The radial probability distribution of the hydrogen 1s orbital is obtained by multiplying the probability density | ψ | 2 by the surface-area element of a sphere with radius r, 4π r 2 . By introducing a characteristic time through the Bohr velocity and interpreting distance as an effective waiting time, the resulting distribution can be mathematically reduced to a Gamma distribution with shape parameter 3. This structure is equivalent to the sum of three independent exponential waiting processes. The origin of the parameter “3” is traced to the factor r2, which arises from the three-dimensional spherical volume element. Thus, the Gamma-distribution structure is closely related to the dimensionality of space. Based on this observation, we propose a mathematical interpretation in which the electron reaching distance (r) corresponds to the accumulation of probability amplitudes along three independent spatial directions, (x), (y), and (z). In higher-dimensional spaces, the Coulomb potential is modified according to Gauss’s law, leading to the well-known “fall-to-the-center” behavior, where stable bound states at the Bohr radius can no longer be maintained. In such cases, the radial distribution no longer reduces to a Gamma distribution. These results suggest that only in three dimensions do the Coulomb potential ( ~1/r ), the exponential wave function ( ~ e ?r ), and the spatial volume element ( ~ r 2 ) become mutually balanced, giving rise to the Gamma-distribution structure.
The radial probability distribution of the hydrogen 1s orbital is obtained by multiplying the probability density | ψ | 2 by the surface-area element of a sphere with radius r, 4π r 2 . By introducing a characteristic time through the Bohr velocity and interpreting distance as an effective waiting time, the resulting distribution can be mathematically reduced to a Gamma distribution with shape parameter 3. This structure is equivalent to the sum of three independent exponential waiting processes. The origin of the parameter “3” is traced to the factor r2, which arises from the three-dimensional spherical volume element. Thus, the Gamma-distribution structure is closely related to the dimensionality of space. Based on this observation, we propose a mathematical interpretation in which the electron reaching distance (r) corresponds to the accumulation of probability amplitudes along three independent spatial directions, (x), (y), and (z). In higher-dimensional spaces, the Coulomb potential is modified according to Gauss’s law, leading to the well-known “fall-to-the-center” behavior, where stable bound states at the Bohr radius can no longer be maintained. In such cases, the radial distribution no longer reduces to a Gamma distribution. These results suggest that only in three dimensions do the Coulomb potential ( ~1/r ), the exponential wave function ( ~ e ?r ), and the spatial volume element ( ~ r 2 ) become mutually balanced, giving rise to the Gamma-distribution structure.
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.
The Bohr model of the hydrogen atom, despite its heuristic success in predicting discrete energy levels, is fundamentally incompatible with the probabilistic interpretation of modern quantum mechanics. In this work, we present a pedagogical derivation of Bohr’s results directly from the time-independent Schrödinger equation in three dimensions. By applying two successive geometric constraints, first, a genuine dimensional reduction to a 2D planar system (not a naive restriction of the spherical Laplacian), and second, a radial localization to a fixed radius, we demonstrate that the resulting radial-azimuthal equation yields the quantized Bohr orbits. We show that the energy levels En = −13.6/n2 eV and orbital radii r[n] = n² ao emerge naturally from the wavefunction normalization combined with the de Broglie standing-wave condition. Special attention is given to the correct form of the 2D Laplacian, which differs from the 3D Laplacian restricted to θ = π/2 by a term (1/r) ∂/∂r. This approach provides a clear conceptual bridge between the diffuse quantum cloud and the classical planetary picture.
In the framework of the quantum model for hydrogen atom, we use the wave functions provided by the solutions of the Schrödinger equation with Coulomb potential. Considering the radial and angular densities of the localization probability, we identify the representative quantum states that play a high-probability role in transitions for H-I spectra. Besides that, we obtain some other complementary results. So, we present the numerical tables for the radial distributions, the electron density functions and some observables like θ nodal angles, nodal surfaces, along with the ordered sequences of the average radii in the sub-shells associated with this n: ⟨r⟩4f = 18a0 < ⟨r⟩4d = 21a0 < ⟨r⟩4p = 23a0 < ⟨r⟩4s = 24a0 (a0 - the first Bohr radius). Further, we have added the significant scientific 3D-visualization of the orbitals shapes and describe them via the nodal values for θ angles.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
