- #1
davidge
- 553
- 21
When considering bound states of potential energy that tends to zero at large ##r##, my book arrives in
$$\frac{d^2}{dr^2} u_{E} = \kappa^2 u \ \ \ \ \kappa^2 \equiv -2mE/ \hbar^2 > 0 \ \ \ \ r \rightarrow \infty$$ from the differential equation satisfied by ##u_{E} \equiv R_{El} (r) / r##, where ##R_{El}(r)## is the radial part of the wave function.
The solution of the above eq. for ##u_{E}## is ##u_{E}(r) \propto e^{- \kappa r}##. Let ##\rho \equiv \kappa r##. The book defines ##u_{El} (\rho)## as
$$u_{El} (\rho) = \rho^{l+1}e^{-\rho} w(\rho)$$ so that the function ##w(\rho)## satisfies the equation $$\frac{d^2}{d\rho^2}w + 2 \bigg(\frac{l+1}{\rho} - 1 \bigg) \frac{dw}{d \rho} + \bigg[ \frac{V}{E} - \frac{2(l+1)}{\rho} \bigg]w = 0$$ and so, one solves this equation for ##w(\rho)## in order to find the radial solutions.
My question is, Is it better defining ##u_{El} (\rho)## in an other way, so that we end up with Legendre's equation instead of the above equation?
$$\frac{d^2}{dr^2} u_{E} = \kappa^2 u \ \ \ \ \kappa^2 \equiv -2mE/ \hbar^2 > 0 \ \ \ \ r \rightarrow \infty$$ from the differential equation satisfied by ##u_{E} \equiv R_{El} (r) / r##, where ##R_{El}(r)## is the radial part of the wave function.
The solution of the above eq. for ##u_{E}## is ##u_{E}(r) \propto e^{- \kappa r}##. Let ##\rho \equiv \kappa r##. The book defines ##u_{El} (\rho)## as
$$u_{El} (\rho) = \rho^{l+1}e^{-\rho} w(\rho)$$ so that the function ##w(\rho)## satisfies the equation $$\frac{d^2}{d\rho^2}w + 2 \bigg(\frac{l+1}{\rho} - 1 \bigg) \frac{dw}{d \rho} + \bigg[ \frac{V}{E} - \frac{2(l+1)}{\rho} \bigg]w = 0$$ and so, one solves this equation for ##w(\rho)## in order to find the radial solutions.
My question is, Is it better defining ##u_{El} (\rho)## in an other way, so that we end up with Legendre's equation instead of the above equation?
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