TF DerivationOfCoulombForce

Poisson's Equation

[math]\nabla^2 \phi(\vec{\xi}) = - \frac{\rho}{\epsilon_0} =- \frac{e}{\epsilon_0} \delta(\vec{\xi})[/math]

Fourier Transform of Poisson's Equation

[math]\frac{1}{(2 \pi)^{3/2}} \int e^{-i \vec{k} \cdot \vec{\xi}} \nabla^2 \phi(\vec{\xi})dV = - \frac{1}{(2 \pi)^{3/2}} \frac{e}{\epsilon_0} \int e^{-i \vec{k} \cdot \vec{\xi}}\delta(\vec{\xi}) dV [/math]

[math]\frac{1}{(2 \pi)^{3/2}} \int e^{-i \vec{k} \cdot \vec{\xi}} \vec{\nabla} \cdot (\vec{\nabla} \phi(\vec{\xi}))dV = - \frac{e}{(2 \pi)^{3/2}\epsilon_0} (1)[/math]

Product rule for dervatives

[math]\frac{1}{(2 \pi)^{3/2}} \int \left \{ \vec{\nabla} \cdot ( e^{-i \vec{k} \cdot \vec{\xi}} \vec{\nabla} \phi ) - (\vec{\nabla} e^{-i \vec{k} \cdot \vec{\xi}}) \cdot (\vec{\nabla} \phi) \right \} dV = - \frac{e}{(2 \pi)^{3/2}\epsilon_0} (1)[/math]

Gauss' Theorem:

[math]\int \vec{\nabla} \cdot ( e^{-i \vec{k} \cdot \vec{\xi}} \vec{\nabla} \phi ) dV = \oint_S e^{-i \vec{k}\cdot \vec{\xi}} \vec{\nabla}\cdot d\vec{A}[/math]

Definition of derivative:

[math](\vec{\nabla} e^{-i \vec{k} \cdot \vec{\xi}}) \cdot (\vec{\nabla} \phi ) = \vec{\nabla} \cdot (\phi \vec{\nabla} e^{-i \vec{k}}) - \phi {\nabla}^2 e^{-i \vec{k} \cdot \vec{\xi}}[/math]

Substituting

[math]\frac{1}{(2 \pi)^{3/2} } \left \{ \oint_S e^{-i \vec{k} \cdot \vec{\xi}} \vec{\nabla} \phi \cdot d\vec{A} - \int \vec{\nabla} \cdot (\phi \vec{\nabla} e^{-i \vec{k} \cdot \vec{\xi}} ) dV + \int \phi {\nabla}^2 e^{-i \vec{k} \cdot \vec{\xi}} dV \right \} = \frac{-e}{(2 \pi)^{3/2} \epsilon_0}[/math]

Gauss' Low:

[math]\int \vec{\nabla}\cdot (\phi \vec{\nabla} e^{-i \vec{k} \cdot \vec{\xi}} ) dV = \oint_S \phi \vec{\nabla} e^{-i k \xi } \cdot d\vec{A}[/math]

[math]\frac{1}{(2 \pi)^{3/2} } \left \{\oint_S \left \{ e^{-i \vec{k} \cdot \vec{\xi}} \vec{\nabla} \phi - \phi \vec{\nabla} e^{-i \vec{k} \cdot \vec{\xi}} \right \} \cdot d\vec{A} + \int \phi {\nabla}^2 e^{-i \vec{k} \cdot \vec{\xi}} dV \right \} = \frac{-e}{(2 \pi)^{3/2} \epsilon_0}[/math]

[math]\frac{1}{(2 \pi)^{3/2} } \int \phi (-ik) (-ik) e^{-i \vec{k} \cdot \vec{\xi}} dV = \frac{-e}{(2 \pi)^{3/2} \epsilon_0}[/math]

[math]-k^2 \frac{1}{(2 \pi)^{3/2} } \int \phi(\xi) e^{-i \vec{k} \cdot \vec{\xi}} dV_{xi} = \frac{-e}{(2 \pi)^{3/2} \epsilon_0}[/math]

