content_image/398629.jpg content_image/398630.jpg content_image/398631.jpg content_image/398628.jpg content_image/398632.jpg content_image/398634.jpg content_image/398633.jpg content_image/398635.jpg content_image/398648.jpg
The notation q denotes a set of quantum numbers n, l, N and L , and Π = ( − 1) l + L is the parity, 

where l and L are coupled to J and m . 

where 

<img src='content_image/398629.jpg'>

<img src='content_image/398630.jpg'>

<img src='content_image/398631.jpg'>

C Jm l ′ m,L ′ 0 is the Clebsch-Gordan coefficients, q ′ is for the set n ′ , l ′ , N ′ and L ′ to be summed up, θ$_{r}$ and φ$_{r}$ are the spherical polar coordinates of r , and so on. 

<img src='content_image/398628.jpg'>

<img src='content_image/398632.jpg'>

is the bracket of transformation between the β − and α − sets (the superscript α is usually ignored), which is called the Talmi-Moshinsky (T-M) coefficients. Their an- alytical expression can be found in [5–7]. In general, the symmetrization would cause the appearance of all three set of coordinates. However, by using the T-M coeffi- cients, Eq. (2) contains only the α − set so as to facil- itate greatly the calculation. Incidentally, since Ψ$_{I}$ is symmetrized, l (included in q ) of Eq. (2) must be even and therefore ( − 1) L = Π. In principle, the right side of Eq. (2) should contain infinite terms. However, the over- lap between Ψ$_{I}$ and higher h.o. states are very small. Say, if, n , N , N$_{c}$ are all smaller than 12 and l , L , L$_{c}$ are all smaller than 23, with the parameters specified below, the overlap of the right side of Eq. (2) with itself is equal to 0 . 99997. This implies that higher h.o. states can be safely ignored. 

## III. HAMILTONIAN AND ITS EIGENSTATES 

The evolution is governed by the Hamiltonian H$_{evol}$ containing U$_{evol}$ ( r ) and the interaction. When the Jacobi 

coordinates are used it can be separated as 

where H$_{c}$ = − 1 6 ∇ 2 R c + 3 $_{2}$R 2 c describes the c.m. motion, and 

<img src='content_image/398634.jpg'>

describes the internal motion. Since the eigenstates of H$_{c}$ are well known, if the eigenstates of H$_{in}$ are also know, the evolution starting from any initial state can be un- derstood. In what follows, the symmetrized eigenstates are obtained via a diagonalization of H$_{in}$ in a limited space. In this way, only approximate solutions can be obtained. Then, we increase the dimension of the space until a better convergency is achieved. 

When Φ$_{Jm}$$_{Π}$$_{q}$( r , R ) as defined in Eq. (4) are used as basis functions (where l is restricted to be even as men- tioned), the matrix elements of H$_{in}$ is 

<img src='content_image/398633.jpg'>

<img src='content_image/398635.jpg'>

<img src='content_image/398648.jpg'>

