This material develops what can deduced about the quantum mechanics of a particle in a central force field. Let r be the distance of the particle from the center of the field and let V(r) be the potential function for that field. If R(r) is the radial component of the wave function for the particle then it satisfies the equation

(1/r²)(d/dr(r²dR/dr) + { −l(l+1)/r² + (2μ/h²)(E-V(r)}R = 0

where l is the angular momentum and E is the energy.

One of the quantities which could be of interest is r²R(r), the radial wave function density as function of r. Let Q(r)=r²R and thus R(r)=(1/r²)Q(r). This means that

dR/dr = (-2/r³)Q + (1/r²)dQ/dr
and hence
r²dR/dr = (-2/r)Q + dQ/dr

Therefore d/dr[r²dR/dr] = (2/r²)Q - (2/r)dQ/dr + d²Q/dr²

The equation satisfied by Q(r) is then

(2/r²)Q - (2/r)dQ/dr + d²Q/dr² + { −l(l+1)/r² + (2μ/h²)(E-V(r)}Q = 0
or, combining the terms involving Q
and factoring out an E from (E-V(r))
d²Q/dr² - (2/r)dQ/dr + { (2 −l(l+1))/r² −2/r + (2μE/h²)(1-V(r)/E)}Q = 0

Now let (2μE/h²)=β and let ρ=βr. Thus r=(1/β)ρ and so d/dρ = (1/β)(d/dr) and d²/dρ² = (1/β)(d²/dr²). The equation for Q(ρ) is then

(1/β²)d²Q/dρ² - (2/ρ)dQ/dρ + {2 −l(l+1)β²/ρ² −2β/ρ + (1-V(ρ)/E)}Q = 0
or, for Q≠0,
(1/Q)(dQ/dρ) = d(ln(Q))/dρ = { −l(l+1)β/ρ² −2/ρ + (1-V(ρ)/E)}

This equation is in terms of the nondimensional variables ρ and V/E. It can be integrated with respect to ρ from +∞ downward to give

ln(Q) = C'-l(l+1)β/ρ + 2ln(ρ) − ρ + ∫(V(z)/E)dz
where C' is an integration constant.

Thus

Q(ρ) = C{exp[−(l(l+1)β/ρ + ρ)]ρ²}exp[∫(V(z)/E)dz]
 

where C is constant deriving from the integration constant; i.e., C=e^C'.

If l>0, this function goes to zero as ρ goes to zero and as ρ goes to +∞.

A Coulomb Force Field

Let V(ρ) = γ/ρ. Then with integration downward from +∞ ∫(V(z)/E)dz = −(γ/E)ln(ρ). Thus

Q(ρ) = C{exp[−(l(l+1)β/ρ + ρ)]ρ²}ρ^−(γ/E)
or, collecting terms
Q(ρ) = C{exp[−(l(l+1)β/ρ + ρ)]ρ^2−(γ/E)}

The value of C is determined such that ∫₀^+∞Q(ρ)dρ = 1,

Since ρ=βr

Q(r) = C{exp[−(l(l+1)/r + βr)](βr)^2−(γ/E)}
or, equivalently
Q(r) = C{exp[−(l(l+1)/r + βr)]β^2−(γ/E)r^2−(γ/E)}}

Since Q(r)=R(r)r²

R(r) = C{exp[−(l(l+1)/r + βr)]β^2−(γ/E)r^−(γ/E)}}
or, in terms of the nondimensional ρ
R(ρ) = C{exp[−(l(l+1)β/ρ + ρ)]β²ρ^−(γ/E)}

There is not a hint of quantization so far in this solution. The conventional procedure, such as in Herbert Strauss' Quantum Mechanics: An Introduction, is to propose a functional form for R(ρ)

R(ρ) = exp[−ρ/2]ρ^lL(ρ)

on the basis that this form satisfies the condition that it goes to zero as ρ→+∞ and also as ρ→0 providing that l>0. If R(ρ) is of this functional form then the function L(ρ) has to satisfy the equation for Laguerre polynomials. The degree of the polynomial has to be finite. The Laguerre equation has solutions only for discrete values of the energy E and this becomes the quantizing condition. But, as was shown above, the functional form of R(ρ) does not have to be of the assumed kind and thus the basis for the Laguerre equation disappears and with it the quantization.

(To be continued.)