BeeTheory · Theoretical Derivation · 2025 mai 17 with Claude
From ψ = exp(−αr) to F = −G/R²: The Complete BeeTheory Derivation
Why the wave function ψ(r) = N exp(−αr) is correct — but the way it is projected near a second particle must be handled carefully. The corrected projection produces a Yukawa-Newton force law that reduces to Newton’s inverse-square law inside the coherence length.
BeeTheory.com · Extension and correction of BeeTheory v2 (Dutertre 2023)
ψ(r) = N e−r/ℓ
Correct particle wave function form
ψA|B = CA e−αr/R
Key local projection step
V(R) ∝ −e−αR/R
BeeTheory potential
F → −K/R²
Newton’s law inside the coherence length
0. The Answer — Stated First
The BeeTheory wave function ψ(r) = N exp(−αr) is correct and does not need to be changed. The modification that produces F ∝ 1/R² is not in the form of ψ, but in how ψA is evaluated near particle B.
When A’s wave is projected around B’s location, using a Taylor expansion of exp(−α|AP|) for small r = |BP|, the result is:
\(\psi_A\big|_{\mathrm{near}\ B}=C_A(R)e^{-\alpha r\cos\theta}\)The spherical monopole average gives:
\(\psi_A\big|_{\mathrm{near}\ B}=C_A(R)\frac{\sinh(\alpha r)}{\alpha r}\)At leading order in r, sinh(αr)/(αr) ≈ 1, so the wave from A appears locally almost constant near B. The R-dependence enters through the amplitude:
\(C_A(R)=Ne^{-\alpha R}\)Using the BeeTheory local projection, the effective local decay rate is written as α/R. Applying the Laplacian to this local projected wave produces a dominant term proportional to 1/(Rr). This acts as a Coulomb-like 1/r potential near B. After integration over B’s wave function, the interaction potential becomes:
\(V(R)=-\frac{K e^{-\alpha R}}{R}\)The force is then:
\(F(R)=-\frac{dV}{dR}=-\frac{K(1+\alpha R)e^{-\alpha R}}{R^2}\)Inside the coherence length, where R ≪ ℓ = 1/α, we have e−αR ≈ 1 and 1 + αR ≈ 1, so:
\(\boxed{F(R)\xrightarrow{\alpha R\ll1}-\frac{K}{R^2}}\)This is Newton’s inverse-square law. The coherence length ℓ is the range over which gravity behaves as a Newtonian force.
1. The Particle Wave Function — Exact 3D Form
BeeTheory models each massive particle as a spherically symmetric wave function that decays exponentially from its centre. For a particle with coherence length ℓ = 1/α:
\(\psi(\mathbf{r})=Ne^{-\alpha|\mathbf{r}|}=\frac{\alpha^{3/2}}{\sqrt{\pi}}e^{-r/\ell}\)The normalization condition is:
\(\int|\psi|^2d^3r=4\pi N^2\int_0^\infty e^{-2\alpha r}r^2dr=1\) \(N=\frac{\alpha^{3/2}}{\sqrt{\pi}}\)This form has a compact, bell-shaped centre, reaches a maximum at r = 0, remains finite everywhere, and decays to zero as r approaches infinity. It represents a localized particle with a wave character that extends beyond its core.
In quantum mechanics, for the hydrogen atom, this is exactly the 1s ground-state wave function with α = 1/a0, where a0 is the Bohr radius. This gives the known ground-state energy E1s = −13.6 eV.
Exact Laplacian in 3D Spherical Coordinates
\(\nabla^2\left(e^{-\alpha r}\right)=\frac{d^2}{dr^2}e^{-\alpha r}+\frac{2}{r}\frac{d}{dr}e^{-\alpha r}\) \(=\alpha^2e^{-\alpha r}-\frac{2\alpha}{r}e^{-\alpha r}=e^{-\alpha r}\left(\alpha^2-\frac{2\alpha}{r}\right)\)What the original paper does and why it loses the R-dependence
The original paper writes ψA(r near B) = C exp(−αr/RAB) and computes the Laplacian as approximately −3α/RAB, a constant. This monopole approximation gives a constant energy rather than a potential, because it loses the R-dependence of the force.
The corrected derivation shows that the −3α/R result can be interpreted as a local coefficient, but the Laplacian must not be evaluated only at r = 0. It must be integrated over the volume of B’s wave function. This is what restores the correct force law.
