# Yukawa-Hooke Equasion

1. Jan 14, 2004

### 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 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?

Last edited: Jan 20, 2004
2. Jan 19, 2004

### 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.