# Free Particle in Spherical Coordinates

### From FSUPhysicsWiki

A free particle is a specific case when of the motion in a uniform potential *V*(*r*) = *V*_{0}, 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

Let Rearranging the equation gives us

If we now let then the equation reduces to the dimensionless form,

where and are the raising and lowering operators,

and

Because it follows that

For *l* = 0,

whose solution is

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

where is a spherical Bessel function and 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:

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

and

The asymptotic forms of the spherical Bessel and Neumann functions as are

and

The first few zeros of the spherical Bessel function for and are

and

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

and