2. Projection of ψA Near B — The Key Step
Place particle A at the origin and particle B at position R along the z-axis. Consider a field point P at position r measured from B, at polar angle θ from the AB axis, with r ≪ R.
2.1 Exact Distance from A to P
\(|AP|^2=R^2+r^2+2Rr\cos\theta\) \(|AP|=R\sqrt{1+\frac{2r\cos\theta}{R}+\frac{r^2}{R^2}}\) \(|AP|\approx R+r\cos\theta\qquad\text{for }r\ll R\)Therefore:
\(\psi_A(P)=Ne^{-\alpha|AP|}=Ne^{-\alpha R}e^{-\alpha r\cos\theta}=C_A(R)e^{-\alpha r\cos\theta}\) \(C_A(R)=Ne^{-\alpha R}\)2.2 Spherical Monopole Average
Averaging over all directions θ, appropriate when B’s wave function is spherically symmetric, gives:
\(\langle\psi_A\rangle_\theta=C_A(R)\frac{1}{2}\int_0^\pi e^{-\alpha r\cos\theta}\sin\theta\,d\theta\) \(=C_A(R)\frac{\sinh(\alpha r)}{\alpha r}\)Inside the coherence length, when r ≪ ℓ = 1/α:
\(\frac{\sinh(\alpha r)}{\alpha r}\approx1+\frac{(\alpha r)^2}{6}+\cdots\approx1\)A’s wave appears locally constant near B. The interaction is dominated by the amplitude CA(R).
The BeeTheory paper uses the local approximation:
\(\psi_A\big|_{\mathrm{near}\ B}(r)=C_A(R)e^{-(\alpha/R)r}\)This treats the effective local decay as βeff = α/R. This is the step that introduces 1/R into the local operator and ultimately generates the inverse-square force.
\(\beta_{\mathrm{eff}}=\frac{\alpha}{R}\)As R grows, the wave from A appears increasingly flat in B’s neighbourhood. This is the BeeTheory mechanism for long-range force.
3. Laplacian of the Projected Wave — Where 1/R² Comes From
3.1 Exact Laplacian of e−βr with β = α/R
\(\nabla^2\left(e^{-\beta r}\right)=e^{-\beta r}\left(\beta^2-\frac{2\beta}{r}\right)\) \(=e^{-\alpha r/R}\left(\frac{\alpha^2}{R^2}-\frac{2\alpha}{Rr}\right)\)This has two structurally different terms:
| Term | Expression | Behaviour | Physical role |
|---|---|---|---|
| Kinetic constant | α²e−αr/R/R² | Finite as r → 0 | Contributes a constant energy shift. |
| Coulomb generator | −2αe−αr/R/(Rr) | Diverges as 1/r | Generates a Coulomb-like local potential with coefficient proportional to 1/R. |
Applying the kinetic operator to A’s local wave near B:
\(\hat{T}\left[C_A(R)e^{-\alpha r/R}\right]=C_A(R)e^{-\alpha r/R}\left[\frac{\hbar^2\alpha}{mR}\frac{1}{r}-\frac{\hbar^2\alpha^2}{2mR^2}\right]\)3.2 Interaction Energy — Integrating Over B’s Volume
The BeeTheory interaction energy is the matrix element of this kinetic operator with B’s wave function:
\(V_{\mathrm{BT}}(R)=C_A(R)\left[\frac{\hbar^2\alpha}{mR}I_1(R)-\frac{\hbar^2\alpha^2}{2mR^2}I_2(R)\right]\)where:
\(I_1(R)=\left\langle\psi_B\middle|\frac{e^{-\alpha r/R}}{r}\middle|\psi_B\right\rangle\) \(I_2(R)=\left\langle\psi_B\middle|e^{-\alpha r/R}\middle|\psi_B\right\rangle\)In atomic units, these integrals are:
\(I_1(R)=\frac{4}{\pi}\int_0^\infty e^{-(2+1/R)r}r\,dr=\frac{4}{\pi(2+1/R)^2}\) \(I_2(R)=\frac{4}{\pi}\int_0^\infty e^{-(2+1/R)r}r^2\,dr=\frac{8}{\pi(2+1/R)^3}\)At large R, these approach constants:
\(I_1(R)\xrightarrow{R\gg\ell}\frac{1}{\pi},\qquad I_2(R)\xrightarrow{R\gg\ell}\frac{1}{\pi}\)The potential becomes:
\(V_{\mathrm{BT}}(R)\xrightarrow{R\gg\ell}-C_A(R)\frac{\hbar^2\alpha}{\pi mR}\left(1-\frac{\alpha}{2R}\right)\) \(V_{\mathrm{BT}}(R)\approx-\frac{\hbar^2\alpha}{\pi m}\frac{Ne^{-\alpha R}}{R}\) \(\boxed{V_{\mathrm{BT}}(R)=-\frac{Ke^{-\alpha R}}{R},\qquad K=\frac{\hbar^2\alpha N}{\pi m}}\)4. The Force — Newton’s Law Emerges
Starting from the BeeTheory potential:
\(V(R)=-K\frac{e^{-\alpha R}}{R}\)The force is:
\(F(R)=-\frac{dV}{dR}=-\frac{d}{dR}\left[-K\frac{e^{-\alpha R}}{R}\right]\) \(\boxed{F(R)=-\frac{K(1+\alpha R)}{R^2}e^{-\alpha R}}\)This single formula contains three regimes.
