Next: Parametrization of the LST
Up: Transformed Harmonic Oscillator Basis
Previous: Local-scaling point transformations
The anisotropic three-dimensional HO potential with three different oscillator
lengths
![\begin{displaymath}L_{k} \equiv \frac{1}{b_k}=\sqrt{\frac{\hbar}{m\omega_{k}}},
\end{displaymath}](img16.gif) |
(5) |
has the form
![\begin{displaymath}U({\mbox{{\boldmath {$r$ }}}})=\frac{\hbar^2}{2m}\left(
\frac...
...}} +\frac{y^{2}}{ L_{y}^{4}}
+\frac{z^{2}}{L_{z}^{4}}\right) .
\end{displaymath}](img17.gif) |
(6) |
Its eigenstates,
the separable HO single-particle wave functions
![\begin{displaymath}\varphi_{\alpha}({\mbox{{\boldmath {$r$ }}}})=\varphi _{n_{x}}(x)\varphi _{n_{y}}(y)
\varphi _{n_{z}}(z) ,
\end{displaymath}](img18.gif) |
(7) |
have a Gaussian asymptotic behavior at large
distances,
![\begin{displaymath}\varphi _{\alpha}({\mbox{{\boldmath {$r$ }}}\rightarrow \inft...
...c{y^{2}}{L_{y}^{2}}
+\frac{z^{2}}{L_{z}^{2}}
\right)
\right] .
\end{displaymath}](img19.gif) |
(8) |
Applying the LST (1) to these wave functions leads to the so-called THO
single-particle wave functions (4),
![\begin{displaymath}\psi_{\alpha}({\mbox{{\boldmath {$r$ }}}})=D^{1/2}\textstyle{...
...\right) \varphi
_{n_{z}}\left( \frac{z}{{r}}f_z({r)}\right)} ,
\end{displaymath}](img20.gif) |
(9) |
whose asymptotic behavior is
![\begin{displaymath}\psi_{\alpha}({\mbox{{\boldmath {$r$ }}}\rightarrow\infty })\...
...y}^{2}r^2} + \frac{z^{2}f^2_z}{L_{z}^{2}r^2} \right)
\right] .
\end{displaymath}](img21.gif) |
(10) |
This suggests that we choose the LST
functions to satisfy the asymptotic conditions
![\begin{displaymath}f_k(r) = \left\{
\begin{array}{cl}
r & \mbox{~~~for small $r$...
...sqrt{2\kappa r} &
\mbox{~~~for large $r$ }.
\end{array}\right.
\end{displaymath}](img22.gif) |
(11) |
With such a choice, the THO wave functions at small r are identical to the
HO wave
functions (note that with (11) one obtains D=1 at small r), while at
large r they have the correct exponential and spherical asymptotic behavior,
![\begin{displaymath}\psi _{\alpha}({\mbox{{\boldmath {$r$ }}}\rightarrow \infty })\sim e^{-\kappa r}.
\end{displaymath}](img23.gif) |
(12) |
Next: Parametrization of the LST
Up: Transformed Harmonic Oscillator Basis
Previous: Local-scaling point transformations
Jacek Dobaczewski
1999-09-13