# Yukawa-Hooke Equasion...

by Orion1
Tags: equasion, yukawahooke
 P: 991 Hooke's Law: $$W(x) = - \frac{kx^2}{2}$$ k - spring force constant Yukawa Potential: $$U(r) = - f^2 \frac{e^- \frac{(r/r_0)}{}}{r}$$ f - interaction strength r0 = 1.5*10^-15 m $$U(r) = W(r)$$ Yukawa-Hooke Equasion: $$-f^2 \frac{e^- \frac{(r/r_0)}{}}{r} = -\frac{kr^2}{2}$$ $$f^2 = \frac{kr^3}{2e^- \frac{(r/r_0)}{}}$$ $$f = \sqrt{ \frac{kr^3}{2e^- \frac{(r/r_0)}{}}}$$ $$r = \sqrt[3]{ \frac{2f^2 e^- \frac{(r/r_0)}{}}{k}}$$ $$E(r) = U(r) + W(r)$$ $$E(r) = -f^2 \frac{e^- \frac{(r/r_0)}{}}{r} - \frac{kr^2}{2}$$ Yukawa Meson Mass-Energy Spectrum: $$\pi ^o (135 Mev) -> \eta ^o (548.8 Mev)$$ r1 = 1.461 Fm -> .359 Fm $$E(r) = W(r)$$ $$- \frac{\hbar c}{r_1} = - \frac{kr_1 ^2}{2}$$ $$k = \frac{2 \hbar c}{r_1 ^3}$$ $$E(r) = U(r)$$ $$- \frac{\hbar c}{r_1} = -f^2 \frac{e^- \frac{(r_1/r_0)}{}}{r_1}$$ $$\hbar c = f^2 e^- \frac{(r_1/r_0)}{}$$ $$f = \sqrt{ \frac{\hbar c}{{e^- \frac{(r_1/r_0)}{} }}$$ How effective is the Yukawa-Hooke Equasion at emulating a Nuclear Force Mediator? What is the depth of such an equasion? and can it be applied to String Theory?
 PF Gold P: 2,884 The issue is relativistic invariance. Can one implement Hooke's law in a relativistic invariant way.? Yukawa force is mediated via a particle of mass 1/R_0, so that relativity can be implemented simply by asking the particle propagator to fullfill it. I am not telling it does not exist a particle interpretation of Hooke's law, just I have never heard of it. Neither of a string interpretation Hooke's law... but it could be, because these strings somehow are relativity-complient.

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