Free Particle in Spherical Coordinates

A free particle is a specific case when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_0=0\!} of the motion in a uniform potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k^2=\frac{2m}{\hbar^2}|E-V|} . Rearranging the equation gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(-\frac{\partial^2}{\partial r^2}+\frac{l(l+1)}{r^2}-k^2\right)u_l(r)=0}

Letting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho=kr\!} gives the terms that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{\ell}^{(1)} = j_{\ell}(z) + in_{\ell}(z) }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_{\ell}^{(2)} = j_{\ell}(z) - in_{\ell}(z) }

The asymptotic form of the spherical Bessel and Neumann functions (as z Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rightarrow} large) are given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_{\ell}(z) = \frac{\sin(z-\frac{\ell \pi}{2})}{z} }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_{\ell}(z) = \frac{\cos(z-\frac{\ell \pi}{2})}{z} }

The first few zeros of the spherical Bessel function:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell = 0: 3.142, 6.283, 9.425, 12.566 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell = 1: 4.493, 7.725, 10.904, 14.066 }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j'_{\ell}(z) = \frac{\ell}{z}j_{\ell}(z) - j_{\ell+1}(z) }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n'_{\ell}(z) = \frac{\ell}{z}n_{\ell}(z) - n_{\ell+1}(z) }