Let us first take a closer look at those free spherical waves.  The Hamiltonian of a free particle is Since [P_i,P_j]=0, (i,j=x,y,z), we have [H₀,P]=0 and we can find common eigenfunctions of H₀, P_x, P_y, and P_z.  The observables P_x, P_y, and P_z form a C.S.C.O. for a spinless particle.  The common eigenstates form a basis {|p>}.

The wave functions associated with the kets |p> are the plane waves

Introducing the wave vector  we define the ket Since |k> is proportional to |p> we have

The eigenstates of H₀ are infinitely degenerate.  The normalization and closure relation for {|k>} are written <k|k'>=d(k-k'), òd³k|k><k|=I.  The Hamiltonian of a free particle also commutes with L² and L_z.  (We can view a free particle as a particle in a central potential of zero strength.)  The observables H₀, L² and L_z form a C.S.C.O. for a spinless particle.  The common eigenstates form a basis {|k,l,m>}.

The wave functions associated with the kets |k,l,m> are the free spherical waves.

For a free particle the radial functions f_kl(r) are solutions to the differential equation

With r=kr, and this equation becomes

This is the spherical Bessel equation.  Its solutions are the spherical Bessel functions j_l(r) and n_l(r) and linear combinations of both.

As   Therefore only j_l(r) is an acceptable wavefunction for a free particle.  Using the orthonormality of the spherical Bessel functions we have

The free spherical waves are orthogonal and complete. Let us take a closer look at j_l(r).

As r®0, the lowest power in r dominates.

Double factorial notation:

As r®¥ the highest power in r dominates.

As we keep on differentiating, the highest power will always come from the derivative of the first term.  We therefore have

The probability of finding a particle in a free spherical wave eigenstate of the Hamiltonian H₀ near the origin, within a solid angle dW₀ and between r and r+dr is proportional to

The probability of finding the particle near the origin decreases as l increases.  If  then the probability of finding the particle between zero and r is negligible.

Note: To show this, you cannot just require that  in the region  because the wavefunction is a continuum wavefunction and cannot be normalized to one.  You have to build a wave packet and show that the wave packet does not penetrate the region Because the wave packet does not penetrate, a particle in an eigenstate |klm> of H₀ is practically unaffected by what happens inside a sphere centered at the origin of radius 

Let us use classical physics as a guide.  In classical mechanics, a free particle with angular momentum L about the origin and momentum p moves in a straight line with closest distance to the origin b=L/p.  b is called the impact parameter relative to the origin.  If

b_l(k) can thus be interpreted semi classically.  In quantum mechanics, b_l(k) is the radius where is equal to the total energy of the free particle.

For r<b_l(k), in the classically forbidden region, the eigenfunctions |klm> can only have an approximately exponentially decaying tail.