# TF DerivationOfCoulombForce

Poisson's Equation

Fourier Transform of Poisson's Equation

Product rule for dervatives

Gauss' Theorem:

Definition of derivative:

Substituting

Gauss' Low:

1.) Coulomb = potential in "k"(momentum) space

To find the potential in "coordinate" space just inverse transform

Coulomb potential
2) Nuclear potential

Consider the force field generated by a point source (nucleon) at location from the origin of a coordinate system.

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:

In terms of Hamiltonian

In a static case

Lets

if then interaction length .

With the source term

As seen before for Coulomb force

: inverse fourier transform

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

C.) Deuteron
, , (d) = a proton-neutron bound state

General properties: L=0 - orbital angular momentum

- (Nuclear spin)

- Binding energy.

Non-relativistic Schrodiger solution

weakly bound system

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

when

Assume  : No angular dependence, only radial dependence.

Schrod. Equation

:

Schrod. Equation becomes

for  :

for  :

:

: spring simple harmonic motion

Boundary condition:

:

New definition of  :

Boundary condition: - finite D=0

Bounding condition

Dividing two equations

But E<0 for bound states

Solving the ??? 59 eqution:

m = reduced mass

:

Find X s.t.

Using ?? 59 of graphing

Spin and Parity

# 66-78 pages

The shrodinger equation for this scattering:

In spherical coordinates this may be written as:

Let

or

General solution:

where

= Bessel function
= Neiman function

1.) Distant scattering: r is large such that neutron "glances" off.

= finite, no cosine term.

Normalizing and simplifying

and are found by applying Boundary conditions:

Example
l=0 special case

Apply Boundary Conditions:

, and R are known. is unknown.

If V=0 : Free neutron; No target; No phase shift.

image70_1


If V>0 :

image70


If V<0 : E>0; Neutron not bound.

If you measure the phase shift you can determine "sign" of V.

Cross-section: differential X-section =

=probability of scattering into the solid angle
probability of scattering in any direction.

Let

From Q.M. particle current density

This comes from the continuity equation

Time dependent Shrodinger equation \Longrightarrow

time derivative of the particle density

If free particle

Cross-section

Check: