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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

Do attractive scattering potentials concentrate particles at the origin in one, two and three dimensions? II. Potentials diverging as — C/rq

G. Barton

Published Aug 8, 1983• 2 citations
Publisher

Abstract

Abstract In classical mechanics (c.m.), and near the semi-classical limit h→0 of quantum mechanics (s.c.l.), the enhancement factors α ≡ ρ0/ρ∞ are found for scattering by attractive central potentials U(r); here ρ0,∞ (and v0,∞) are the particle densities (and speeds) at the origin and far upstream in the incident beam. For finite potentials (U(0) > — ∞), and when there are no turning points, the preceding paper found both in c.m., and near the s.c.l. (which then covers high v∞ ), α1 = v∞/v0, α2 = 1, α3 = v0/v∞ respectively in one dimension (1D), 2D and 3D. The argument is now extended to potentials (still without turning points), where U(r→0) ~ ─ C/rq, with 0 < q < 1 in ID (where r ≡ |x|), and 0 < q < 2 in 2D and 3D, since only for such q can classical trajectories and quantum wavefunctions be defined unambiguously. In c.m., α1(c.m.) = 0, α3(c.m.) = ∞, and α2(c.m.) = (1 —½q) N, where N = [integer part of (1 ─½q)-1]is the number of trajectories through any point (r, θ) in the limit r → 0. All features of U(r) other than q are irrelevant. Near the s.c.l. (which now covers low v∞) a somewhat delicate analysis is needed, matching exact zero-energy solutions at small r to the ordinary W.K.B. approximation at large r; for small v∞/u it yields the leading terms α1(s.c.l.) = Λ1(q)v∞/u, α2(s.c.I) = (1 ─½q)-1, α3(s.c.l.)= Λ3(q)u/v∞, where u ≡ (C/hqm1-q)1/(2-q) is a generalized Bohr velocity. Here Λ1,3 are functions of q alone, given in the text; as q →0 the α (s.c.l.) agree with the α quoted above for finite potentials. Even in the limit h = 0, α2 (s.c.l.) and α2 (c.m.) differ. This paradox in 2D is interpreted loosely in terms of quantal interference between the amplitudes corresponding to the N classical trajectories. The Coulomb potential ─ C/r is used as an analytically soluble example in 2D as well as in 3D. Finally, if U(r) away from the origin depends on some intrinsic range parameter α(e.g. U = ─ Cexp (─r/a)/rq), and if, near the s.c.l., v∞/u is regarded as a function not of h but more realistically of v∞, then the expressions α (s.c.l.) above apply only in an intermediate range 1/a ≪ mv∞/h ≪ (mC/h2)1/(2-q) which exists only if a ≫ (h2/mC)1/(2-q)).

DOI
10.1098/rspa.1983.0090

Related papers

Do attractive scattering potentials concentrate particles at the origin in one, two and three dimensions? III. High energies in quantum mechanics

G. Barton

Abstract The enhancement factor αD ≡ 1+ δαD is defined (in D dimensions) as the ratio of the particle density at the origin to the density far upstream in the incident beam. At high incident momentum p (and for regular potentials) the first Born approximation is known to be adequate in 3D and 1D, and is assumed to be adequate also in 2D; it entails δα1 = ─δα3 if U(r) in 3D equals U(x) in 1D when r = x. For central potentials U(r) with U(0) finite, previous work implies δα3 ~ ─δα1 ~ ─mU(0)/p2, and δα2 = 0 to the same order h°. It is shown that if U(r→0) ~ U(0) + rU'(0) + ..., then δα2 ~ (2mU'(0)h/p3)(π/16), the value of U(0) being irrelevant. IfU(r→0) ~ ─C/rq, with 0 < q < 2, then δα3 ~ ─δα1 ~ (2mC/hqp2-q) π½Γ(1-½ q)/2Γ(½+½q); and, with -1 < q < 2, δα2 ~ (2mC/hqp2-q)π½Γ2(1-½q)/2Γ(½q)Γ(3/2-½q). The only mathematics needed in 3D and 1D is the standard asymptotic estimation of Fourier integrals; but in 2D one needs to develop corresponding methods for integrals where the sine or cosine has been replaced by a product J0Y0 of two Bessel functions.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Aug 8, 1983•8 citations

