Yukawa-Hooke Equasion...


by Orion1
Tags: equasion, yukawahooke
Orion1
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#1
Jan14-04, 06:57 AM
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P: 991


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?

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arivero
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#2
Jan19-04, 07:20 AM
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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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