Mathematical Summary (sum of e−αre^{-\alpha r}e−αr waves)

1) Ansatz (two particles A and B)

Model each particle as a monochromatic, localized, isotropic source of a complex scalar field (the “matter wave”): ψA(r,t)=A e−α∣r−rA∣ e−iω1t,ψB(r,t)=B e−β∣r−rB∣ e−iω2t,\psi_A(\mathbf r,t)=A\,e^{-\alpha|\mathbf r-\mathbf r_A|}\,e^{-i\omega_1 t},\qquad \psi_B(\mathbf r,t)=B\,e^{-\beta|\mathbf r-\mathbf r_B|}\,e^{-i\omega_2 t},ψA​(r,t)=Ae−α∣r−rA​∣e−iω1​t,ψB​(r,t)=Be−β∣r−rB​∣e−iω2​t,

and superpose: Ψ(r,t)=ψA(r,t)+ψB(r,t).\Psi(\mathbf r,t)=\psi_A(\mathbf r,t)+\psi_B(\mathbf r,t).Ψ(r,t)=ψA​(r,t)+ψB​(r,t).

Switch to spherical coordinates around BBB: write r=rB+s\mathbf r=\mathbf r_B+\mathbf sr=rB​+s with r=∣s∣≪Rr=|\mathbf s|\ll Rr=∣s∣≪R, and define R≡∣rB−rA∣,∣r−rB∣=r,∣r−rA∣=∣R−s∣.R\equiv|\mathbf r_B-\mathbf r_A|,\qquad |\mathbf r-\mathbf r_B|=r,\qquad |\mathbf r-\mathbf r_A|=|\mathbf R-\mathbf s|.R≡∣rB​−rA​∣,∣r−rB​∣=r,∣r−rA​∣=∣R−s∣.

For r≪Rr\ll Rr≪R, ∣R−s∣≈R−rcos⁡θ+O(r2/R),|\mathbf R-\mathbf s|\approx R- r\cos\theta + O(r^2/R),∣R−s∣≈R−rcosθ+O(r2/R),

so near BBB: ψA(r,t)≈A e−αR e+αrcos⁡θ e−iω1t,ψB(r,t)=B e−βr e−iω2t.\psi_A(\mathbf r,t)\approx A\,e^{-\alpha R}\,e^{+\alpha r\cos\theta}\,e^{-i\omega_1 t},\qquad \psi_B(\mathbf r,t)=B\,e^{-\beta r}\,e^{-i\omega_2 t}.ψA​(r,t)≈Ae−αRe+αrcosθe−iω1​t,ψB​(r,t)=Be−βre−iω2​t.

At the point B0B_0B0​ (i.e. r=0r=0r=0), the contribution from AAA is simply ψA(B0,t)=A e−αR e−iω1t.\psi_A(B_0,t)=A\,e^{-\alpha R}\,e^{-i\omega_1 t}.ψA​(B0​,t)=Ae−αRe−iω1​t.

2) Which wave equation to use?

The correct free Schrödinger equation is iℏ ∂tΨ=−ℏ22m ∇2Ψ.i\hbar\,\partial_t\Psi = -\frac{\hbar^2}{2m}\,\nabla^2\Psi.iℏ∂t​Ψ=−2mℏ2​∇2Ψ.

Its stationary states are oscillatory plane/spherical waves; an envelope e−αre^{-\alpha r}e−αr alone is not an exact free-Schrödinger solution.

To obtain genuinely exponential profiles, use the modified Helmholtz (massive mediator) or Poisson (massless mediator) equation for a mediator field ϕ\phiϕ: (∇2−μ2) ϕ(r,t)=−4π S(r) e−iωt    ⇒    Gμ(r)=e−μr4πr.(\nabla^2-\mu^2)\,\phi(\mathbf r,t)= -4\pi\,S(\mathbf r)\,e^{-i\omega t} \;\;\Rightarrow\;\; G_\mu(r)=\frac{e^{-\mu r}}{4\pi r}.(∇2−μ2)ϕ(r,t)=−4πS(r)e−iωt⇒Gμ​(r)=4πre−μr​.

