In the quantum mechanics of atoms one quantum number represents the angular momentum of the system. Zero is an allowable value and this is interpreted as an enigma from the viewpoint of classical mechanics. However there is a puzzle only if the subatomic particles are taken to be solid units enclosing their centers of mass and charge. However if either of the particles has an empty center the particles can pass through each other without a collision. In the case of the hydrogen atom, suppose the electron is a point particle and the proton is a triadic structure of quarks such that the center is empty. Then the electron can pass back and forth through the proton without collision and the system would have zero angular momentum.

Such zero momentum systems are one dimensional; i.e., have only one degree of freedom. As such they are relatively easy to analyze mathematically. Let f(x) is the attractive force between the particles where x is the separation distance. The dynamics of the particle is given by the equation

m(d²x/dt²) = f(x)

A conservative force is given by the negative of the gradient of a potential function V(x); i.e.,

f(x) = −(dV(x)/dx)
and thus integrating down
from +∞ to s>0 gives
∫_s^∞(−f(x))dx = V(+∞) − V(s)

The asymptotic value of the potential function may be taken to be zero and hence

∫_s^∞(−f(x))dx = − V(s)
or, equivalently
V(s) = −∫_s^∞(−f(x))dx
or V(s) = ∫_s^∞f(x)dx

Now consider two particles with masses m₁ and m₂. Let the center of mass be at the origin of the coordinate system and the particles located at distances r₁ and r₂ from that center. This means that

m₁r₁ = m₂r₂
and therefore
r₂/r₁ = m₁/m₂

Thus

s = r₁ + r₂ = r₁(1 + r₂/r₁)
s = r₁(1+m₁/m₂) = r₁(m₂+m₁)/m₂
and therefore
r₁ = [m₂/(m₁+m₂)]s
and likewise
r₂ = [m₁/(m₁+m₂)]s

The dynamics equations for the two particles with respect to the conservative force are

m₁(d²r₁ /dt²) = f(s)
and
m₂(d²r₂ /dt²) = f(s)
which, when the expressions for r₁ and r₂
in terms of s are substituted into the equations,
(m₁m₂/(m₁+m₂))(d²s/dt²) = f(s)
(m₂m₁/(m₁+m₂))(d²s/dt²) = f(s)

These last two equations are the same.

The expression (m₁m₂/(m₁+m₂)) is just the reduced mass μ since

1/μ = 1/m₁ + 1/m₂
and thus
μ = 1/[1/m₁ + 1/m₂]
which upon multiplication of
the numerator and denominator
by m₁m₂ gives
μ = m₁m₂/(m₁+m₂)

Thus the dynamics of the system is given by the single equation

μ(d²s/dt²) = f(s) = −dV/ds

Note now that

dV(s)/dt = (dV(s)/ds)(ds/dt)

When both sides of this equation are multiplied by (ds/dt) the result is

μ(ds/dt)(d²s/dt²) = −(dV/ds)(ds/dt)

Note now that

dV(s)/dt = (dV(s)/ds)(ds/dt)

This means that the previous equation can be represented as

μ(ds/dt)(d²/dt²) = − dV/dt
or, letting (ds/dt) be denoted as v
μv(dv/dt) = −(dV/dt)
or, equivalently
μ(d(½v²)/dt) = −(dV/dt)

When this equation is integrated (indefinitely) with respect to t the result is

½μv² = E − V(s(t))

where E is a constant.

Rewritten, the above equation is the familiar conservation of energy equation

½μv² + V(s(t)) = E

In words, the kinetic energy, ½μv², plus the potential energy V is equal to a constant at all times. However there is seemingly a problem in that the kinetic energy of the two particles might perhaps not be equal to ½μ(ds/dt)².

The kinetic energy of the two particles is given by

½m₁(dr₁/dt)² + ½m₂(dr₂/dt)²
which expressed in terms of (ds/dt) is
½[(m₁m₂² + m₁²m₂)/(m₁+m₂)²](ds/dt)²
which may be factored to
½[(m₁m₂)(m₁+m₂)/(m₁+m₂)²](ds/dt)²
which reduces to
½[(m₁m₂)/(m₁+m₂)](ds/dt)²
which is none other than
½μ(ds/dt)²

The equation ½μv² + V(s(t)) = E can be solved for v=(ds/dt) to give

v = ds/dt = [(2/μ)(E−V(s))]^½
and hence
ds/[1−V(s)/E)]^½ = (2E/μ)^½dt

The quadrature (integration) of this last equation gives the trajectory of s as a function of time t. Once s(t) is found then the radial momentum μ(ds/dt) can be determined for application of the Wilson-Sommerfeld quantization condition.

The Wilson-Sommerfeld Quantization Condition

According to the Wilson-Sommerfeld quantization condition

(p_sds) = n_sh

where the integration is over a complete cycle, n_s is an integer, h is Planck's constant and p_s is the momentum associated with s, μ(ds/dt).

Since for a system in which energy is conserved

½μ(ds/dt)² + V(s) = E
and hence
(ds/dt) = [(2/μ)(E−V(s))]^½
and thus the momentum is
p_s = μ(ds/dt) = [(2μ(E−V(s))]^½

The quantization condition is then

[(2μ(E−V(s))]^½ds = n_sh

This condition can be put into a nondimensional form. First both sides of the equation are divided by h and then h is taken into the square root expression to give

[(2μE/h² − 2μV(s)/h²)]^½ds = n_s

Note that both E and V are negative with V≤E. Let V(s) be expressed as −CW(s) where C is force formula constant with dimensions [ML³/T²] and W(s) is a positive function with dimenion [1/L].