Rutherford scattering in two dimensions

G. Barton

Parabolic coordinates allow Schrödinger’s equation to be solved analytically for a particle confined to the xy plane and incident with velocity v on a potential C/r. In sharp contrast to the three-dimensional case, the cross section assumes its classical value only in the rationally foreseeable classical regime ‖C‖/ℏv≫1, but not otherwise; and in this low-velocity regime the enhancement factor (probability density at the origin) for an attractive potential tends to 2 (independently of the value of ‖C‖), instead of diverging as in three dimensions like some inverse power of v.

American Journal of Physics•May 1, 1983•39 citations

Near-threshold scattering, quantum reflection, and quantization in two dimensions

Florian Arnecke, Harald Friedrich, Patrick Raab

We study the near-threshold behavior of scattering phase shifts, quantum reflection amplitudes, and quantization functions for systems described by a circularly symmetric potential in two spatial dimensions. In contrast to the three-dimensional case, the centrifugal potential $({m}^{2}\ensuremath{-}\frac{1}{4}){\ensuremath{\hbar}}^{2}∕(2M{r}^{2})$ is nonvanishing and even attractive for $s$ waves, $m=0$, and the leading near-threshold energy dependence of phase shifts and amplitudes for scattering and quantum reflection is logarithmic in this case. The near-threshold behavior of the $s$-wave phase shifts and amplitudes can nevertheless be characterized by a well-defined scattering length (for potentials falling off faster than $1∕{r}^{2}$) and an effective range (for potentials falling off faster than $1∕{r}^{4}$). For a potential with a bound state at energy ${E}_{n}=\ensuremath{-}{\ensuremath{\hbar}}^{2}{\ensuremath{\kappa}}_{n}^{2}∕(2M)$ very near threshold, the scattering length obeys $a\stackrel{{\ensuremath{\kappa}}_{n}\ensuremath{\rightarrow}0}{\ensuremath{\sim}}2{e}^{\ensuremath{-}{\ensuremath{\gamma}}_{\mathrm{E}}}∕{\ensuremath{\kappa}}_{n}+O({\ensuremath{\kappa}}_{n})$, with no term of order ${\ensuremath{\kappa}}_{n}^{0}$---in contrast to the three-dimensional case. Analytical results are derived for homogeneous potentials, and the necessary modification of the effective-range expansion is given for potentials proportional to $1∕{r}^{4}$. For $m\ensuremath{\ne}0$ we give analytical expressions for the near-threshold behavior of phase shifts as well as scattering and quantum reflection amplitudes, which are generally valid, even when $m$ is neither integer nor half-integer.

Physical Review A•Nov 24, 2008•15 citations

Two- and three- dimensional few-body systems in the universal regime

F. F. Bellotti

Macro properties of cold atomic gases are driven by few-body correlations, even if the gas has thousands of particles. Quantum systems composed of two and three particles with attractive zero\=/range pairwise interactions are considered for general masses and interaction strengths in two and three dimensions (2D and 3D). The Faddeev decomposition is used to derive the equations for the bound state, which is the starting point for the investigation of universal properties of few\=/body systems, i.e. those that all potentials with the same physics at low energy are able to describe in a model\=/independent form. In 2D, the number of bound states in a three\=/body system increases without bound as the mass of one particle becomes much lighter than the other two. The analytic form of an effective potential between the heavy particles explains the mass\=/dependence on the number of bound energy levels. An exact analytic expression for the large\=/momentum asymptotic behaviour of the spectator function in the Faddeev equation is presented. The spectator function and its asymptotic form define the two- and three\=/body contact parameters. The two\=/body parameter is found to be independent of the quantum state in some specific 2D systems. The 2D and 3D momentum distributions have a distinct sub\=/leading form whereas the 3D term depends on the mass of the particles. A model that interpolates between 2D and 3D is proposed and a sharp transition in the energy spectrum of three-body systems is found.

arXiv:1411.2937•Nov 11, 2014•0 citations

The Wave-Optical Description Of Potential Scattering

Peter Fröbrich, R. Lipperheide

Abstract In the preceding chapter we have discussed the semiclassical approximation to quantum scattering theory, where the dynamics is described in terms of classical trajectories. By analytic continuation of these trajectories into the complex plane the treatment has even been extended to the scattering by complex, i.e. absorptive, potentials.

