3. THE QUANTUM MODEL 8 In quantization of the theory, we obtain the Wheeler-DeWitt equation or So, we get a Schr¨ odinger like equation where the new variable plays the role of a dynamical time for the theory. Explicitly, the Schr¨ odinger equation can be written as: where we have ignored the potential term in this regime. In fact, in quantum cosmology, the Universe is described by a single wave function ψ providing puzzling interpretations when analyzing the differences between ordinary quantum mechanics and quantum cosmology. If we consider the universe wave function as: then, by using separation of variables method we obtain where l 2 = m 2 + n 2 + k $^{2}$, k 2 = 24 πE ℏ 2 and E , k , m , n and l are arbitrary constants. The solutions to these equations are: A ( a ) = √$_{a}$$_{(}$$_{c}$$_{1}$$_{′′}$$_{cos(}$ √ (1 − 4 l $^{2}$) ln a 2 ) + c$_{2}$ ′′ sin( √ (1 − 4 l $^{2}$) ln a 2 )) , (53) B − ( β − ) = c$_{1}$ ′ cos( nβ − ) + c$_{2}$ ′ sin( nβ − ) , (52) B$_{+}$ ( β$_{+}$ ) = c$_{1}$ cos( mβ$_{+}$ ) + c$_{2}$ sin( mβ$_{+}$ ) , (51) T ( T ) = exp( − iET ℏ ) , (50) a 2 A d $^{2}$A da 2 + l 2 = 0 , (49) 1 B − d $^{2}$B − dβ 2 − + n 2 = 0 , (48) 1 B$_{+}$ d $^{2}$B$_{+}$ dβ 2 + + m 2 = 0 , (47) i ℏ 1 T d T dT = E, (46) ψ ( a, β$_{+}$, β − ; T ) = A ( a ) B$_{+}$ ( β$_{+}$ ) B − ( β − ) T ( T ) , (45) 1 24 [ a 2 ∂ 2 ∂a 2 − ∂ 2 ∂β 2 + − ∂ 2 ∂β 2 − ] ψ = i ℏ ∂ ∂T ψ, (44) ̂ H ∗ ψ = i ℏ ∂ ∂T ψ, (43) ̂ H ∗ ψ = − Π$_{T}$ ψ. (42) ̂ H $_{1}$ψ = 0 , (41)