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Previous: T-signature or T-simplex
Signature or simplex
Let us now consider operator
which is even or odd
with respect to one of the six linear D
operators,
signatures
or simplexes
,
k=x,y,z, which have
squares equal to minus unity, (
=
=
,
where
denotes
one of them), i.e.,
![\begin{displaymath}
\hat{\cal{X}}^\dagger\hat{\cal{O}}\hat{\cal{X}}= \epsilon_X\hat{\cal{O}},
\qquad \epsilon_X=\pm 1.
\end{displaymath}](img145.gif) |
(18) |
Since every operator
commutes with the time-reversal
,
one can always
choose a basis in which Eq. (9) holds,
and
![\begin{displaymath}
\hat{\cal{X}}\vert\mbox{{\boldmath {$n$ }}}\,\zeta\rangle = i\zeta\vert\mbox{{\boldmath {$n$ }}}\,\zeta\rangle,
\end{displaymath}](img146.gif) |
(19) |
where again
=
1, so r=
=
is the signature
(for
=
)
or simplex s=
=
(for
=
)
quantum number.
Table 4 lists
such bases constructed for the HO states
|nxnynz,sz=
.
A similar construction
is possible for any other single-particle basis. Note that
one can arbitrarily change the phases of states
=
,
and still fulfill Eqs. (9) and (19); we shall use this
freedom in Secs. 3.1.8 and 3.2.
From Eqs. (18) and (19) one gets:
![\begin{displaymath}
\langle \mbox{{\boldmath {$n$ }}}\,\zeta\vert\hat{\cal{O}}\...
...\hat{\cal{O}}\vert
\mbox{{\boldmath {$n$ }}}'\,\zeta'\rangle,
\end{displaymath}](img149.gif) |
(20) |
so the matrix
has the form
![\begin{displaymath}
\left(\begin{array}{cc}
A & 0 \\
0 & B
\end{array} \rig...
...egin{array}{cc}
0 & Y \\
Y^\dagger & 0
\end{array} \right)
\end{displaymath}](img150.gif) |
(21) |
for
=+1 and
=-1, respectively,
with A and B hermitian, and Y arbitrary complex matrix.
Note that in
order to diagonalize
one only needs to
diagonalize: (i) for
=+1, two hermitian matrices
with complex, in general, submatrices
in twice smaller dimension, which gives real
eigenvalues with no additional restrictions,
or (ii) for
=-1, one hermitian
matrix
,
again with complex submatrices in twice smaller dimension,
which gives pairs of real non-zero eigenvalues with opposite signs.
Table 4:
Eigenstates
of the signature or
simplex operators,
or
for
k=x, y, or z, [cf. Eqs. (9) and (19)],
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.
Phases of eigenstates are fixed so as to fulfill condition
(41).
k |
![$\zeta$](img106.gif) |
![$\vert n_xn_yn_z\,\zeta\rangle_k$](img153.gif) |
| | |
x |
+1 |
|
![$\frac{1}{\sqrt{2}}$](img138.gif) |
(|nxnynz,sz=
![$\frac{1}{2}\rangle $](img139.gif) | - |
(-1)Nx |
|nxnynz,sz=
![$-\frac{1}{2}\rangle\,)$](img140.gif) |
x |
-1 |
|
![$\frac{(-1)^{N_x}}{\sqrt{2}}$](img154.gif) |
(|nxnynz,sz=
![$\frac{1}{2}\rangle $](img139.gif) | + |
(-1)Nx |
|nxnynz,sz=
![$-\frac{1}{2}\rangle\,)$](img140.gif) |
y |
+1 |
|
![$\frac{i^{N_y}}{\sqrt{2}}$](img155.gif) |
(|nxnynz,sz=
![$\frac{1}{2}\rangle $](img139.gif) | - |
i(-1)Ny |
|nxnynz,sz=
![$-\frac{1}{2}\rangle\,)$](img140.gif) |
y |
-1 |
|
![$\frac{i^{N_y-1}}{\sqrt{2}}$](img156.gif) |
(|nxnynz,sz=
![$\frac{1}{2}\rangle $](img139.gif) | + |
i(-1)Ny |
|nxnynz,sz=
![$-\frac{1}{2}\rangle\,)$](img140.gif) |
z |
+1 |
=
![$+\frac{1}{2}(-1)^{N_z+1}\rangle$](img158.gif) |
| | |
z |
-1 |
=
![$-\frac{1}{2}(-1)^{N_z+1}\rangle$](img160.gif) |
| | |
Comparing result (21) with that obtained in
Sec. 3.1.2, one sees that the antilinear symmetries allow for
using real matrices, while linear symmetries give special
block-diagonal forms for complex matrices.
Next: Parity
Up: Single-particle bases for conserved
Previous: T-signature or T-simplex
Jacek Dobaczewski
2000-02-05