b$_{NL}$ = ∫ R $^{2}$dR f (2 / 3) NL ( R ) e − η $_{3}$R 2 × ∫ d ̂ R Y ∗ L $_{0}$( ̂ R ) e η (arccos θ$_{R}$ − 3 a $^{2}$/ 4) (7) Φ$_{Jm}$$_{Π}$$_{q}$( r , R ) ≡ [ φ (1 / 2) nl ( r ) φ (2 / 3) NL ( R )$_{Jm}$] (4) G$_{Jm}$$_{Π}$$_{q}$ = ∑ q ′ a$_{n}$ ′ l ′ $_{m}$b$_{N}$ ′ L ′ C Jm l ′ m,L ′ $_{0}$(1 + ( − 1) l $^{′}$) × [ δ$_{qq}$ ′ + A q $^{′}$J ( β → α ) + A q $^{′}$J ( γ → α )] (5) a$_{nlm}$ = ∫ r $^{2}$dr f (1 / 2) nl ( r ) $_{e}$− η $_{4}$r 2 × ∫ $^{d}$̂ r Y ∗ $^{$_{lm}$(}$̂ r ) e η $_{$_{2}$(}$√$_{3 arcsin}$$_{θ}$$_{r}$ cos φ$_{r}$ − 3 a $^{2}$/ 2) (6) A q $^{′}$J ( β → α ) ≡ 〈 [ φ (1 / 2) nl ( r ) φ (2 / 3) NL ( R )]$_{J}$ | [ φ (1 / 2) n ′ l ′ ( r $^{β}$) φ (2 / 3) N ′ L ′ ( R $^{β}$)]$_{J}$ 〉 (8) H$_{evol}$ = H$_{c}$ + H$_{in}$ (9) H$_{in}$ = −∇ 2 $_{r}$+ 1 4 r 2 − 3 4 ∇ 2 $_{R}$+ 1 3 R $^{2}$+ ∑ i<j V ( | r i − r j | ) (10) 〈 Φ$_{J}$ ′ m ′ Π ′ q ′ ( r , R ) | H$_{in}$ | Φ$_{Jm}$$_{Π}$$_{q}$( r , R ) 〉 = δ$_{J}$ ′ $_{J}$δ$_{m}$ ′ $_{m}$δ$_{Π}$ ′ Π [ δ$_{q}$ ′ $_{q}$(2 n + 2 N + l + L + 3) + δ$_{l}$ ′ $_{l}$δ$_{N}$ ′ $_{N}$δ$_{L}$ ′ $_{L}$V$_{n}$ ′ nl +2 ∑ q ′′ ,q ′′′ δ$_{l}$ ′′′ l ′′ δ$_{N}$ ′′′ N ′′ δ$_{L}$ ′′′ L ′′ A q $^{′}$J q ′′′ ( α → β ) A qJ q ′′ ( α → β ) V$_{n}$ ′′′ n ′′ l ′′ ] (11) 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 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. The evolution is governed by the Hamiltonian H$_{evol}$ containing U$_{evol}$ ( r ) and the interaction. When the Jacobi 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. 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 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. where where l and L are coupled to J and m . The notation q denotes a set of quantum numbers n, l, N and L , and Π = ( − 1) l + L is the parity, 2 III. HAMILTONIAN AND ITS EIGENSTATES V$_{n}$ ′ nl = ∫ r $^{2}$dr f (1 / 2) n ′ l ( r ) V ( r ) f (1 / 2) nl ( r ) (12)