Free Particle in Spherical Coordinates

A free particle is a specific case when $V_{0}=0\!$ of the motion in a uniform potential $V(r)=V_{0}\!$. So it's more useful to consider a particle moving in a uniform potential. The Schrödinger equation for the radial part of the wave function is:

$\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}+V_{0}\right)u_{l}(r)=Eu_{l}(r)$

let $k^{2}={\frac {2m}{\hbar ^{2}}}|E-V|$. Rearranging the equation gives

$\left(-{\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {l(l+1)}{r^{2}}}-k^{2}\right)u_{l}(r)=0$

Letting $\rho =kr\!$ gives the terms that ${\frac {1}{r^{2}}}={\frac {k^{2}}{\rho ^{2}}}$ and ${\frac {\partial ^{2}}{\partial r^{2}}}=k^{2}{\frac {\partial ^{2}}{\partial \rho ^{2}}}$. Then the equation becomes:

$\left(-{\frac {\partial ^{2}}{\partial \rho ^{2}}}+{\frac {l(l+1)}{\rho ^{2}}}\right)u_{l}(\rho )=u_{l}(\rho )=d_{l}d_{l}^{+}u_{l}(\rho )$

where $d_{l}\!$ and $d_{l}^{\dagger }\!$ become the raising and lowering operators:

$d_{l}={\frac {\partial }{\partial \rho }}+{\frac {l+1}{\rho }},$

$d_{l}^{\dagger }=-{\frac {\partial }{\partial \rho }}+{\frac {l+1}{\rho }}$

Being $d_{l}^{\dagger }d_{l}=d_{l+1}d_{l+1}^{\dagger }$, it can be shown that

$d_{l}^{\dagger }u_{l}(\rho )=c_{l}u_{l+1}(\rho )$

For $\ell =0$, $-{\frac {\partial ^{2}}{\partial \rho ^{2}}}u_{0}(\rho )=u_{0}(\rho )$, gives the solution as:

$u_{0}(\rho )=A\sin(\rho )-B\cos(\rho )\!$

The raising operator can be applied to the ground state in order to find high orders of $\ u_{0}(\rho )$;

$d_{0}^{\dagger }u_{0}(\rho )=\left(-{\frac {\partial }{\partial \rho }}+{\frac {l+1}{\rho }}\right)u_{0}(\rho )=c_{0}u_{1}(\rho )$

By this way, we can get the general expression:

$f_{l}(\rho )={\frac {u_{l}(\rho )}{\rho }}=A_{l}j_{l}(\rho )+B_{l}n_{l}(\rho )$,

where $j_{l}(\rho )\!$ is spherical Bessel function and $n_{l}(\rho )\!$ is spherical Neumann function.

Explicit Forms of the Spherical Bessel and Neumann Functions

$j_{0}(z)={\frac {\sin(z)}{z}}$

$j_{1}(z)={\frac {\sin(z)}{z^{2}}}-{\frac {\cos(z)}{z}}$

$j_{2}(z)=\left({\frac {3}{z^{3}}}-{\frac {1}{z}}\right)\sin(z)-{\frac {3}{z^{2}}}\cos(z)$

$n_{0}(z)=-{\frac {\cos(z)}{z}}$

$n_{1}(z)=-{\frac {\cos(z)}{z^{2}}}-{\frac {\sin(z)}{z}}$

$n_{2}(z)=-\left({\frac {3}{z^{3}}}-{\frac {1}{z}}\right)\cos(z)-{\frac {3}{z^{2}}}\sin(z)$

The spherical Hankel functions of the first and second kind can be written in terms of the spherical Bessel and spherical Neumann functions, and are defined by:

$h_{\ell }^{(1)}=j_{\ell }(z)+in_{\ell }(z)$

and

$h_{\ell }^{(2)}=j_{\ell }(z)-in_{\ell }(z)$

The asymptotic form of the spherical Bessel and Neumann functions (as z $\rightarrow$ large) are given by:

$j_{\ell }(z)={\frac {\sin(z-{\frac {\ell \pi }{2}})}{z}}$

and

$n_{\ell }(z)={\frac {\cos(z-{\frac {\ell \pi }{2}})}{z}}$

The first few zeros of the spherical Bessel function:

$\ell =0:3.142,6.283,9.425,12.566$

$\ell =1:4.493,7.725,10.904,14.066$

The derivatives of the spherical Bessel and Neumann functions are defined by:

$j'_{\ell }(z)={\frac {\ell }{z}}j_{\ell }(z)-j_{\ell +1}(z)$

and

$n'_{\ell }(z)={\frac {\ell }{z}}n_{\ell }(z)-n_{\ell +1}(z)$