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The Solution of the Radial Equation
for the Wave Function of a Single
Particle in a Central Field

Consider a single particle in a central field which has a potential energy function V(r), where r is the distance of the particle from the center of the field. The Schrödinger Equation can be derived from the Hamiltonian function for this system. This is a second order partial differential equation involving the spherical coordinates (r, θ, φ). The technique of Separation of Variables yields an ordinary differential equation for each of the coordinates. The equations for θ and φ lead to the existence of two quantum numbers, l and m. The quantum number m must lie in the range from -l to l. This means that there are 2l+1 possible values of m. If l can range from 0 up to some maximum then the number of possible states is the sum of the first odd numbers, which is equal to n² for n being an integer. All of this comes without specifying the functional form of the potential energy function V(r). Empirically it is found that the number of possible states of the electrons in an atom is equal to 2n². The factor of 2 is explained by there being two possible spin orientations for an electron.

The prediction that the capacities of the electronic shells are equal to 2n² for some integer n is borne out by examining the atomic numbers of the so-called noble gases. Those are helium (2), neon (10), argon (18), krypton (36), xenon (54), radon (86). The differences of these numbers are 2-0=2*1², 10-2=2*2². 18-10=2*2², 36-18=2*3², 54-36=2*3², 86-54=2*4². Since the deduction that the shell capacities are of the form 2n² is independent of the functional form of the potential energy function of the central field and hence of the force form, it means that the model does not hold for the shells of nuclei. The capacities of the neutron and the proton shells for nuclei are {2, 4, 8, 14, 22, 32, 44} or {2, 6, 12, 8, 22, 32, 44} depending upon whether 6 and 14 are used as the nuclear magic numbers or 8 and 20. In either case some of the capacities are of the form 2n² and some of which are not.

The Radial Equation

The equation for the radial distance r which comes out of the Separation of Variables procedure is

(1/r²)d(r²(dR/dr))/dr + (2μ/h²)[E − V(r)]R = l(l+1)R/r²
or, expanding the first term
d²R/dr² + (2/r)(dR/dr)) + (2μ/h²)[E − V(r)]R = l(l+1)R/r²

where R is the radial component of the wave function, μ is the reduced mass of the system and h is Planck's constant divided by 2π. The parameter E is the total energy of the system and for a stable system E is negative.

A little can be deduced about the solution to the above equation without specifying the functional form of V(r). However it can be assumed that V(r)→0 as r→∞. This means that certain of the terms in the above equation become insignificant for large values of r. This leads to the equation

d²R/dr² + (2μ/h²)ER = 0
or, denoting (2μ/h²)as γ
and γE as −β²
d²R/dr² − β²R = 0

The term h² has the dimensions [L4M2/T2], therefore γ has the dimensions of [L-4M-1T2]. Hence γE has the dimensions of [L-2] and β the dimensions of [1/L].

The previous equation for R has solutions of the form

R(r) = exp(±βr)
but the positive value
can be rejected, thus
R(r) = exp(−βr)

This suggests that the solution to the general equation for R is of the form

R(r) = exp(−βr)S(r)

and further that it would be worthwhile to express the independent variable as ρ=βr.


dR/dr = −βexp(−βr)S(r) + exp(−βr)(dS/dr)
d²R/dr² = β²exp(−βr)S(r) − 2βexp(−βr)(dS/dr) + exp(−βr)d²S/dr²

When these expressions are substituted into the radial equation and that result divided by exp(−βr) the end result is

β²S − 2β(dS/dr) + d²S/dr² + (2/r)[−βS + (dS/dr)] − [β² + γV(r)]S = l(l+1)S/r²
which reduces to
d²S/dr² − 2(β−1/r)(dS/dr) − [(2β/r) + γV(r) + l(l+1)/r²]S = 0

As suggested previously, some simplification can be achieved by transforming the independent variable r to the dimensionless variable ρ=βr. Thus

dR/dr = (dR/dρ)(dρ/dr) = (dR/dρ)β
d²R/dr² = (d²R/dρ²)β²

Thus the radial equation is

(d²S/dρ²)β² − 2(β−β/ρ)β(dS/dρ) − [(2β²/ρ) + γV(ρ) + l(l+1)β²/ρ²]S = 0
which, by division by β², reduces to
d²S/dρ² − 2(1−1/ρ)(dS/dρ) − [(2/ρ) − V(ρ)/E + l(l+1)/ρ²]S = 0
noting that γ is the same as −β²/E

A further resolution of signs in the above equation results in

d²S/dρ² + 2(1/ρ−1)(dS/dρ) + [V(ρ)/E − (2/ρ) − l(l+1)/ρ²]S = 0

A possible form for the solution of the above equation is a series solution of the form

S(ρ) = ρλΣk=0akρk

The analysis cannot go any further without specifying a functional form for V(ρ). For the electrostatic field V(ρ) is equal to −K/ρ, where K is a constant. For that case λ must be equal to l. Furthermore for the electrostatic case there is an integer n such that the series solution terminates. That value of n is called the principal quantum number of the system.

(To be continued.)

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