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Let's have potential $$ U(r) = -U_{0}e^{-\frac{r}{a}}. $$ I need to find energy levels for particles moving in this field (for an arbitrary values of orbital number $l$). This task isn't exactly solvable, so I need some method which can help to find approximate energy levels.

What to do?

I reduced the Schrodinger equation to the form (with normalized $r \to \frac{r}{a}$ and $\Psi (r, \varphi , \theta ) = \kappa e^{-\beta r}r^{l}\kappa (r) Y_{lm}(\theta , \varphi )$) $$ r\kappa '' + \kappa {'}(2l + 2 - 2\beta r) + \kappa (\alpha^{2}r e^{-r} - 2\beta (l + 1)) = 0, $$ where $$ \alpha^{2} = \frac{2mU_{0}a^{2}}{\hbar^{2}}, \quad \beta^{2} = \frac{2m |E|a^{2}}{\hbar^{2}}. $$ It would be tempting to use the approximation $$ r \approx \frac{1 - e^{-r}}{e^{-r}}, $$ but $r e^{-r}$ didn't reduce to the normal form.

The exact solution for $l = 0$ is existed.

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  • $\begingroup$ @KyleKanos : unfortunately, it is harder, because there is $r, e^{-r}$ both in the equation. I tried to express $r$ as $r \approx \frac{1 - e^{-r}}{e^{-r}}$ near zero, but it didn't help to solve equation exactly. Also I can use quasiclassical approximation, but I believe that there is some other method. $\endgroup$ – Andrew McAddams Dec 17 '13 at 20:41
  • $\begingroup$ Well, there's always the variational method $\endgroup$ – Kyle Kanos Dec 17 '13 at 20:52
  • $\begingroup$ @KyleKanos : But it may help only for the ground state, doesn't it? $\endgroup$ – Andrew McAddams Dec 17 '13 at 21:01
  • $\begingroup$ Correct. Guess I didn't read the "arbitrary values of $\ell$" bit. If you need an approximation, might $\exp(-x)\simeq1-x$ be too approximate an approximation? $\endgroup$ – Kyle Kanos Dec 17 '13 at 21:06

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