Single-particle bases and matrix structure of the single-particle hermitian operators

Throughout this section we restrict our analysis to hermitian single-particle operators, and we study their matrix elements in the single-particle space. Therefore, we are here concerned with the odd number of particles (one), and hence we have to consider the double group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$. As discussed in[6], the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ operators are either linear or antilinear, and they can have squares equal to either unity or minus unity, as summarized in Table II of Ref.[6]. This gives us four categories of operators with markedly different properties, which here will be used for different purposes.

In the discussion which follows, we assume that the single-particle basis is composed of pairs of time-reversed states, and in addition we assume that the spatial wave functions are real, i.e., not affected by the time-reversal,

$$\hat{\cal{T}}\vert\mbox{{\boldmath {$n$ }}}\,\zeta\rangle = \zeta\vert\mbox{{\boldmath {$n$ }}}\,-\!\zeta\rangle,$$ (9)

where $\zeta$=$\pm$1 represents the intrinsic-spin degree of freedom, and $\mbox{{\boldmath {$n$ }}}$ represents the set of quantum numbers corresponding to space coordinates. In particular, for the harmonic-oscillator (HO) basis, $\mbox{{\boldmath {$n$ }}}$= (n_x,n_y,n_z) are the numbers of quanta in three Cartesian directions. Assumption (9) does not preclude whether or not the time-reversal is a conserved operator; it only defines the property of the single-particle basis in which the dynamic problem is to be solved. (In principle, the discussion below can be mutatis mutandis repeated with $\hat{\cal{T}}$ replaced by $\hat{\cal{P}}^T$, however, the use of the time-reversal operator is more appropriate in practical applications.)

From now on we also assume that the basis is ordered in such a way that its first half corresponds to the $\zeta$=+1 states, and the second half is composed of their time-reversed $\zeta$=-1 partners. In fact, we are entirely free to choose states in the first half of the basis ($\zeta$=+1), and then Eq. (9) defines those which belong to the second half ($\zeta$=-1). In such basis, the single-particle matrix elements corresponding to an arbitrary hermitian operator $\hat{\cal{O}}$ have the form

$${\cal{O}}= \left(\begin{array}{cc} A & Y \\ Y^\dagger & B \end{array} \right)$$ (10)

where A and B are hermitian matrices, Y is arbitrary, and all submatrices are, in general, complex.