In Sec. 3.1 we have shown how one can simplify the
matrix elements of operators by using a given phase convention,
i.e, by fixing phases of single-particle basis states in a given
way. Whenever an antilinear D
operator is conserved,
one can always construct a phase convention for which the matrix elements
of the mean-field Hamiltonian are real numbers. However,
from technical point of view, it can be more advantageous to
fix the phase convention in yet another way. Indeed, whenever the
calculation of matrix elements is more time consuming than
the diagonalization of the Hamiltonian matrix, one may use
the phase convention to facilitate the former task, at the expense
of diagonalizing complex matrices. Moreover, such a strategy
allows for keeping the simplicity of performing the former task
even in cases when there is no antilinear conserved symmetry
available, and when one has to diagonalize complex matrices anyhow.
In the present section
we show constructions of phase conventions which facilitate
calculations of the space-spin matrix elements.
Representation (9), which separates space and spin
degrees of freedom, is convenient in applications pertaining to
deformed single-particle states, as those discussed in the
present study. This is because, each hermitian operator can be
represented as a sum of four components of the form
Matrix elements of operators
can be made purely real
or purely imaginary if phases of single-particle basis states are
chosen in such a way that, for one Cartesian direction
l=x, y, or z, one has
As is usual for antilinear operators, there is
a lot of freedom in finding bases (36) of eigenstates of
.
We can use this freedom to fulfill other useful conditions.
For example, since
and
anticommute for
l
k, one can find bases (36) which are at the
same time the eigenstates of signature or simplex operators. In fact,
phases of eigenstates listed in Table 4, has been
chosen in such a way that,
l |
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||
x |
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= |
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+1 | +1 | -1 | -1 |
y |
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= |
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+1 | -1 | +1 | -1 |
z |
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= |
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+1 | -1 | -1 | +1 |