Thus a point source SA δ(r−rA)S_A\,\delta(\mathbf r-\mathbf r_A)SA​δ(r−rA​) yields ϕA(r,t)=SA4π e−μ∣r−rA∣∣r−rA∣ e−iω1t.\phi_A(\mathbf r,t)=\frac{S_A}{4\pi}\,\frac{e^{-\mu|\mathbf r-\mathbf r_A|}}{|\mathbf r-\mathbf r_A|}\,e^{-i\omega_1 t}.ϕA​(r,t)=4πSA​​∣r−rA​∣e−μ∣r−rA​∣​e−iω1​t.

In the quasi-static limit μ→0\mu\to 0μ→0 (massless mediator), G0(r)=1/(4πr)G_0(r)=1/(4\pi r)G0​(r)=1/(4πr): you recover the 1/r1/r1/r behavior.

3) Effective potential and the 1/R1/R1/R law

If BBB couples to AAA’s field with coupling gBg_BgB​ (and AAA with gAg_AgA​), the interaction energy at BBB is proportional to the received field: VAB(R,t)=gB ℜ ϕA(B,t)=gAgB4π e−μRR cos⁡(ω1t+φ).V_{AB}(R,t)= g_B\,\Re\,\phi_A(B,t) = \frac{g_A g_B}{4\pi}\,\frac{e^{-\mu R}}{R}\,\cos(\omega_1 t+\varphi).VAB​(R,t)=gB​ℜϕA​(B,t)=4πgA​gB​​Re−μR​cos(ω1​t+φ).

After time averaging (or if ω1≃ω2\omega_1\simeq\omega_2ω1​≃ω2​), the envelope is  VAB(R)∝e−μRR \boxed{\,V_{AB}(R)\propto \dfrac{e^{-\mu R}}{R}\,}VAB​(R)∝Re−μR​​

with radial force F(R)=−∂RV R^=−gAgB4π e−μR ⁣(1R2+μR)R^.\mathbf F(R)=-\partial_R V\,\hat{\mathbf R} = -\frac{g_A g_B}{4\pi}\,e^{-\mu R}\!\left(\frac{1}{R^2}+\frac{\mu}{R}\right)\hat{\mathbf R}.F(R)=−∂R​VR^=−4πgA​gB​​e−μR(R21​+Rμ​)R^.

In the long-range limit μR≪1\mu R\ll1μR≪1 (“quasi-massless” mediator), this reproduces a 1/R21/R^21/R2 gravity-like law.

4) Useful identities (quick validation)

• Laplacian of radial exponentials:

∇2 ⁣(e−αr)=e−αr ⁣(α2−2αr).\nabla^2\!\big(e^{-\alpha r}\big)= e^{-\alpha r}\!\left(\alpha^2-\frac{2\alpha}{r}\right).∇2(e−αr)=e−αr(α2−r2α​).

• Green’s function:

∇2 ⁣(e−μrr)=μ2e−μrr−4π δ(r).\nabla^2\!\left(\frac{e^{-\mu r}}{r}\right)=\mu^2\frac{e^{-\mu r}}{r}-4\pi\,\delta(\mathbf r).∇2(re−μr​)=μ2re−μr​−4πδ(r).

The 1/r1/r1/r singularity (and thus the far-field 1/R1/R1/R) comes from the Green’s function structure G(r)∼1/rG(r)\sim 1/rG(r)∼1/r, not from a bare e−αre^{-\alpha r}e−αr without the 1/r1/r1/r factor.

In two lines

• Superpose localized waves: Ψ=ψA+ψB\Psi=\psi_A+\psi_BΨ=ψA​+ψB​ with envelopes e−αre^{-\alpha r}e−αr.
• To obtain a potential ∼1/R\sim 1/R∼1/R (and force ∼1/R2\sim 1/R^2∼1/R2), the mediator field must obey Poisson/Helmholtz, whose Green’s function is G(r)∼e−μr/rG(r)\sim e^{-\mu r}/rG(r)∼e−μr/r. Then VAB(R)∝e−μR/RV_{AB}(R)\propto e^{-\mu R}/RVAB​(R)∝e−μR/R, and μ→0⇒V∝1/R\mu\to 0\Rightarrow V\propto 1/Rμ→0⇒V∝1/R.