Nuclear Models

A goal of nuclear physics is to account for the properties of nuclei in terms of mathematical models of their structure and internal motion.  A nucleus must be treated quantum-mechanically.  We have many interacting quantum-mechanical particles.  The Schroedinger equation for a particle in one dimension is

 (-ħ2/(2m))∂2ψ(x,t)/∂x2 + U(x)ψ(x,t)  = iħ∂ψ(x,t)/∂t.

In 3 dimensions it becomes  

(-ħ2/(2m))2ψ(r,t) + U(r)ψ(r,t)  = iħ∂ψ(r,t)/t.

where 2ψ(r,t) = ∂2ψ(r,t)/x2 + 2ψ(r,t)/y2 + 2ψ(r,t)/z2.

For many particles in 3 dimensions we have to write

Σi[(-ħ2/(2mi))i2ψ(r1,r2,r3, ..., t)] + U(r1,r2,r3, ...)ψ(r1,r2,r3, ..., t) = iħ∂ψ(r1,r2,r3, ..., t)/t.

This is a many-body equation that cannot be solved exactly.  Approximations are necessary. 
In addition, we do not know the nuclear potential
U(r1,r2,r3, ...) very well.  It is mainly due to the strong nuclear force and the repulsive electrostatic Coulomb force.  But nucleons have spins and magnetic moments and since they are very close packed, magnetic interactions also become important.

Two approximations are often the starting point for calculation.  They are then refined by adding extra terms as small perturbations.

So there are two basic types of simple nuclear models.

The Shell Model and the Collective Model represent the two extremes of the behavior of nucleons in the nucleus.  More realistic models, known as unified models, attempt to include both shell and collective behaviors.