ter of mass motion requires the calculation of the invariant matrix elements of the pseudo-Hamiltonian

which is added to the Hamiltonian of the system. The square of the center of mass momentum operator P will require the evaluation of one and two-body ma

2

→2

2

trix elements (viz. P and P11). On the other hand the square of the center of mass coordinate R is a sum of many-body operators which we will transform into products of one- and two-body operators as discussed in Section 1.4, Eqs. (1.48) and (1.49).

2

For a system made of Bosons and Fermions the C.M. momentum is given in terms of the field operators ( and ¥ respectively) by

the annihilation part of the real Boson fields, Eqs. (3.24, 3.25).

In order to evaluate the one-body invariant matrix elements of the quadratic terms (diagonal in the total angular momentum of the particle) we introduce the number operator N(p) in the p representation

Thus in terms of fields the single particle momentum operator

For the spin 1/2 Fermion field the number operator is (on similar li

with

[veg|p2 \v'ej] = 2 [velp2 Iv'el

in terms of the matrix element of Eq. (4.5).

(4.7)

For spin 1 Bosons with isospin t and multipolarity κ we obtain along

the same line as in Eq. (3.239) for the number operator

The radial integrals entering expression (4.5) are readily evaluated

when using the harmonic oscillator basis (3.31) with parameter απ (mw) -1

α

C

δ

1/2 (√√(x+e+1/2) 8,'v-1 + √ (v+1) (v + e +3 /2)
(v+1) (v +e+3/2) 8', v+1)

The coefficients C are defined in Eq. (3.32).

ve

. (4.10)

the

We evaluate now the invariant two-body matrix elements by expressing

scalar product P1.P1 in terms of a product of invariant operators

We consider first the single particle linear momentum operator p ex

pressed in terms of the field operators.

Figure 4.2

We are making use on this diagram of the same simplified graphical representation as in figure 3.6, to represent the coupling of the n's to the isospin part of the B's.