of all possible final states with different values of the magnetic numbers $m_{f}$ at the same orbital number $l_{f}$. Let’s assume that in the initial state we have $l_{i}=m_{i}=0$ and the radial component of wave function $\varphi_{f} (r)$ does not depend on $m_{f}$ for selected $l_{f}$. We write wave functions so: $$\begin{array}{lcl}
  \psi_{i} (\mathbf{r}) & = & \varphi_{i} (r) \: Y_{00}({\mathbf n}_{r}^{i}), \\
  \psi_{f} (\mathbf{r}) & = & \varphi_{f} (r) \: \displaystyle\sum\limits_{m} Y_{l_{f}m}({\mathbf n}_{r}^{f})
\end{array}
\label{eq.2.4.2}$$ and obtain: $$p\,(k_{i}, k_{f}) = \sqrt{2\pi}\: \sum\limits_{l} \: (-i)^{l}\, \sqrt{2l+1} \: \Bigl[ p_{l}^{M} -ip_{l}^{E} \Bigr],
\label{eq.2.4.3}$$ where $$\begin{array}{ll}
  p_{l}^{M} = \sum\limits_{\mu = -1, 1} \mu\, h_{\mu}\: p_{l\mu}^{M}, &
  p_{l}^{E} = \sum\limits_{\mu = -1, 1} \mu^{2} h_{\mu}\: p_{l\mu}^{E}
\end{array}
\label{eq.2.4.4}$$ and $$\begin{array}{lcl}
  p_{l\mu}^{M} & = &
        \displaystyle\int\limits^{+\infty}_{0} dr
        \displaystyle\int d\Omega \: r^{2} \,
        \psi^{*}_{f}(\mathbf{r}) \,
        \biggl( \displaystyle\frac{\partial}{\partial \mathbf{r}}\, \psi_{i}(\mathbf{r}) \biggr) \,
        \mathbf{A}_{l\mu}^{*} (\mathbf{r}, M), \\

  p_{l\mu}^{E} & = &
        \displaystyle\int\limits^{+\infty}_{0} dr
        \displaystyle\int d\Omega \: r^{2} \,
        \psi^{*}_{f}(\mathbf{r}) \,
        \biggl( \displaystyle\frac{\partial}{\partial \mathbf{r}}\, \psi_{i}(\mathbf{r}) \biggr)\,
        \mathbf{A}_{l\mu}^{*} (\mathbf{r}, E).
\end{array}
\label{eq.2.4.5}$$

Using *gradient formula* (see (2.56), p. 46 in [@Eisenberg.1973]): $$\displaystyle\frac{\partial}{\partial \mathbf{r}}\:
    f(r)\, Y_{lm}({\mathbf n}_{r}) =
  \sqrt{\displaystyle\frac{l}{2l+1}}\:
    \biggl( \displaystyle\frac{df}{dr} + \displaystyle\frac{l+1}{r} f \biggr)\,
    \mathbf{T}_{l l-1, m}({\mathbf n}_{r}) -
  \sqrt{\displaystyle\frac{l+1}{2l+1}}\:
    \biggl( \displaystyle\frac{df}{dr} - \displaystyle\frac{l}{r} f \biggr)\,
    \mathbf{T}_{l l+1, m}({\mathbf 