•Jul 4, 1996•0 citations

The Kinematics of Quantum Space A Classical Proof of Orbital Stability and Trajectory Continuity (Part II)

Mario Kozar

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.

•May 12, 2026•0 citations
Abstract In classical mechanics (c.m.), and near the semi-classical limit h→0 of quantum mechanics (s.c.l.), the enhancement factors α ≡ ρ0/ρ∞ are found for scattering by attractive central potentials U(r); here ρ0,∞ (and v0,∞) are the particle densities (and speeds) at the origin and far upstream in the incident beam. For finite potentials (U(0) > — ∞), and when there are no turning points, the preceding paper found both in c.m., and near the s.c.l. (which then covers high v∞ ), α1 = v∞/v0, α2 = 1, α3 = v0/v∞ respectively in one dimension (1D), 2D and 3D. The argument is now extended to potentials (still without turning points), where U(r→0) ~ ─ C/rq, with 0 < q < 1 in ID (where r ≡ |x|), and 0 < q < 2 in 2D and 3D, since only for such q can classical trajectories and quantum wavefunctions be defined unambiguously. In c.m., α1(c.m.) = 0, α3(c.m.) = ∞, and α2(c.m.) = (1 —½q) N, where N = [integer part of (1 ─½q)-1]is the number of trajectories through any point (r, θ) in the limit r → 0. All features of U(r) other than q are irrelevant. Near the s.c.l. (which now covers low v∞) a somewhat delicate analysis is needed, matching exact zero-energy solutions at small r to the ordinary W.K.B. approximation at large r; for small v∞/u it yields the leading terms α1(s.c.l.) = Λ1(q)v∞/u, α2(s.c.I) = (1 ─½q)-1, α3(s.c.l.)= Λ3(q)u/v∞, where u ≡ (C/hqm1-q)1/(2-q) is a generalized Bohr velocity. Here Λ1,3 are functions of q alone, given in the text; as q →0 the α (s.c.l.) agree with the α quoted above for finite potentials. Even in the limit h = 0, α2 (s.c.l.) and α2 (c.m.) differ. This paradox in 2D is interpreted loosely in terms of quantal interference between the amplitudes corresponding to the N classical trajectories. The Coulomb potential ─ C/r is used as an analytically soluble example in 2D as well as in 3D. Finally, if U(r) away from the origin depends on some intrinsic range parameter α(e.g. U = ─ Cexp (─r/a)/rq), and if, near the s.c.l., v∞/u is regarded as a function not of h but more realistically of v∞, then the expressions α (s.c.l.) above apply only in an intermediate range 1/a ≪ mv∞/h ≪ (mC/h2)1/(2-q) which exists only if a ≫ (h2/mC)1/(2-q)).
Abstract The enhancement factor αD ≡ 1+ δαD is defined (in D dimensions) as the ratio of the particle density at the origin to the density far upstream in the incident beam. At high incident momentum p (and for regular potentials) the first Born approximation is known to be adequate in 3D and 1D, and is assumed to be adequate also in 2D; it entails δα1 = ─δα3 if U(r) in 3D equals U(x) in 1D when r = x. For central potentials U(r) with U(0) finite, previous work implies δα3 ~ ─δα1 ~ ─mU(0)/p2, and δα2 = 0 to the same order h°. It is shown that if U(r→0) ~ U(0) + rU'(0) + ..., then δα2 ~ (2mU'(0)h/p3)(π/16), the value of U(0) being irrelevant. IfU(r→0) ~ ─C/rq, with 0 < q < 2, then δα3 ~ ─δα1 ~ (2mC/hqp2-q) π½Γ(1-½ q)/2Γ(½+½q); and, with -1 < q < 2, δα2 ~ (2mC/hqp2-q)π½Γ2(1-½q)/2Γ(½q)Γ(3/2-½q). The only mathematics needed in 3D and 1D is the standard asymptotic estimation of Fourier integrals; but in 2D one needs to develop corresponding methods for integrals where the sine or cosine has been replaced by a product J0Y0 of two Bessel functions.