Abstract The Kinematics of Quantum Space: A Classical Proof of Orbital Stability and Trajectory Continuity (Part II) Building upon the dense, discrete, nonlocal medium defined in Part I (Quantum Space, density ratio ρ = 6.54×10¹²), we prove that the electron in a hydrogen atom follows a strictly deterministic, continuous, and classical trajectory. No probabilistic interpretation, wave function collapse, or irreducible uncertainty is required. 1. Orbital velocity – six independent derivations Starting from the Bohr radius (a₀ = 5.291772×10⁻¹¹ m) and the electron mass (mₑ = 9.109384×10⁻³¹ kg), we derive the ground state orbital velocity through six independent physical paths: Galilean kinematics (v = s/t) Bohr quantization condition (mₑ v a₀ = ħ) Fine‑structure constant (v = α c, α = 1/137.036) Kinetic energy (ionization energy 13.6 eV) Coulomb–centripetal force balance Direct experimental transit time (attosecond measurements) All six converge identically to: v=e22ϵ0h=αc≈2.188×106 m/sv=2ϵ0he2=αc≈2.188×106m/s The electron is a real satellite of the nucleus, not a probability cloud. 2. Orbital stability – zero radiation from destructive interference The classical objection (Larmor's formula predicts radiative collapse within ≈47 ps) assumes a point charge. For a standing wave charge distribution satisfying 2πr = nλ (Bohr condition), we compute the Liénard–Wiechert potentials and the far‑field Poynting vector explicitly: S=1μ0Erad×Brad=0,Ptotal=∮S⋅dA=0S=μ01Erad×Brad=0,Ptotal=∮S⋅dA=0 The ground state does not radiate. Energy is locked in the standing wave. Radiation occurs only during transitions (n→m), exactly as observed in atomic spectroscopy. 3. Quantum jump as a vertical ballistic throw – continuity proven The transition n=1 → n=2 (energy ΔE = 10.2 eV) is treated as a classical vertical throw in an effective constant acceleration field g_q: gq=FCoulomb(a0)me=kee2mea02≈9.043×1022 m/s2gq=meFCoulomb(a0)=mea02kee2≈9.043×1022m/s2 Jump height: h = r₂ - r₁ = 1.5875×10⁻¹⁰ mEffective initial velocity (constant‑g approximation): v₀ ≈ 5.36×10⁶ m/sTime of ascent: t_up = v₀/g_q ≈ 59 asTotal flight time: t_total = 118 as The position function y(t) = v₀ t - ½ g_q t² is continuous and differentiable (C¹ class). The illusion of an instantaneous "quantum jump" arises solely because t_total is shorter than the temporal resolution of most instruments (Δt_instr ≫ 118 as). The trajectory exists – we are simply not fast enough to see it. 4. Spin as toroidal vortex kinematics – no superluminal speed Spin (S_z = ±½ħ) is not a rigid sphere rotation. We model the electron as a toroidal charge distribution with major radius R. The no‑superluminal condition v = ωR < c yields a lower radius bound: Rmin=ℏ2mec≈1.93×10−13 mRmin=2mecℏ≈1.93×10−13m This is physically reasonable (≈300× smaller than the Bohr radius). Quantization to ±½ħ follows from resonance stability: only prograde (+) and retrograde (–) synchronous rotations are stable, analogous to the Moon (synchronous) and Venus (retrograde). 5. Measurement as ballistic collision – Heisenberg is technical, not fundamental A typical measurement photon (λ = 0.1 nm, E ≈ 12.4 keV) carries momentum: pphoton=hλ≈6.626×10−24 kg⋅m/spphoton=λh≈6.626×10−24kg⋅m/s The electron's orbital momentum is: pelectron=mev≈1.993×10−24 kg⋅m/spelectron=mev≈1.993×10−24kg⋅m/s Ratio: p_photon / p_electron ≈ 3.33. The photon delivers more momentum than the electron possesses in its orbit. Measurement is not passive observation – it is a ballistic collision (Compton scattering). The post‑collision electron momentum is fully determinable if both the energy and scattering angle of the outgoing photon are measured precisely. "Uncertainty" is a technical deficiency of incomplete measurement, not a law of nature. 6. Five falsifiable predictions Electron transit time across one Bohr diameter: 48.4 ± 5 as (testable with attosecond spectroscopy). n=1 → n=2 transition duration: 100–120 as (continuous, not instantaneous). Heisenberg product Δx·Δp → 0 as detector precision → ∞ (no fundamental lower bound). Bell inequality violations decrease as measurement impulse strength decreases (weak measurement regime). Residual "quantum randomness" equals unmodeled environmental force noise; full force accounting restores classical determinism. 7. Conclusion All calculations use only standard constants and classical equations (Newton, Maxwell, Coulomb, Compton). No new postulates are introduced. The electron's path exists, is continuous, and follows deterministic mechanics. Quantum mysticism – wave function collapse, irreducible uncertainty, non‑locality as magic – is replaced by scale‑inverted classical physics operating within the dense geometry of Quantum Space defined in Part I. Keywords: orbital stability, Larmor paradox, destructive interference, Poynting vector, quantum jump as ballistic throw, attosecond transit time, toroidal spin, Compton scattering measurement, Bell inequalities reinterpretation, classical determinism, falsifiable predictions, scale inversion. PACS: 03.65.Sq (Semiclassical theories), 03.65.Ta (Foundations), 31.15.xk (Classical treatments of atomic structure), 32.80.t (Atomic transitions), 42.50.Ct (Quantum optics, measurement).MSC: 81Q05 (Closed and approximate solutions), 70F05 (Two‑body problem, Kepler), 78A35 (Motion of charged particles), 83A05 (Special relativity).arXiv: quant-ph, physics.class-ph, physics.atom-ph.
The electronic transition energies from the electron clouds are the same as predicted by the Bohr model. The hydrogen atom, most probable radius of the first quantum mechanical electron cloud is identical to the radius of the first Bohr orbit. The quantum mechanical description goes much further, however. Not only does it predict a series of increasingly larger spherical electron clouds, but other electron cloud series as well. Because of their three dimensional shapes, these electron clouds sometimes are said to describe space states. The analogy to planetary orbits ends when we grasp the fact that quantum mechanical electron clouds are three-dimensional, where a Bohr orbit would be a circle lying entirely in a plane. Quantum mechanics also predicts the ionization energy of the hydrogen atom.