Yukawa-Hooke Equasion...

Orion1

Hooke's Law:
[tex]W(x) = - \frac{kx^2}{2}[/tex]
k - spring force constant

Yukawa Potential:
[tex]U(r) = - f^2 \frac{e^- \frac{(r/r_0)}{}}{r}[/tex]
f - interaction strength
r₀ = 1.5*10^-15 m

[tex]U(r) = W(r)[/tex]

Yukawa-Hooke Equasion:
[tex]-f^2 \frac{e^- \frac{(r/r_0)}{}}{r} = -\frac{kr^2}{2}[/tex]

[tex]f^2 = \frac{kr^3}{2e^- \frac{(r/r_0)}{}}[/tex]

[tex]f = \sqrt{ \frac{kr^3}{2e^- \frac{(r/r_0)}{}}}[/tex]

[tex]r = \sqrt[3]{ \frac{2f^2 e^- \frac{(r/r_0)}{}}{k}}[/tex]

[tex]E(r) = U(r) + W(r)[/tex]
[tex]E(r) = -f^2 \frac{e^- \frac{(r/r_0)}{}}{r} - \frac{kr^2}{2}[/tex]

Yukawa Meson Mass-Energy Spectrum:
[tex]\pi ^o (135 Mev) -> \eta ^o (548.8 Mev)[/tex]
r₁ = 1.461 Fm -> .359 Fm

[tex]E(r) = W(r)[/tex]

[tex]- \frac{\hbar c}{r_1} = - \frac{kr_1 ^2}{2}[/tex]

[tex]k = \frac{2 \hbar c}{r_1 ^3}[/tex]

[tex]E(r) = U(r)[/tex]
[tex]- \frac{\hbar c}{r_1} = -f^2 \frac{e^- \frac{(r_1/r_0)}{}}{r_1}[/tex]

[tex]\hbar c = f^2 e^- \frac{(r_1/r_0)}{}[/tex]

[tex]f = \sqrt{ \frac{\hbar c}{{e^- \frac{(r_1/r_0)}{} }}[/tex]

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?

arivero

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.