolfram Demonstrations by Klaus von Bloh
Wolfram Demonstrations by Klaus von Bloh
Wolfram Community posts by Other People
Wolfram Community posts by Other People
Wolfram Community posts by Klaus von Bloh
Wolfram Community posts by Klaus von Bloh
Articles
Articles
I have used square brackets for in-text citations and included a link to the original source for each reference. Some of the formulations, mathematical equations, and translations within this essay were generated or verified with the assistance of AI tools such as ChatGPT, Google Gemini, and DeepL. The development of certain functions and Mathematica codes, including their underlying mathematical principles and the verification of equations, also benefited from the support of these intelligent tools, but any errors or inaccuracies remain solely my responsibility. I have incorporated some sentences verbatim from my Wolfram demonstrations without properly attributing them as quotations.
[1] G. Frobenius, “Ueber die Integration der linearen Differentialgleichungen durch Reihen.” Journal für die reine und angewandte Mathematik, 76, 1873 pp. 214-235. EuDML Documents
or Göttinger Digitalisierungszentrum, (Language: German).
or Göttinger Digitalisierungszentrum, (Language: German).
[2] E. Schrödinger, “A Method of Determining Quantum-Mechanical Eigenvalues and Eigenfunctions,” Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, 46, 1939 pp. 9-16. JSTORE.
[3] E. Schrödinger, “Quantisierung als Eigenwertproblem (Zweite Mitteilung)”. Annalen der Physik 384(6) 1926 pp. 489-527. doi: 10.1002/andp.19263840602,(Language: German).
[4] L. Infeld and T. E. Hull,”The Factorization Method,” Reviews of Modern Physics, 23(21), 1951 pp. 22-68. doi:10.1103/RevModPhys.23.21
[5] L. I. Schiff, Quantum Mechanics, 2nd ed., New York: McGraw-Hill, 1955.
[6] E. Merzbacher, Quantum Mechanics, New York: Wiley, 1961.
[7] P. R . Holland, The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics, New York: Cambridge University Press, 1993. Cambridge University Press.
[8] D. Bohm, Quantum Theory, New York: Dover Publications (Reprint of the Prentice-Hall), 1951.Dover Publications.
[9] D. Bohm, “A suggested Interpretation of the Quantum Theory in Terms of Hidden Variables, I,” Physical Review 85 (2), 1952 pp. 166–179. doi:10.1103/PhysRev.85.166.
[10] D. Bohm, “A suggested Interpretation of the Quantum Theory in Terms of Hidden Variables, II,” Physical Review 85 (2), 1952 pp. 180–193. doi:10.1103/PhysRev.85.18.
[11] D. Bohm and B. J. Hiley, The Undivided Universe, New York: Routledge, 1993. doi:10.4324/9780203980385.
[12] R. E. Wyatt, Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics, Springer Science & Business Media, 2005. ISBN: 978-0-387-22964-5.
[13] D.Bohm and J.P. Vigier, “Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid with Irregular Fluctuations,” Physical Review 96(20), 1954 pp. 208–216. doi:10.1103/PhysRev.96.208.
[14a] C. Dewdney, A Kyprianidis and J.P. Vigier, “Illustration of the causal model of quantum statistics,” Journal of Physics A: Mathematical and General 17(14), 1984 pp. 741–744. doi: 10.1088/0305-4470/17/14/003.
[14b] J.P. Vigier, “Causal stochastic interpretation of quantum statistics,” Pramana-Journal of Physics 25, 1985 pp. 397–418. doi: 10.1007/BF02846766.
[15] S. Goldstein. “Bohmian Mechanics.” The Stanford Encyclopedia of Philosophy. (Jan 4, 2025) plato.stanford.edu/entries/qm-bohm.
[17] J. Faye, “Copenhagen Interpretation of Quantum Mechanics,” The Stanford Encyclopedia of Philosophy (Summer 2024 Edition), E. N. Zalta & U. Nodelman (eds.), https://plato.stanford.edu/entries/qm-copenhagen/.
[18] J. P. da Silva and R. O. Lenzi and C. H. de Oliveira Barreto and D. G. Tedesco, “Two particle dynamics in a Calogero-Moser type potential: Bohm and non-locality,” Revista Brasileira de Iniciação Científica, 11, 2024 pp 1-28. e024047, (Language: Portuguese).
[19] Wikipedia. “Conjugate variables.”(Jan 4, 2025) https://en.wikipedia.org/wiki/Conjugate_variables.
[20] E. Weisstein “Dirac Notation.” From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DiracNotation.html
[23] Wikipedia. “Hermite polynomials.”(Jul 4, 2025) https://en.wikipedia.org/wiki/Hermite_polynomials.
[26] Wikipedia. “Rodrigues’ formula.”(Jul 4, 2025) https://en.wikipedia.org/wiki/Rodrigues%27_formula.
[29] Wikipedia. “Canonical commutation relation.”(Jan 4, 2025) https://en.wikipedia.org/wiki/Canonical_commutation_relation .
The pictures are taken from Wikipedia (the free encyclopedia ) and Alchetron (free social encyclopedia for the world): David Bohm, Louis de Broglie, Erwin Schrödinger and Erwin Madelung , Ferdinand Georg Frobenius , Leopold Infeld (Aug 8, 2023).
Other Demonstrations and internet sources
Other Demonstrations and internet sources
YouTube Internet Sources:
[1] K. von Bloh. The 3D Harmonic Oscillator in Spherical Coordinates in the Causal Interpretation of Quantum Theory. [Video]. (Aug 28, 2025). https://youtu.be/s0SP4osQa5c
[2] K. von Bloh. Continuous Transitions Between Classical and Quantum Motion for a Harmonic Oscillator Potential. [Video]. (Aug 28, 2025). https://youtu.be/5qvGZF6I2Hc
[3] K. von Bloh. Continuous Transition Between Classical and Quantum Motion for an Harmonic Oscillator Potential II. [Video]. (Aug 28, 2025). https://youtu.be/Fuxf4khXv0E
[4] K. von Bloh. 3D Quantum Trajectory for a Particle in a Harmonic Potential. [Video]. (Jan 28, 2025). https://youtu.be/BBpGEGZ9Zt0
[5] K. von Bloh. The Quantum Harmonic Oscillator with Time-Dependent Boundary Condition in the Causal Interpretation. [Video]. (Aug 28, 2025). https://youtu.be/REtj1QaD6ks
[6] K. von Bloh. Exploring the 2D Schrödinger Equation in a Ring-Shaped Potential using Bohmian Mechanics. [Video]. (Aug 28, 2025). https://youtu.be/NF_UfGOYpHw
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Causal origin of quantum statistics via Bohmian trajectories
by Klaus von Bloh
Wolfram Community, STAFF PICKS, October 21, 2025
https://community.wolfram.com/groups/-/m/t/3563610
by Klaus von Bloh
Wolfram Community, STAFF PICKS, October 21, 2025
https://community.wolfram.com/groups/-/m/t/3563610