# 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) = V0, so it is more useful to consider a particle moving in a uniform potential. We will make use of these results in the next section to discuss the spherical potential well. The Schrödinger equation for the radial part of the wave function is

$\left(-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}+V_0\right)u_l=Eu_l.$

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

$\left(-\frac{d^2}{dr^2}+\frac{l(l+1)}{r^2}-k^2\right)u_l=0.$

If we now let $\rho=kr,\!$ then the equation reduces to the dimensionless form,

$\left(-\frac{d^2}{d\rho^2}+\frac{l(l+1)}{\rho^2}\right)u_l(\rho)=u_l(\rho)=d_ld_l^+u_l(\rho),$

where $d_l\!$ and $d_l^{\dagger}\!$ are the raising and lowering operators,

$d_l=\frac{d}{d\rho}+\frac{l+1}{\rho}$

and

$d_l^\dagger=-\frac{d}{d\rho}+\frac{l+1}{\rho}.$

Because $d_l^{\dagger}d_l=d_{l+1}d_{l+1}^{\dagger},$ it follows that

$d_l^\dagger u_l(\rho)=c_l u_{l+1}(\rho).$

For l = 0,

$-\frac{d^2}{d\rho^2} u_0(\rho)=u_0(\rho),$

whose solution is

$u_0(\rho)=A\sin{\rho}-B\cos{\rho}.\!$

The raising operator may now be applied to this state in order to find the solutions for higher values of $l.\!$ By repeated application of this operator, we obtain the wave function for all values of $l:\!$

$f_l(\rho)=\frac{u_l(\rho)}{\rho}=A_lj_l(\rho)+B_ln_l(\rho),$

where $j_l(\rho) \!$ is a spherical Bessel function and $n_l(\rho) \!$ is a spherical Neumann function, or spherical Bessel functions of the first and second kinds, respectively.

## Properties of the Spherical Bessel and Neumann Functions

Explicit forms of the first few spherical Bessel and Neumann functions:

$j_0(z) = \frac{\sin(z)}{z}$ $j_1(z) = \frac{\sin(z)}{z^2} - \frac{\cos(z)}{z}$ $j_2(z) = \left( \frac{3}{z^3} - \frac{1}{z}\right) \sin(z) - \frac{3}{z^2}\cos(z)$

$n_0(z) = -\frac{\cos(z)}{z}$ $n_1(z) = -\frac{\cos(z)}{z^2} - \frac{\sin(z)}{z}$ $n_2(z) = - \left( \frac{3}{z^3} - \frac{1}{z}\right) \cos(z) - \frac{3}{z^2}\sin(z)$

We may also define spherical Hankel functions of the first and second kind in terms of the spherical Bessel and Neumann functions:

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

and

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

The asymptotic forms of the spherical Bessel and Neumann functions as $z\rightarrow\infty$ are

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

and

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

The first few zeros of the spherical Bessel function for $l=0\!$ and $l=1\!$ are

$l = 0: 3.142, 6.283, 9.425, 12.566, \ldots$

and

$l = 1: 4.493, 7.725, 10.904, 14.066, \ldots$

The derivatives of the spherical Bessel and Neumann functions are given by

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

and

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