Parabolic coordinates allow Schrödinger’s equation to be solved analytically for a particle confined to the xy plane and incident with velocity v on a potential C/r. In sharp contrast to the three-dimensional case, the cross section assumes its classical value only in the rationally foreseeable classical regime ‖C‖/ℏv≫1, but not otherwise; and in this low-velocity regime the enhancement factor (probability density at the origin) for an attractive potential tends to 2 (independently of the value of ‖C‖), instead of diverging as in three dimensions like some inverse power of v.
We study the near-threshold behavior of scattering phase shifts, quantum reflection amplitudes, and quantization functions for systems described by a circularly symmetric potential in two spatial dimensions. In contrast to the three-dimensional case, the centrifugal potential $({m}^{2}\ensuremath{-}\frac{1}{4}){\ensuremath{\hbar}}^{2}∕(2M{r}^{2})$ is nonvanishing and even attractive for sss waves, m=0m=0m=0, and the leading near-threshold energy dependence of phase shifts and amplitudes for scattering and quantum reflection is logarithmic in this case. The near-threshold behavior of the sss-wave phase shifts and amplitudes can nevertheless be characterized by a well-defined scattering length (for potentials falling off faster than 1∕r21∕{r}^{2}1∕r2) and an effective range (for potentials falling off faster than 1∕r41∕{r}^{4}1∕r4). For a potential with a bound state at energy ${E}_{n}=\ensuremath{-}{\ensuremath{\hbar}}^{2}{\ensuremath{\kappa}}_{n}^{2}∕(2M)$ very near threshold, the scattering length obeys $a\stackrel{{\ensuremath{\kappa}}_{n}\ensuremath{\rightarrow}0}{\ensuremath{\sim}}2{e}^{\ensuremath{-}{\ensuremath{\gamma}}_{\mathrm{E}}}∕{\ensuremath{\kappa}}_{n}+O({\ensuremath{\kappa}}_{n})$, with no term of order ${\ensuremath{\kappa}}_{n}^{0}$---in contrast to the three-dimensional case. Analytical results are derived for homogeneous potentials, and the necessary modification of the effective-range expansion is given for potentials proportional to 1∕r41∕{r}^{4}1∕r4. For $m\ensuremath{\ne}0$ we give analytical expressions for the near-threshold behavior of phase shifts as well as scattering and quantum reflection amplitudes, which are generally valid, even when mmm is neither integer nor half-integer.
Macro properties of cold atomic gases are driven by few-body correlations, even if the gas has thousands of particles. Quantum systems composed of two and three particles with attractive zero\=/range pairwise interactions are considered for general masses and interaction strengths in two and three dimensions (2D and 3D). The Faddeev decomposition is used to derive the equations for the bound state, which is the starting point for the investigation of universal properties of few\=/body systems, i.e. those that all potentials with the same physics at low energy are able to describe in a model\=/independent form. In 2D, the number of bound states in a three\=/body system increases without bound as the mass of one particle becomes much lighter than the other two. The analytic form of an effective potential between the heavy particles explains the mass\=/dependence on the number of bound energy levels. An exact analytic expression for the large\=/momentum asymptotic behaviour of the spectator function in the Faddeev equation is presented. The spectator function and its asymptotic form define the two- and three\=/body contact parameters. The two\=/body parameter is found to be independent of the quantum state in some specific 2D systems. The 2D and 3D momentum distributions have a distinct sub\=/leading form whereas the 3D term depends on the mass of the particles. A model that interpolates between 2D and 3D is proposed and a sharp transition in the energy spectrum of three-body systems is found.
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.