[math]-k^2 \phi(k) = \frac{-e}{(2 \pi)^{3/2} \epsilon_0}[/math]

1.) Coulomb [math]\phi(k) = \frac{e}{(2 \pi)^{3/2} \epsilon_0} \frac{1}{k^2}[/math] = potential in "k"(momentum) space

To find the potential in "coordinate" [math](\xi)[/math] space just inverse transform

[math]\phi (\xi) = \frac{1}{(2 \pi)^{3/2} } \int e^{+ i \vec{k} \cdot \vec{\xi}} \phi (k) dV_k[/math]

[math]= \frac{1}{(2 \pi)^{3/2} } \int e^{i \vec{k} \cdot \vec{\xi}} \frac{e}{2 \pi)^{3/2} \epsilon_0} \frac{1}{k^2} dV_k[/math]

[math]= \frac{e}{(2 \pi)^{3} \epsilon_0} \int \frac{e^{i \vec{k} \cdot \vec{\xi}}}{k^2} dV_k[/math]

[math]dV_k=k^2 sin{\theta}_k d{\theta}_k d{\phi}_k dk[/math]

[math]=\frac{e}{(2 \pi)^{3} \epsilon_0} {{\int}_0}^{2\pi} d{\phi}_k {{\int}_0}^{\pi} d{\theta}_k {{\int}_0}^{\infty} dk \times k^2 sin{\theta}_k e^{i \vec{k} \cdot \vec{\xi}}[/math]

[math]=\frac{e}{(2\pi)^2 \epsilon_0} {{\int}_0}^{\pi} {{\int}_0}^{\infty} sin{\theta}_k e^{ik \xi cos{\theta}_k} k^2 dk[/math]

[math]u=cos\theta[/math]

[math]du=sin\theta d\theta[/math]

[math]\phi(\xi) = \frac{e}{4 {\pi}^2 \epsilon_0} {{\int}_0}^infty {{\int}_{-1}}^1 \frac{e^{ik\xi u}}{k^2} du k^2 dk[/math]

[math]=\frac{e}{4 {\pi}^2 \epsilon_0} {{\int}_0}^infty \frac{e^{ik \xi} - e^{-ik\xi}}{ik\xi} dk[/math]

[math]=\frac{e}{4 {\pi}^2 \epsilon_0} \frac{1}{i\xi} (i\pi) = \frac{e}{4 {\pi}^2 \epsilon_0} \frac{1}{\xi}[/math]

[math]=\frac{e}{4 {\pi}^2 \epsilon_0} \frac{1}{|\vec{r} - \vec{r}^'|} =[/math] Coulomb potential

2) Nuclear potential

Consider the force field generated by a point source (nucleon) at location [math]\vec{r}[/math] from the origin of a coordinate system.

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Assume a particle of mass m is e charged to generate the field (In Coulomb force particle was m=o photon).

Definition of relativistic Energy:

[math]E^2=(mc^2)^2 + (cp)^2[/math]

In terms of Hamiltonian

[math]-\hbar \frac{d^2}{dt^2} \phi(\vec{r}) = [ (mc^2)^2 + (\frac{c\hbar \vec{\nabla}}{i})^2 ] \phi (r)[/math]

In a static case

[math][(mc^2)^2 - c^2 {\hbar}^2 {\nabla}^2]\phi(r)=0[/math]

[math][ {\nabla}^2 - (\frac{mc}{\hbar})^2]\phi(r)=0[/math]

Lets [math]\mu = \frac{mc}{\hbar} = \frac{(140 \frac{MeV}{c^2})c}{6.6 \times 10^{-16}eV \cdot s} (\frac{10^6 eV}{MeV})[/math]

[math]=\frac{2.1 \times 10^23}{(3\times 10^8 m)} (\frac{10^{-15}m}{fm}) =\frac{0.7}{fm} (200 MeV fm) = 140 MeV[/math]

[math]\hbar c=(6.6 \times 10^{-16} eV \cdot s) (3 \times 10^8 \frac{m}{s}) (\frac{10^{15} fm}{m}) (\frac{1 MeV}{10^6 eV}) = 198 MeV \cdot fm \approx 200 MeV \cdot fm[/math]

[math]\mu = \frac{mc^2}{\hbar c} = \frac{mc^2}{200 MeV \cdot fm}[/math] [math]\Longrightarrow[/math] if [math]mc^2\approx 200 MeV[/math] then interaction length [math]\sim 1 fm[/math].

