Next: Continuity equation for densities
Up: Time-dependent density functional theory
Previous: Time-dependent density functional theory
Continuity equation for the scalar-isoscalar density
The CE now results from specifying
to the
local gauge transformation [13,Dobaczewski and Dudek(1995)] that is defined as
![\begin{displaymath}
\psi'_\alpha({\bm r}\sigma\tau) \equiv (U\psi_\alpha)({\bm r...
...gma\tau)
=e^{i\gamma({\bm r})}\psi_\alpha({\bm r}\sigma\tau)
.
\end{displaymath}](img66.png) |
(21) |
Then, Eq. (13) gives:
![\begin{displaymath}
\rho'({\bm r}\sigma\tau,{\bm r}'\sigma'\tau',t)
= e^{i\left(...
... r}')\right)}
\rho({\bm r}\sigma\tau,{\bm r}'\sigma'\tau',t)
.
\end{displaymath}](img67.png) |
(22) |
Matrix elements of the local-gauge angle
are
given by local integrals,
![\begin{displaymath}
\gamma_{\alpha\beta} = \int {\rm d}^3\bm{r}\sum_{\sigma\tau}...
...m r}\sigma\tau)
\gamma({\bm r})\psi_\beta({\bm r}\sigma\tau)
;
\end{displaymath}](img69.png) |
(23) |
therefore, from Eq. (13) again, the average value of the
gauge angle,
, depends on the scalar-isoscalar local density
,
that is,
![\begin{displaymath}
\langle\gamma\rangle = \int {\rm d}^3\bm{r}
\gamma({\bm r})\rho_0^0({\bm r},t)
.
\end{displaymath}](img72.png) |
(24) |
Now, the assumed local-gauge invariance of the potential energy
implies the equation of motion for the average value
,
which from Eq. (20) reads
![\begin{displaymath}
\frac{{\rm d}}{{\rm d} t}\langle\gamma\rangle
= -\frac{\hba...
...^3\bm{r}\gamma({\bm r})
\bm{\nabla}\cdot\bm{j}_0^0(\bm{r},t)
,
\end{displaymath}](img74.png) |
(25) |
where the standard scalar-isoscalar current is defined as [16]
.
We note here [13,Dobaczewski and Dudek(1995)], that the gauge invariance
that corresponds to a specific dependence of the gauge angle on
position,
, represents the
Galilean invariance of the potential energy for the system boosted to
momentum
. Then, equation of motion (25) simply
represents the classical equation for the center-of-mass velocity,
![\begin{displaymath}
\frac{{\rm d}}{{\rm d} t}\frac{\langle{\bm r}\rangle}{A}
\e...
...le}{mA}
\equiv \frac{\langle -i\hbar{\bm\nabla}\rangle}{mA}
.
\end{displaymath}](img78.png) |
(26) |
In the general case, that is, when the potential energy is gauge-invariant
and the gauge angle
is an arbitrary function of
,
Eq. (25) gives the CE that reads
![\begin{displaymath}
\frac{{\rm d}}{{\rm d} t} \rho_0^0({\bm r},t)
= -\frac{\hbar}{m} \bm{\nabla}\cdot\bm{j}_0^0(\bm{r},t)
.
\end{displaymath}](img80.png) |
(27) |
Thus for a gauge-invariant potential energy
density, the TDHF or TDDFT equation of motion implies the CE, that is,
the gauge invariance is a sufficient condition for the validity of the CE.
By proceeding in the opposite direction, we can prove that it is also a necessary
condition. Indeed, the CE of Eq. (27) implies the first-order
condition (19), and then the full gauge invariance stems from
the fact that the gauge transformations form local U(1) groups.
Next: Continuity equation for densities
Up: Time-dependent density functional theory
Previous: Time-dependent density functional theory
Jacek Dobaczewski
2011-11-11