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Next: Parametrization of the LST Up: Transformed Harmonic Oscillator Basis Previous: Local-scaling point transformations

Transformed harmonic oscillator wave functions

 

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} (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} (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} (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} (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} (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} (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} (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} (12)


next up previous
Next: Parametrization of the LST Up: Transformed Harmonic Oscillator Basis Previous: Local-scaling point transformations
Jacek Dobaczewski
1999-09-13