With the source term

[math][{\nabla}^2 - {\mu}^2]\phi(\xi) = -{{\xi}_0}^' \delta(\xi)[/math]

As seen before for Coulomb force

[math]\frac{1}{(2\pi)^{3/2}} \int e^{-ik \xi} [{\nabla}^2 - {\mu}^2] \phi (\xi) dV = -\frac{{{\xi}_0}^'}{(2\pi)^{3/2}} \int e^{-i \vec{k} \cdot \vec{\xi}} \delta (\xi) dV[/math]

[math][-k^2 -{\mu}^2]\phi(\vec{k}) = -\frac{{{\xi}_0}^'}{(2\pi)^{3/2}}[/math]

[math]\Longrightarrow \phi(\vec{k}) = \frac{{{\xi}_0}^'}{(2\pi)^{3/2}} \frac{1}{k^2 + {\mu}^2}[/math]

[math]\phi(\vec{\xi}) = \frac{1}{(2 \pi)^{3/2}} \int \frac{e^{+i \vec{k} \cdot \vec{\xi}}}{(2 \pi)^{3/2}} \frac{{{\xi}_0}^'}{k^2 + {\mu}^2} dV_k[/math] : inverse fourier transform

[math]= \frac{{{\xi}_0}^'}{(2\pi)^3} \int \frac{e^{+i \vec{k} \cdot \vec{\xi}}}{k^2 + {\mu}^2} k^2 sin\theta d\theta d\phi dk[/math]

[math]\phi(\xi) = \frac{{{\xi}_0}^'}{(2\pi)^2} \int \frac{e^{i\vec{k} \cdot \vec{\xi}}}{(k^2 + {\mu}^2)} k^2 d[cos\theta] (2\pi) dk[/math]

[math]= \frac{{{\xi}_0}^'}{(2\pi)^2} {{\int}_0}^{\infty} \frac{k^2}{k^2 + {\mu}^2} {{\int}_0}^{\pi} e^{ik\xi cos\theta} d[cos\theta] dk[/math]

[math]= \frac{{{\xi}_0}^'}{(2\pi)^2} {{\int}_0}^{\infty} \frac{k^2}{k^2 + {\mu}^2} dk \frac{e^{ik\xi \mu}}{ik\xi} {|_{-1}}^1[/math]

[math]= \frac{{{\xi}_0}^'}{(2\pi)^2} \frac{1}{i\xi} {{\int}_0}^{\infty} \frac{k}{k^2 + {\mu}^2} (e^{ik\xi} - e^{-ik\xi})[/math]

[math]= \frac{{{\xi}_0}^'}{(2\pi)^2}\frac{1}{i\xi} {{\int}_{-\infty}}^{\infty} \frac{e^{ik\xi}kdk}{k^2 + {\mu}^2}[/math]

[math]{{\int}_{-\infty}}^{\infty} \frac{e^{ik\xi}kdk}{k + i{\mu}}{k - i{\mu}} = i\pi e^{-\mu \xi}[/math]

[math]\phi(\xi) = \frac{{{\xi}_0}^'}{(2\pi)^2} \frac{e^{-\mu \xi}}{\xi}[/math]

[math]\Longrightarrow[/math]

[math]\phi(\vec{r}) = \frac{{{\xi}_0}^'}{(2\pi)^2} \frac{1}{|\vec{r} - {\vec{r}}^'|}[/math]

[math]{\phi}_{EM}(\vec{r}) = \frac{e}{4\pi \epsilon_0} \frac{1}{|\vec{r} - {\vec{r}}^'|}[/math]

Coupling constants are:

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Summary

There are now at least two forces which act between Nucleons, the Coulomb force and the Nucleon force. We can write the force in terms of a potential