I. Gravitational Regime: R ≪ ℓ
\(e^{-\alpha R}\approx1,\qquad 1+\alpha R\approx1\) \(F(R)\approx-\frac{K}{R^2}\)This is Newton’s inverse-square law. Gravity appears as 1/R² at scales smaller than the coherence length.
II. Transition Regime: R ∼ ℓ
\(F(R)=-\frac{K(1+\alpha R)}{R^2}e^{-\alpha R}\)The exponential factor begins to suppress the force. This is the regime where deviations from Newtonian scaling become measurable.
III. Yukawa Regime: R ≫ ℓ
\(F(R)\approx-\frac{K\alpha}{R}e^{-\alpha R}\)The force becomes exponentially suppressed. This is the short-range Yukawa regime.
4.1 Numerical Verification: F(R) · R²
For a perfect Newtonian inverse-square law, the product F(R) · R² should be constant. The BeeTheory correction factor is:
\(\frac{F(R)R^2}{K}=(1+\alpha R)e^{-\alpha R}=\left(1+\frac{R}{\ell}\right)e^{-R/\ell}\)When R/ℓ is small, this factor remains close to 1.
| R/ℓ | e−R/ℓ | (1 + R/ℓ)e−R/ℓ | Error vs pure 1/R² | Regime |
|---|---|---|---|---|
| 0.01 | 0.9900 | 0.9999 | <0.01% | Newtonian |
| 0.05 | 0.9512 | 0.9988 | 0.12% | Newtonian |
| 0.10 | 0.9048 | 0.9953 | 0.47% | Newtonian |
| 0.30 | 0.7408 | 0.9631 | 3.7% | Transition begins |
| 0.50 | 0.6065 | 0.9098 | 9.0% | Transition |
| 1.00 | 0.3679 | 0.7358 | 26.4% | Mixed regime |
| 2.00 | 0.1353 | 0.4060 | 59.4% | Yukawa dominant |
| 5.00 | 0.0067 | 0.0403 | 96% | Exponential decay |
Suggested graph: Plot F(R) · R² / K versus R for different coherence lengths ℓ. For very large ℓ, the curve remains nearly flat at 1, showing Newtonian behavior. For smaller ℓ, the curve drops exponentially.
5. Complete Equations — All Steps
Step 1 — Particle Wave Function
\(\psi(r)=\frac{\alpha^{3/2}}{\sqrt{\pi}}e^{-\alpha r},\qquad \alpha=\frac{1}{\ell},\qquad N=\frac{\alpha^{3/2}}{\sqrt{\pi}}\) \(\psi(0)=N,\qquad \psi(\infty)=0,\qquad \int|\psi|^2d^3r=1\)Step 2 — Projection of A’s Wave Near B
\(\psi_A\big|_{\mathrm{near}\ B}(r)=\underbrace{Ne^{-\alpha R}}_{C_A(R)}\underbrace{e^{-(\alpha/R)r}}_{\text{local shape}}\) \(\beta_{\mathrm{eff}}=\frac{\alpha}{R}\)Step 3 — Exact Laplacian of the Local Wave
\(\nabla^2\left(e^{-\alpha r/R}\right)=e^{-\alpha r/R}\left(\frac{\alpha^2}{R^2}-\frac{2\alpha}{Rr}\right)\)The −2α/(Rr) term is the origin of the local Coulomb-like potential and therefore of the inverse-square force.
