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T-signature or T-simplex
As is well known[17,6], for each of the six antilinear D
operators, i.e., for T-signature
or T-simplex
,
k=x,y,z, which have
squares equal to unity, (
=
,
where
denotes one
of them), one can construct a basis composed of eigenvectors of
with eigenvalue equal to 1,
![\begin{displaymath}
\hat{\cal{Z}}\vert\mbox{{\boldmath {$n$ }}}\,\zeta\rangle = \vert\mbox{{\boldmath {$n$ }}}\,\zeta\rangle.
\end{displaymath}](img121.gif) |
(14) |
Moreover, since every
operator
commutes with the time-reversal
,
such basis can
always be chosen so as to fulfill condition
(9) at the same time. Table 3 lists
examples of such bases, constructed for the HO states
|nxnynz,sz=
.
A similar construction
is possible for any other single-particle basis, and has the explicit
form shown in Table 3 provided
the space and spin degrees of freedom are separated. Note that
any linear combination of states
=
and
=
,
with real coefficients, is
another valid eigenstate of
with eigenvalue 1.
For operators even or odd with respect to
,
![\begin{displaymath}
\hat{\cal{Z}}^\dagger\hat{\cal{O}}\hat{\cal{Z}}= \epsilon_Z\hat{\cal{O}}, \qquad \epsilon_Z=\pm1,
\end{displaymath}](img126.gif) |
(15) |
one then has
![\begin{displaymath}
\langle \mbox{{\boldmath {$n$ }}}\,\zeta\vert\hat{\cal{O}}\...
...\hat{\cal{O}}\vert\mbox{{\boldmath {$n$ }}}'\,\zeta'\rangle^*.
\end{displaymath}](img127.gif) |
(16) |
Hence, in bases fulfilling Eqs. (9) and (14),
matrix
is purely real
(
=+1) or purely imaginary (
=-1). This gives
matrix
in the form (tilde stands for the transposition)
![\begin{displaymath}
\left(\begin{array}{cc}
A & Y \\
\widetilde{Y} & B
\end...
...{cc}
A' & Y' \\
-\widetilde{Y}' & -B'
\end{array} \right),
\end{displaymath}](img130.gif) |
(17) |
for
=+1 and
=-1, respectively,
where all submatrices are real, A and B are symmetric, A' and B' are
antisymmetric, and Y and Y' are arbitrary.
Note that in order to diagonalize
one only needs to diagonalize a real matrix with unrestricted
eigenvalues (for
=+1),
or an imaginary matrix with pairs of opposite non-zero
eigenvalues (for
=-1).
Table 3:
Examples of eigenstates
of
the T-signature,
,
or
T-simplex,
operators,
k=x, y, or z, with eigenvalue 1
[cf. Eqs. (9) and (14)],
determined
for the harmonic oscillator states
|nxnynz,sz=
.
Symbols
(Nx,Ny,Nz) refer to
(nx,ny,nz) for
operators, and to (ny+nz,nx+nz,nx+ny) for
operators.
k |
![$\zeta$](img106.gif) |
![$\vert n_xn_yn_z\,\zeta\rangle_k^T$](img131.gif) |
| | |
x |
+1 |
=
![$+\frac{1}{2}\rangle$](img133.gif) |
| | |
x |
-1 |
=
![$-\frac{1}{2}\rangle$](img135.gif) |
| | |
y |
+1 |
=
![$+\frac{1}{2}\rangle$](img133.gif) |
| | |
y |
-1 |
=
![$-\frac{1}{2}\rangle$](img135.gif) |
| | |
z |
+1 |
|
![$\frac{1}{\sqrt{2}}$](img138.gif) | (
|nxnynz,sz=
![$\frac{1}{2}\rangle $](img139.gif) | + |
i(-1)Nz |
|nxnynz,sz=
![$-\frac{1}{2}\rangle\,)$](img140.gif) |
z |
-1 |
|
![$\frac{i(-1)^{N_z}}{\sqrt{2}}$](img141.gif) | (
|nxnynz,sz=
![$\frac{1}{2}\rangle $](img139.gif) | - |
i(-1)Nz |
|nxnynz,sz=
![$-\frac{1}{2}\rangle\,)$](img140.gif) |
Next: Signature or simplex
Up: Single-particle bases for conserved
Previous: Time-reversal
Jacek Dobaczewski
2000-02-05