[math]V_{EM} = \frac{e}{4\pi \epsilon_0} \frac{1}{|\vec{r} - {\vec{r}}^'|}[/math]

[math]V_{Nuc} = \frac{{{\xi}_0}^'}{(2\pi)^2} \frac{e^{-\mu |\vec{r} - {\vec{r}}^'|}}{|\vec{r} - {\vec{r}}^'|}[/math]

[math]V_{tot} = \frac{1}{4\pi} \left \{ \frac{Q_1 Q_2}{\epsilon_0 e } - \frac{{\xi}_0}{\pi} e^{-\mu |\vec{r} - {\vec{r}}^'|} \right \} \frac{1}{|\vec{r} - {\vec{r}}^'|}[/math]

[math]\mu=\frac{m_{\pi}c}{\hbar}=\frac{1}{R} \sim (1.5 fm)^{-1}[/math]

C.) Deuteron

[math]({H_1}^2)[/math], [math](D^2)[/math], (d) = a proton-neutron bound state

General properties: L=0 - orbital angular momentum

[math]J^{\pi} = 1^+[/math] - (Nuclear spin)

[math]\sqrt{r^2}=2.095 fm[/math] - Mean radius.

[math]B.E.=2.2246 MeV[/math] - Binding energy.

Non-relativistic Schrodiger solution

[math]B.E. = 2.2246 MeV \lt \lt 1.8 GeV m(H^2)c^2[/math]

[math]\Longrightarrow[/math] weakly bound system

Instead of Dirac equation try 3-D Square Well Schrod. Eq. approximation for Deuteron wavefunction.

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[math]V(r) = -V_0[/math] when [math]r \leq R = 2.2[/math]

[math]V(r) = 0[/math] [math]r\gt R[/math]

Assume [math]\psi = Y_{00} \frac{V(r)}{r}[/math] : No angular dependence, only radial dependence.

Schrod. Equation

[math]-\frac{\hbar^2}{2m} {\nabla}^2 \psi + V(r)\psi = E \psi[/math]

[math]{\nabla}^2 = \frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial}{\partial r}) + \frac{1}{r^2 sin\theta} \frac{\partial}{\partial \theta} (sin\theta \frac{\partial}{\partial \theta}) + \frac{1}{r^2 sin^2 \theta} \frac{{\partial}^2}{\partial {\phi}^2} [/math]

[math]-\frac{\hbar^2}{2m} Y_{00} \frac{1}{r^2} \frac{\partial}{\partial r} r^2 \frac{\partial (U(r)/r)}{\partial r} + \frac{1}{r^2 sin\theta}\frac{U(r)}{r}(\frac{\partial}{\partial \theta} sin \theta \frac{\partial Y_{00}}{\partial \theta})[/math]

[math]+ \frac{U(r)}{r^3 sin^2 \theta} \frac{{\partial}^2}{\partial {\phi}^2} Y_{00} + \frac{V(r)Y_{00} U(r)}{r} = \frac{E U(r)}{r}[/math]

[math]\frac{1}{r^2} \frac{\partial}{\partial r} r^2 \frac{\partial}{\partial r} \frac{U(r)}{r} = \frac{1}{r^2} \frac{\partial}{\partial r} r^2 [\frac{1}{r}\frac{\partial U(r)}{r} - \frac{U(r)}{r^2}][/math]

[math]= \frac{1}{r^2} [\frac{\partial U(r)}{\partial r} + r \frac{\partial^2 U(r)}{\partial r^2} - \frac{2rU(r)}{r^2} - \frac{r^2}{r^2} \frac{\partial U(r)}{\partial r} - r^2 U(r)\frac{-2}{r^3}][/math]

[math]= \frac{1}{r}\frac{\partial^2 U(r)}{\partial r^2} + \frac{1}{r^2} \frac{\partial U(r)}{\partial r} - \frac{1}{r^2} \frac{\partial U(r)}{\partial r} - \frac{2U(r)}{r^2} + \frac{2U(r)}{r^2}[/math]

[math]= \frac{1}{r}\frac{\partial^2 U(r)}{\partial r^2}[/math]