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Expansion of the dilation operator
A scaling wavefunction
(see Eq.(6.1)) whose
unscaled wavenumber is
will be rescaled to a new wavenumber
, where
is the wavenumber shift.
Its value at location
is
![\begin{displaymath}
\psi(k_\mu + \delta,{\mathbf r}) \ = \ \psi(\frac{k_\mu + \...
...k_\mu}
{\mathbf r} ) \ = \
\psi({\mathbf r} + {\mathbf a}) ,
\end{displaymath}](img1872.gif) |
(I.1) |
where the translation vector is
.
Formally this translation can be performed by exponentiation of the
generator of translation [174,7]:
![\begin{displaymath}
\psi(k_\mu + \delta,{\mathbf r}) \ = \ e^{{\mathbf a}\cdot\...
...\cdot\nabla)
({\mathbf a}\cdot\nabla) + \cdots \right] \psi .
\end{displaymath}](img1874.gif) |
(I.2) |
Thus we have written the scaled value in terms of local derivatives of the
unscaled wavefunction.
This becomes a definition of dilation only when the
-dependence
of
is finally substituted.
Note that
is held constant as far as
is concerned.
(This differs from other possible definitions of dilation [14],
where the commutator
plays a role;
however my form
will be simpler to expand to high order).
For instance in Cartesian coordinates,
is to be interpreted as
(Einstein summation
assumed), because in Cartesian coordinates
.
Figure I.1:
Curvilinear coordinates
used to represent points
near the
billiard boundary
.
The orthogonal unit vectors are the vector fields
and
.
![\begin{figure}\centerline{\epsfig{figure=fig_ap/curvi.eps,width=0.7\hsize}}\end{figure}](img1881.gif) |
Next: Curvilinear boundary coordinates
Up: Appendix I: Scaling expansion
Previous: Appendix I: Scaling expansion
Alex Barnett
2001-10-03