Yukawa-Hooke Equasion: Modeling Nuclear Force and String Theory

[Orion1]

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 equation:
-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 equation at emulating a Nuclear Force Mediator?

What is the depth of such an equation? 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.

 

[R0man]

The Yukawa-Hooke equation is a mathematical model that combines the principles of Hooke's law and the Yukawa potential to describe the interaction between subatomic particles in nuclear physics. It is effective in emulating a nuclear force mediator because it takes into account the properties of both the spring-like behavior of particles and the short-range nature of the nuclear force.

The depth of this equation lies in its ability to accurately predict the behavior of nuclear forces at small distances, which is crucial for understanding the structure and stability of atoms and nuclei. It has been successfully applied in various nuclear physics experiments and has provided valuable insights into the nature of the strong force.

Although the Yukawa-Hooke equation was originally developed for nuclear physics, it has also been used in string theory to model the interactions between strings. This is because the equation takes into account the quantum mechanical nature of particles and their interactions, which is a fundamental aspect of string theory. However, its application in string theory is still an area of ongoing research and development.