## 1. Introduction

Our universe consists of substance. Atoms and molecules are basic components of material. Each atom contains a nucleus which is spread in a small area of atom, and electrons. Also, a nucleus contains Neutrons and protons. It is well known today that electrons in atom and Neutrons and protons in the nucleus are interacting together through different forces. It is clear today that the source of different interactions are composed of four basic forces of the universe, namely gravitational, coulomb, strong and weak nuclear interactions.

In quantum mechanics, to study a particle, it is necessary to have knowledge about its interaction with the surrounding media. The Schrödinger equation is a second-order differential equation that is solved to obtain energy spectrum and wave functions of a particle in quantum mechanics. For a many-body system such as atom or nucleus, it is not possible to solve a set of Schrödinger equations to obtain energy spectrum and wave functions analytically. Therefore in such situations, it is necessary to use an average potential which is a mean potential of all interacting forces acting upon a single particle. Then the Schrödinger equation is should be solve for a single particle. This procedure is called the mean field method [1, 2, 3].

To review this method consider a system consisting of N identical interacting particles. The Hamiltonian of system composed of kinetic energy,

where

where

is the mean field Hamiltonian of the system and

is the mean residual interaction. It should be noted that the residual interaction is related to the strength of the actual interaction and can be reduced if the mean field potential is close to the actual potential of the system.

Actually, the mean field method is an approximation in which each particle of system moves under an external field generated by the remaining

It is important to know that the time average idea was considered only for clearance of the subject and not applicable in practice unless one studies the thermo-dynamical behavior of nucleus.

Therefore the idea of using mean field theory capable of reducing many particles interacting system in to a system of non-interacting (quasi-particles) considered in an external field

The corresponding

The wave function

Substituting this ansatz in to the Schrödinger equation (6) yields

With the quasi-particle energy,

The wave function of the many- body system is thus an anti symmetric product of single-particle wave functions which are one-particle wave functions of an external potential well. In summary the mean field theory reduces the complicated many-body problem in to a simple one-particle system.

The main idea in this approach is to determine the mean field potential or in particular, an appropriate mean field potential in which the residual interactions between the quasi-particles should be optimal. To do so, one may seek an optimal set

As a starting point, one may construct an ansatz wave function. It is customary to use a product of single particle wave functions as Eigen function of the system,

It is an anti-symmetrized product ansatz wave function following the Hartree-Fock method and is called the Slater determinant of the given single particle states

Here

The energy

which can be transformed in to an unconstrained one by minimizing the energy for normalized wave function,

One powerful method to address such uncertainties is the following Hartree consistent equation [4,5],

This equation is like the Schrödinger equation except that the simple potential term, V

Here, the Hartree mean field potential,

The differential equation (14) is nonlinear and therefore, much more difficult to solve than the regular Schrödinger equation. The solution can only be carried using consistent iteration method. In this procedure, one can start using a complete set of guessed single-particle states

This procedure is repeated to achieve self-consistency for wave functions (or Eigen energies). This means that after each loop the resultant wave function or Eigen energies compared with the starting wave function or Eigen energies and when their difference becomes less than a given preset limit, i.e.

the procedure is repeated, otherwise, it will be automatically terminated. Where the

The results of each run, contain a self-consistent mean field,

In our discussions, the generated mean-field potential is a central one, that is only a function of

In some convenient way to avoid self-consistency loops, a phenomenological potential like a simple square well with finite depth, simple harmonic oscillator well and complicated Woods-Saxon with considerable parameters that can be determined using the fit of potential with experimental data, is introduced.

## 2. Applications of mean-field theory in nuclear physics

Over the years after Rutherford's valuable experiments that suggest nuclei for atom, many theoretical and experimental attempts have been done to obtain knowledge about the stability of nuclei. It is clear today that a nucleus of mass number

As mentioned above, a conventional approach is to select a particular type of mean field potential to avoid the steps leading to self-consistency. The selected mean field potential and considered remaining residual interactions as approximations produce the preciseness of the obtained results. The simplest custom potential is the three-dimensional harmonic oscillator potential well

where

where

The + and – signs are considered for protons and neutrons, respectively. In the case when there is no distinction between protons and neutrons a suitable average value of

The Woods–Saxon potential,

### 2.1. The spin–orbit interaction

Sometimes in 1949, Meyer and independently, Haxel, Jensen, and Swees showed that if in addition to mean field central potential,

The non-central potential due to the interaction between the spin of nucleons with the angular momentum of orbital that nucleons located on it, is called spin-orbit interaction. As a result of spin-orbit interaction [7, 8], the nuclear energy level for a given

Consider that the harmonic oscillator central potential is produced only for the first three observed magic numbers 2, 8, and 20. To obtain the remaining numbers 28, 50, 82 and 126, it is necessary to add a spin-orbit interaction potential to the Schrödinger equation.

The origin of the spin–orbit interaction is not the same in atoms and nucleus. The atomic spin–orbit force is due to a well-known electromagnetic interaction, and the scale of energy separation is in the order of milli-electronvolts, while the energy difference of sublevels separated because of the nuclear spin-orbit interaction is in the order of million electronvolts and its origin is not well understood yet. In most cases, this force is considered phenomenologically. For the spin–orbit term, we use [9]

The second pair of parentheses guarantees that the derivative does not operate on the wave function when substituted in the radial Schrödinger equation. The *r* dependence of this interaction arises from its central nature.

The derivative part of this potential is often neglected for simplicity and

To obtain the strength of the spin-orbit part, we use

and its expectation value for the nuclear wave equation made

In addition to the mean field plus spin–orbit interaction, protons in nuclei interact together via the coulomb force, which is defined by the following relation, considering nuclei as a sphere with a constant charge density [10]

To obtain the energy spectrum and wave functions for neutrons, one needs to solve the radial Schrödinger equation for the Woods-Saxon and spin-orbit potentials. Such second-order differential equation cannot be solved analytically. To solve this complicated differential equation, it is necessary to introduce some new variables and use reasonable approximations. By introducing new variable [11]

where

Likewise, the Taylor expansion is applicable for

where

This type of expansion has been widely used for differential equations resulting from the Schrödinger equation with different potentials [12].

By means of these expansions, the spin-orbit term transforms into

and the centrifugal term is obtained

By using these expansions, the spin-orbit term transforms into

This equation can be transformed into the following simple form,

Equation (20) can be transformed into the well-known form of hypergeometric differential equation or, alternatively Nikiforo-Avorono (NU) type [13]. The obtained results using the NU method are

where Γ is the well-known gamma function, and C is the normalization constant. λ, μ, and η^{'} are defined as follows

Note that λ is valid only for the *l=0,* the solution reduces to its simple form. Also, the energy eigenvalues are obtained as a function of z satisfying the following relation

which *Ф* (z) can be evaluated using a graphical method.

Note that

and

Finally,

The results obtained in this special case are in agreement with the results obtained using other methods [14].

## 3. Conclusions

In this chapter we briefly discussed the idea of mean field theory as an improvable approximation method for many-body problems of identical particles like atoms and nucleus that cannot be solved analytically. We have shown that for a system of A - nucleons nucleus by considering a suitable potential using this model, one is able to obtain energy spectrum and wave equations. However, the obtained results cannot reproduce the measured nuclear spectroscopy, but one may hope to become successful by considering an accurate potential in the Schrödinger equation.