Free spherical waves

Let us first take a closer look at free spherical waves.  The Hamiltonian of a free particle is H₀ = P²/(2m).  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>} with normalization and closure relations  <p|p'> = δ(p - p'), ∫d³p|p><p| = I.
P|p> = p|p>,   H₀|p> = [p²/(2m)]|p>.
The wave functions associated with the kets |p> are the plane waves
<r|p> = (2πħ)^-3/2exp(i(p/ħ)∙r).
Introducing the wave vector k = p/ħ we define the ket |k> = ħ^3/2|p>.
Since |k> is proportional to |p> we have P|k> = k|k>,  H₀|k> = [ħ²k²/(2m)]|k>.

The eigenstates of H₀ are infinitely degenerate.  The normalization and closure relations for {|k>} are written <k|k'> = δ(k - k'), ∫d³k|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>}.
H|k,l,m> = [ħ²k²/(2m)]|k,l,m>,  L²|k,l,m> = ħ²l(l + 1)|k,l,m>,  L_z|k,l,m> = mħ|k,l,m>.

The wave functions associated with the kets |k,l,m> are the free spherical waves.
<r|k,l,m> = f_kl(r)Y_lm(θ,φ).
For a free particle the radial functions f_kl(r) are solutions to the differential equation
[-(ħ²/(2m))(1/r)(∂²/∂r²)r + ħ²l(l + 1)/(2mr²)]f_kl(r) = E_klf_kl(r).

With ρ = kr, and E_kl = E_k = ħ²k²/(2m) this equation becomes
[(1/ρ)(∂²/∂ρ²)ρ + 1 - l(l + 1)/ρ²]f_kl(ρ) = 0.
This is the spherical Bessel equation.  Its solutions are the spherical Bessel functions j_l(ρ) and n_l(ρ) and linear combinations of both.

j_l(ρ) = (-1)^lρ^l[(1/ρ)(∂/∂ρ]^l )sin(ρ)/ρ,  or  2^lρ^lΣ_s=0^∞(-1)^s(s + l)! ρ^2s/[s!(2s + 2l + 1)!].
n_l(ρ) = -(-1)^lρ^l[(1/ρ)(∂/∂ρ)]^l cos(ρ)/ρ, or  2^-l (-1/ρ)^l+1Σ_s=0^∞(-1)^s(s - l)! ρ^2s/[s!(2s - 2l)!].

As ρ --> 0, j_l(ρ) ∝ ρ^l,  n_l(ρ) ∝ ρ^-(l+1).
Therefore only j_l(ρ) is an acceptable wave function for a free particle.  Using the orthonormality of the spherical Bessel functions we have
f_kl(r)Y_lm(θ,φ) =  (2k²/π)^½j_l(kr)Y_lm(θ,φ) = <r|k,l,m>.
The free spherical waves are orthogonal and complete. Let us take a closer look at j_l(ρ).

j_l(ρ) = 2^lρ^lΣ_s=0^∞(-1)^s(s + l)! ρ^2s/[s!(2s + 2l + 1)!].
As r --> 0, the lowest power in ρ dominates.
j_l(ρ --> 0) = 2^lρ^l (l)! /(2l + 1)! =  ρ^l/(2l + 1)!!.

Double factorial notation:
(2l)!! = 2l*(2l - 2)*(2l - 4)* ... *6*4*2 = 2^l*(l)!,
(2l + 1)!! = (2l + 1)*(2l - 1)*(2l - 3)* ... *5*3*1 = (2l + 1)!/(2^l*(l)!).

As r --> ∞ the highest power in ρ dominates.
j_l(ρ) = (-1)^lρ^l[(1/ρ)(∂/∂ρ)]^l sin(ρ)/ρ = (-1)^lρ^l[(1/ρ)(∂/∂ρ)]^l-1 (cos(ρ)/ρ² - sin(ρ)/ρ²).
As we keep on differentiating, the highest power will always come from the derivative of the first term.  We therefore have
j_l(ρ-->∞) = (-1)^lρ^l(1/ρ)^l+1(∂/∂ρ)^l sin(ρ).
(∂/∂ρ) sin(ρ) = cos(ρ) = -sin(ρ - π/2).
(∂/∂ρ)^l sin(ρ) = (-1)^l sin(ρ - lπ/2).
j_l(ρ-->∞) = (1/ρ) sin(ρ - lπ/2).

The probability of finding a particle in a free spherical wave eigenstate of the Hamiltonian H₀ near the origin, within a solid angle dΩ₀ and between r and r + dr is proportional to
 k²r²|j_l(kr)|²|Y_lm(θ,φ)|²drdΩ₀.
j_l(kr) = (kr)^l/(2l + 1)!! as kr --> 0.
 k²r²|j_l(kr)|² = (kr)^2l+2/[(2l + 1)!!]² as kr --> 0.
The probability of finding the particle near the origin decreases as l increases.  If r < (l(l + 1))^½/k, then the probability of finding the particle between zero and r is negligible.

Note:  To show this, you cannot just require that  k²r²|j_l(kr)|² = 0 in the region  r < (l(l + 1))^½/k, because the wave function is a continuum wave function and cannot be normalized to one.  You have to build a wave packet and show that the wave packet does not penetrate the region r < (l(l + 1))^½/k.  Because the wave packet does not penetrate, a particle in an eigenstate |k,l,m> of H₀ is practically unaffected by what happens inside a sphere centered at the origin of radius  b_l(r) < (l(l + 1))^½/k.

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 L = ħ(l(l + 1))^½ and p = ħk, then b_l(k) = (1/k)(l(l + 1))^½.
b_l(k) can thus be interpreted semi classically.  b_l(k) is the radius where the angular momentum barrier potential, U_eff(r) = l(l + 1)ħ²/(2mr²) is equal to the total energy of the free particle.
l(l + 1)ħ²/(2m(b_l(k))²) = ħ²k²/(2m),  b_l(k) = (1/k)(l(l + 1))^½.
In Quantum Mechanics, for r < b_l(k), in the classically forbidden region, the eigenfunctions |k,l,m> can only have an approximately exponentially decaying tail.