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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

r_{0}= 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]

r_{1}= 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?

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# Yukawa-Hooke Equasion

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