Step 4 — Matrix Elements Over B’s Wave Function
\(\left\langle\psi_B\middle|\frac{e^{-\alpha r/R}}{r}\middle|\psi_B\right\rangle=\frac{4}{\pi(2+\alpha/R)^2}\) \(\left\langle\psi_B\middle|e^{-\alpha r/R}\middle|\psi_B\right\rangle=\frac{8}{\pi(2+\alpha/R)^3}\) \(\xrightarrow{R\gg\ell}\frac{1}{\pi},\qquad \frac{1}{\pi}\)Step 5 — Interaction Potential and Force
\(V_{\mathrm{BT}}(R)=-\frac{Ke^{-\alpha R}}{R}\) \(K=\frac{\hbar^2\alpha N}{\pi m}=\frac{\hbar^2}{\pi m\ell}\frac{1}{\sqrt{\pi}\ell^{3/2}}\) \(\boxed{F(R)=-\frac{K(1+\alpha R)e^{-\alpha R}}{R^2}=-\frac{K}{R^2}\left(1+\frac{R}{\ell}\right)e^{-R/\ell}}\) \(\boxed{R\ll\ell\quad\Longrightarrow\quad F(R)=-\frac{K}{R^2}}\)With:
\(K=\frac{\hbar^2\alpha^{5/2}}{\pi^{3/2}m},\qquad \alpha=\frac{1}{\ell}=\frac{mv_{\mathrm{wave}}}{\hbar}\)5.1 Identifying Newton’s Constant G
For two masses m1 and m2, the Newtonian limit requires:
\(F(R)=-\frac{Gm_1m_2}{R^2}\)Comparing with the BeeTheory limit F = −K/R² gives:
\(G=\frac{K}{m_1m_2}=\frac{\hbar^2\alpha^{5/2}}{\pi^{3/2}m\,m_1m_2}\)For m1 = m2 = m:
\(G=\frac{\hbar^2}{\pi^{3/2}m^2\ell^{5/2}}\)Solving for ℓ:
\(\ell=\left(\frac{\hbar^2}{\pi^{3/2}Gm^2}\right)^{2/5}\)For the proton mass mp = 1.67 × 10−27 kg:
\(\ell_p=\left(\frac{(1.055\times10^{-34})^2}{\pi^{3/2}\times6.674\times10^{-11}\times(1.67\times10^{-27})^2}\right)^{2/5}\approx1.2\times10^{13}\,\mathrm{m}\approx80\,\mathrm{AU}\)This is the gravitational coherence length of a proton in this simplified scaling. For macroscopic bodies, the effective coherence length would scale with the aggregate wave field of all constituent particles.
6. Summary: Original Paper vs Corrected Derivation
BeeTheory v2 Paper
ψ = N exp(−αr): correct form.
Near B: CA(R) exp(−αr/R): correct projection idea.
Laplacian approximation: ∇²[exp(−αr/R)] ≈ −3α/R. This evaluates only a local coefficient and discards the 1/r term.
The conclusion F ∝ 1/R² is physically right, but the derivation is incomplete.
Corrected Derivation
ψ = N exp(−αr): unchanged.
Near B: CA(R) exp(−αr/R): retained as the effective local projection.
\(\nabla^2(e^{-\alpha r/R})=e^{-\alpha r/R}\left(\frac{\alpha^2}{R^2}-\frac{2\alpha}{Rr}\right)\)The full derivation integrates the operator over B’s wave function and obtains:
\(V(R)=-\frac{Ke^{-\alpha R}}{R}\) \(F(R)=-\frac{K(1+\alpha R)e^{-\alpha R}}{R^2}\)The conclusion of the paper is correct — the derivation needed completion.
BeeTheory v2 reaches the right physical answer through a correct intuition, but the monopole approximation must be completed by retaining the 1/r term in the Laplacian and integrating over the second particle’s wave function.
References
- Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, BeeTheory.com v2, 2023.
- Yukawa, H. — On the Interaction of Elementary Particles, Proc. Phys.-Math. Soc. Japan 17, 48, 1935.
- Abramowitz, M., Stegun, I. A. — Handbook of Mathematical Functions, Dover, 1972.
- Jackson, J. D. — Classical Electrodynamics, 3rd ed., Wiley, 1999.
- Griffiths, D. J. — Introduction to Quantum Mechanics, 2nd ed., Pearson, 2005.
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