Thus

[(2μE/h² + (2μC/h²W(s))]^½ds = n_s

Planck's constant, h, has dimensions [ML²/T] so h²/(2μC) has dimension [L]. It is a scale length. For the hydrogen atom and using h instead of h it is the Bohr radius of the ground state electron orbit. Let h²/(2μC) be denoted as σ. Since a force constant such as k as the same dimensions as hc or hc, C may be expressed as ζhc, where ζ is called a structure constant. Thus

σ = h²/(2μC) = h²/(2μζhc) = h/(2μζc) = hc/(2ζμc²)

where μc² is related to the rest mass energy of the particles in the system.

The quantization now has the form

[(2μE/h² + W(s)/σ)]^½ds = n_s
which can be more conveniently expressed as
[((2μE/h²)(σ²/σ²) + σW(s))/σ²]^½ds = n_s
which, upon factoring out a (1/σ) term
from the square root, becomes
[((2μE/h²)σ² + σW(s))]^½(ds/σ) = n_s

The term

(2μE/h²)σ²
reduces to
2Eh²/(μC²)
and further to
(2E/C)σ
and still further to
(2E/(ζhc))(hc/(2μc²) = (E/(μc²)/ζ

This is a negative dimensionless constant. Let it be denoted as −ψ and let z denote s/σ. Further let σW(s) be denoted as the dimensionless function U(z). Thus the quantization condition is

[(U(z) − ψ]^½dz = n_s

Let S be the maximum separation. The cycle may be considered to consist of four phases: From +S to 0, from 0 to −S, from −S to 0 and from 0 to +S. These phases are essentially identical so the quantization condition may be reduced to

4∫₀^S([(2μ(E−V(s))]^½ds) = n_sh
or, in terms of the previous formulation,
4∫₀^S/σ([U(z)−ψ]^½dz) = n_s

The maximum separation distance S is defined by the condition that V(S)=E; i.e., the kinetice energy is zero and hence velocity is also equal to zero. Empirically its value should be equal or proportional to the measured diameter of the particle. Thus the above condition would establish the quantization of ψ and thereby the quantization of the energy E.

The Average Values of the Quantities Which Vary

Although total energy may be quantized the kinetic and potential energies vary throughout the cycle. The time the system spends at separation distance s is inversely proportional to |(ds/dt)|. By integrating 1/(ds/dt) over a cycle one can construct a distribution function q(s) and from it compute the average values of kinetic and potential energy; i.e.,

K = q(s)K(s)ds

Since K(s)=½μ(ds/dt)² the evaluation of K reduces to evaluating the integral |μ(ds/dt)|ds which by the Wilson-Sommerfeld condition is quantized. Since K+V=E this means the average potential energy V is also quantized.

The Electrostatic Force Function

The formula for the electrostatic force is k/s² where k is a constant incorporating unit charges. The constants of force functions have the dimensions of Planck's constant h, or hc, which is Planck's constant divided by 2π. Thus k may be expressed as ζhc where ζ is the fine structure constant 1/137.036. For the electrostatic force

V(s) = −ζhc/s

The nondimensional version of the quantization condition is

4∫₀^S/σ[1/z −ψ]^½dz = n_s

The integral in the above equation has an analytic evaluation, but it is instructive to compute the value numerically in preparation for the evaluating the corresponding integral for the nuclear force which does not have an analytic solution.

The problem in the numeric evaluation is how to handle the singularity at z=0.

The first task is to evaluate the parameter σ. Its value is 1.0430952l×10^-9 m.

The value of the maximum separation is such that

|E| = k/S
and thus
S = k/|E|
and hence
S/σ = (k/σ)/|E|
which reduces to
S/σ = (2ζμc²/hc)/|E| = 2ζ/ψ

Now the quantization condition takes the form

4∫₀^2ζ/ψ[1/z −ψ]^½dz = n_s

The left-hand side of the above equation is to be evaluated as a function of ψ with ζ being 1/(2π137.036).

If φ is defined to be (2ζ)/ψ and w to be (2ζ)z then the quantization condition takes the form

(4/(2ζ)^½)∫₀^φ[1/w −1/φ]^½dw = n_s

The value of the coefficient of the integral is 82.995.

The integral is to be evaluated for a number of different values of φ. For a specific value of φ the range from 0 to φ is divided up into intervals of width Δz=φ/N. Because of the singularity at z=0 a value of δz will be used for the first point. Ultimately the behavior of the estimate must be investigated both for N→∞ and δz→0.

Here is the table of values involved in estimating the integral ∫₀^φ(1/w−1/φ)^½dz for φ=0.1 and δz=0.0001.

The trapezoidal rule for the estimation of J(φ)=∫_a^bf(z)dz leads to the formula 0.5f(a)Δz+Σf(z_i)Δz+0.5f(b)Δz. Because of the singularity at z=a this formula has to be modified to 0.5f(a+δz)(Δz-δz)+(Δz-δz)f(a+2Δz)+Σf(z_i)Δz+0.5f(b)Δz .

For the data in the above table this leads to

J(φ) = 0.5(99.94998749)(0.00499)
+0.5(13.78404875)(0.00999)
+64.83547042(0.005)
+0.5(0.72547625)(0.005)
= 14.28599702

When this value is multiplied by the coefficient 82.995 the result is 1185.666323. Because this is not an integer it means that φ=0.1 is not a quantum level for φ.

The relationship between J and φ is displayed below in terms of the common logarithms of the two quantities.

(To be continued.)

Consider the effect of different values of δz.

(To be continued.)

A Nuclear Force Function

Now consider a force of the form

f(x) = −H*e^−λ*x/x²

(To be continued.)