Abstract
In this chapter, we relax the condition that electrons inside the atomic target \(\mathfrak{T}\), or after leaving it, do not experience any SDIs. Hence, the spin-states of a photoelectron plus a photoion, or of two electrons, are now generated in the presence of both the C + SOIs. A proper description of the physical situation arising from the presence of both of these two important interactions demands [10, 60] that the required density matrices be now calculated in the j-j coupling scheme of angular momenta. This chapter, therefore, first presents a reformulation in j-j coupling of the density matrices for each of the processes of 1-SPI (1.1), 1-DPI (1.2), and 2-DPI (1.3), hitherto calculated in the L-S coupling in the respective Chaps. 4–6.
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Notes
- 1.
See the discussion on page 70 related to (3.40).
- 2.
One can readily write, using either of the definitions (A.26) and (A.27a),
$$\displaystyle\begin{array}{rcl}{ \rho }^{\mbox{ ($e_{p}$)}}\ =\ \mbox{ Tr}_{{ \mbox{ $\mathfrak{T}$}}^{1+}}\Big({\rho }^{\mbox{ (1-SPI)}}\Big)& & \end{array}$$(7.6a)and, hence,
$$\displaystyle\begin{array}{rcl} & & \langle \mu _{p}\,\hat{\boldsymbol{u}}_{p}\,\boldsymbol{k}_{p}{\vert \,\rho }^{\mbox{ ($e_{p}$)}}\,\vert \mu _{p}^{\,{\prime}}\,\hat{\boldsymbol{u}}_{p}\,\boldsymbol{k}_{p}\rangle \\ & & \quad =\displaystyle\sum _{ M_{{ 1}^{+}}=-J_{{1}^{+}}}^{J_{{1}^{+}}}\langle J_{{ 1}^{+}}M_{{1}^{+}};\,\mu _{p}\,\hat{\boldsymbol{u}}_{p}\,\boldsymbol{k}_{p}{\vert \,\rho }^{\mbox{ (1-SPI)}}\,\vert J_{{ 1}^{+}}M_{{1}^{+}};\,\mu _{ p}^{\,{\prime}}\,\hat{\boldsymbol{u}}_{p}\,\boldsymbol{k}_{p}\rangle \end{array}$$(7.6b)in view of (A.27b). Here, superscript e p means that the density operator and matrix in (7.6a) and (7.6b), respectively, describe only the observed photoelectron.
- 3.
For definition of \(\boldsymbol{j}_{t}\), see, for example, footnote (1) on page 151.
- 4.
In the present case, one has
$$\displaystyle\begin{array}{rcl}{ \rho }^{\mbox{ (${\mbox{ $\mathfrak{T}$}}^{1+}$)}}\ =\ \displaystyle\int \bigg [\mbox{ Tr}_{e_{ p}}\Big({\rho }^{\mbox{ (1-SPI)}}\Big)\bigg]\,\mbox{ d}\hat{\boldsymbol{k}}_{p}.& & \end{array}$$(7.7a)The corresponding density matrix is, therefore, given by
$$\displaystyle\begin{array}{rcl} & & \langle J_{{1}^{+}}M_{{1}^{+}}{\vert \,\rho }^{\mbox{ (${\mbox{ $\mathfrak{T}$}}^{1+}$)}}\,\vert J_{{ 1}^{+}}M_{{1}^{+}}^{\,{\prime}}\rangle \\ & & \quad =\displaystyle\int \Bigg (\displaystyle\sum _{\mu _{ p}=-\frac{1} {2} }^{+\frac{1} {2} }\langle J_{{ 1}^{+}}M_{{1}^{+}};\,\mu _{p}\,\hat{\boldsymbol{u}}_{p}\,\boldsymbol{k}_{p}{\vert \,\rho }^{\mbox{ (1-SPI)}}\,\vert J_{{ 1}^{+}}M_{{1}^{+}}^{\,{\prime}};\,\mu _{ p}^{\,{\prime}}\,\hat{\boldsymbol{u}}_{p}\,\boldsymbol{k}_{p}\rangle \Bigg)\,\mbox{ d}\hat{\boldsymbol{k}}_{p}. \end{array}$$(7.7b)The superscript \({\mbox{ $\mathfrak{T}$}}^{1+}\), in (7.7), is used to indicate physical quantities related to only the photoion in the process (1.1).
- 5.
For a brief description of circular dichroism, see, for example, footnote (3) in Chap. 11.
- 6.
See footnote (6) on page 39.
- 7.
Partial transpose of a Hermitian matrix is also Hermitian.
- 8.
In the present example of (7.21), each of both Xe and \({\mbox{ Xe}}^{2+}\) is in \(^{1}\mbox{ S}_{0}\) electronic state.
- 9.
It, in other words, means that there is no circular dichroism [see footnote (3) in Chapter 11] in the fine-structure entanglement between the spins of \((e_{p},\,e_{a})\) in the presently being considered experimental geometry (7.22).
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Chandra, N., Ghosh, R. (2013). Fine-Structure Entanglement: Bipartite States of Flying Particles with Rest Mass Different from Zero. In: Quantum Entanglement in Electron Optics. Springer Series on Atomic, Optical, and Plasma Physics, vol 67. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24070-6_7
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