[NOTES/QM-16004] Free Particle Solution in Polar Coordinates

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The radial equation for a free particle, $V(r)=0$, for all $r$ is \begin{equation} \frac{1}{r^2} \pp{r} \left(r^2\pp[R]{r}\right) + \left( k^2 - \frac{\ell(\ell+1)}{r^2}\right)R =0, \end{equation} where $k^2 =\dfrac{2mE}{\hbar^2}$. The solution of the radial equation has the most general form \begin{equation} R(r) = A j_\ell(kr) + B n_\ell(kr) \end{equation} but we must set $B=0$ because $n_\ell(kr) \to \infty$ as $r \to 0$. Hence we get \begin{equation} R_\ell(r) = A j_\ell(kr) \end{equation} and the full free particle wave function is \begin{equation} \Psi(r,\theta,\phi) = N j_\ell(kr) Y_{\ell m}(\theta,\phi) \end{equation} For a given value of energy $E$, $\ell$ can take all values $0,1,2,\ldots.$ and $m$ has $(2\ell+1)$ values from $-\ell$ to $\ell$. Therefore, for every energy value $E>0$ there are infinite number of solutions. If we take linear combinations of solutions with fixed energy $E$ we get most general form of the solution for a given energy as \begin{equation} \Phi(\vec{r}) = \sum C_{\ell m} j_\ell(kr) Y_{\ell m}(\theta,\phi). \end{equation} In Cartesian coordinates the free particle solutions for energy $E$ are plane waves $ \exp(i\vec{k}\cdot\vec{r}).$ Thus it is possible to write each of these two type of solutions as a linear combination of the other type. In particular we have \begin{equation} \exp(\vec{k}\cdot\vec{r}) = \sum C_{\ell m} j_\ell(kr) Y_{\ell m}(\theta,\phi) \end{equation} As a particular case of this equation, when $k$ is along the $z$ axis and $ \vec{k}\cdot\vec{r}=kz $, we have the expansion of plane waves \begin{equation} \exp(ikz) = \sum_0^\infty (2\ell+1)i^\ell j_\ell(kr) P_\ell(\cos\theta) \end{equation} Note that only $m=0$ terms contribute in the above equation.