Next: Properties of the D
Up: Symmetry operators
Previous: Double group D for
Cartesian harmonic oscillator basis
One often uses the Cartesian harmonic oscillator (HO) basis
to solve the self-consistent equations when neither spherical
nor axial symmetry is assumed, see, e.g., Refs. [17,10].
The Cartesian HO states are identified by
the numbers of oscillator quanta, nx, ny, and nz, in the three
Cartesian directions,
and by the spin projection sz=
on the z axis.
For the standard HO phase convention, this basis is real,
,
where according to our standard
convention the script symbol
denotes the coordinate-space
complex-conjugation operator acting in the odd-fermion-number
space
,
see Sec. 2.1.
For the HO states the following relations hold [18]:
![$\displaystyle \hat{\cal{P}}\vert n_xn_yn_z,s_z\rangle$](img151.gif) |
= |
![$\displaystyle \phantom{i}(-1)^{n_x+n_y+n_z}\vert n_xn_yn_z,s_z\rangle ,$](img152.gif) |
(47) |
![$\displaystyle \hat{\cal{T}}\vert n_xn_yn_z,s_z\rangle$](img153.gif) |
= |
![$\displaystyle \phantom{i}(-1)^{\frac{1}{2}-s_z}\vert n_xn_yn_z,-s_z\rangle ,$](img154.gif) |
(48) |
![$\displaystyle \hat{\cal{R}}_{x}\vert n_xn_yn_z,s_z\rangle$](img155.gif) |
= |
![$\displaystyle i(-1)^{n_y+n_z+1}\vert n_xn_yn_z,-s_z\rangle ,$](img156.gif) |
(49) |
![$\displaystyle \hat{\cal{R}}_{y}\vert n_xn_yn_z,s_z\rangle \!$](img157.gif) |
= |
![$\displaystyle \!\! \phantom{i}(-1)^{n_x+n_z+\frac{1}{2}-s_z}\vert n_xn_yn_z,-s_z\rangle ,$](img158.gif) |
(50) |
![$\displaystyle \hat{\cal{R}}_{z}\vert n_xn_yn_z,s_z\rangle$](img159.gif) |
= |
![$\displaystyle i(-1)^{n_x+n_y+\frac{1}{2}+s_z}\vert n_xn_yn_z,s_z\rangle ,$](img160.gif) |
(51) |
![$\displaystyle \bar{\cal{E}}\vert n_xn_yn_z,s_z\rangle$](img161.gif) |
= |
![$\displaystyle - \vert n_xn_yn_z,s_z\rangle ,$](img162.gif) |
(52) |
from where one can find similar equations for all the remaining
operators of group D
.
Since the HO Hamiltonian
is symmetric under D
,
its eigenstates can be
classified according to the ircoreps of D
.
It is easily seen that
the HO states
form bases of the spinor, pseudospinor,
antispinor and antipseudospinor ircoreps for
=nx+ny+nz, Ny=nx+
being {even, odd},
{odd, odd}, {even, even} and {odd, even}, respectively (see Appendix). The
entire HO basis would have belonged to the spinor and pseudospinor
ircoreps only, if the basis states and phase convention were chosen differently,
see Ref.[14].
Next: Properties of the D
Up: Symmetry operators
Previous: Double group D for
Jacek Dobaczewski
2000-02-05