4 Transition Amplitudes

Greens function methods may also be used to solve the coupled equations, and furthermore suggest various approximations which simplify the solution methods in many special cases of interest. First, therefore, we present in this section several exact results using T-matrix integrals derived from Greens function analysis. In the following section 5 we examine various consequent approximations that are often still accurate. If the full wave function $\Psi$ were known, then the scattering amplitude f_i for the $i_0\rightarrow i$ reaction may be found from the T-matrix by the equivalence

$$T_{i_0i} = - \frac{2 \pi \hbar^2}{\mu_i} f_i$$ (20)

so that, written in terms of the transition amplitudes, the expression for the cross section becomes

$$d\sigma_i(\theta_i)/d\Omega = \frac{\mu_i\mu_{i_0}}{(2\pi\hbar^2)... ...i}{k_{i_0}} ~\frac{n_{pi_0}!n_{ti_0}!}{n_{pi}!n_{ti}!} ~ \vert T_{i_0i}\vert ^2$$ (21)

Expressions for the T transition amplitudes may be derived by using either plane waves or distorted waves in the exit channel. In addition, for transfer reactions where the channel Hamiltonians are different in the initial and final channels, we have a further choice of using either post or prior forms of the coupling. The post form uses the form of H_i for the exit channel, and the prior form the H_i0 from the entrance channel. The plane-wave post matrix element is

$$\langle \phi_{pi}\phi_{ti}e^{i{\bf k}_i{\bf R}_i} \vert H_i-E \vert\sum_j \phi_{pj}\phi_{tj}\psi_j^{(+)} \rangle$$ T_i0i = (22)

$$\langle \phi_{pi}\phi_{ti}e^{i{\bf k}_i{\bf R}_i} \vert V_i \vert\sum_j \phi_{pj}\phi_{tj}\psi_j^{(+)} \rangle$$ = (23)

where the ⁽⁺⁾ superscript in $\psi_j^{(+)}$ indicates that they are found with plane incoming waves in the i₀ channel. A prior matrix element uses the Hamiltonian H_i0 of the initial channel. Direct substitution in Eq. (22), however, affords no simplifications, so we insert it in the matrix element for the time-reversed reaction, and derive

$$T_{i_0i} = \langle \sum_j \phi_{pj}\phi_{tj}\psi_j^{(-)} \vert V_{i_0} \vert \phi_{pi_0}\phi_{ti_0}e^{i{\bf k}_{i_0}{\bf R}_{i_0}}\rangle$$ (24)

where the ^(-) superscript in $\psi_j^{(-)}$ indicates that it has an incident plane wave along ${\bf k}_{i_0}$ and incoming spherical waves e^-ik_iR_i in all channels. Distorted-wave expressions may be found by replacing the exponential factors on the left sides by one-channel scattering waves ( $\chi_i^{(+)}({\bf R}_i)$ on the right sides and $\chi_i^{(-)}({\bf R}_i)$ on the left), found with some distorting potential W_i by $[T_i + W_i - E_i]\chi_i({\bf R}_i)=0$. The distorted-wave post matrix element is then

$$T_{i_0i} = \langle \phi_{pi}\phi_{ti}\chi_i^{(-)} \vert V_i-W_i \vert\sum_j \phi_{pj}\phi_{tj}\psi_j^{(+)} \rangle$$ (25)

and the equivalent prior form is

$$T_{i_0i} = \langle \sum_j \phi_{pj}\phi_{tj}\psi_j^{(-)} \vert V_{i_0}-W_{i_0} \vert \phi_{pi_0}\phi_{ti_0}\chi_i^{(+)}\rangle$$ (26)

The distorting potential W_i may be real or complex without affecting the validity of these matrix elements. All these four expressions are so far identical, and exactly equivalent to solving the coupled equations directly and using Eq. (20).