content_image/371061.jpg content_image/371062.jpg content_image/371059.jpg content_image/371057.jpg content_image/371058.jpg
where c$_{1}$ , c$_{2}$ , c$_{1}$ ′ , c$_{2}$ ′ , c$_{1}$ ′′ and c$_{2}$ ′′ are normalisation coefficients. 

For the Hamiltonian operator to be self adjoint the wave function must satisfy one of the following boundary conditions [22], [23] 

<img src='content_image/371061.jpg'>

<img src='content_image/371062.jpg'>

The first boundary condition is satisfied while the second one leads to infinity. Obviously, the wave function is not square integrable and in order to obtain a possible physical solution we may construct wave packets as [23] 

<img src='content_image/371059.jpg'>

However, the wave packets are also problematic and a satisfactory solution to the WDW equation is not easy to obtain. 

## Isotropic Bianchi type IX : 

In the case of isotropic universe, the associated Wheeler-DeWitt equation, i.e., the Schr¨ odinger-like equation is 

The solution is 

<img src='content_image/371057.jpg'>

<img src='content_image/371058.jpg'>

where c$_{3}$ and c$_{4}$ are constants of integrations, J$_{+}$$_{ν}$ and J − ν are Bessel functions and ν = √$_{1}$ − 96 E/ 4. Since J − ν grows exponentially to infinity as a goes to zero, one must set c$_{4}$ = 0, and consequently the first boundary condition, (54), is satisfied. In this isotropic case the wave function is still not square integrable and even using wave packets does not give us a satisfactory wave function for the universe. 

## 4. CONCLUSION 

In this work, we present the Hamiltonian formulation of the Bianchi type IX cosmological model minimally coupled to a scalar field. We show that the dynamics of the metric functions can be obtained using a time variable, T ( t ), as a function of the scalar field. 

