where λ is a parameter subject to physical measurement. This form for the potential was somewhat surprising but a little reflection indicates that if the force was carried by particles the functional form has to be of the nature of the one Yukawa proposed.

To see this, consider the case of radiation from a point source. Radiation creates pressure propagated by photons. The intensity of the radiation is inversely proportional to the square of the distance from the source; i.e., A/r². The number of photons (and consequentially the energy) on a wave front of radius r₁ is the intensity A/r₁² multiplied by the area of the spherical wave front 4πr₁² or A. The wave front would have the same number of photons (and energy) when it expands to a radius of r₂. And of course the number of photons and amount of energy have to be the same because there is conservation of energy and photons. Where would any reduction in energy go? But this strict inverse distance squared dependence is a special consequence of photons not decaying.

If the strong force is carried by particles which decay then the intensity of the strong force will diminish with distance not only as the inverse of r squared but also because force-carrying particles decay over time. The number of particles remaining in a wave front after time t is N₀exp(-αt) where t is time and α is the decay rate. If v is the velocity of the particles then the number remaining at distance r is N₀exp(-(α/v)r). Thus the intensity of the strong force at distance r is of the form:

A·exp(-λr)/r²

where λ=α/v.

For any other functional form it would be impossible to account for energy differences as a function of distance.

The principle involved is that the force carried by particles has to be inversely proportion to the distance squared to account for the spreading of the particles over a larger spherical surface and must also be multiplied by an exponential factor to take into account the decay of the particles with time and hence distance. The potential function for a force is the function such that the negative of its gradient gives the force as a function of distance. Yukawa's potential function does not quite satisfy this condition but it is an approximation to one that does.

Yukawa's Analysis

Yukawa notes that the potential U=±g²/r satisfies the wave equation

(∇² - (1/c²)∂²/∂t²)U = 0.

The potential function he postulates, U=±g²e^-λr/r satisfies the equation:

(∇² - (1/c²)∂²/∂t² - λ²)U = 0.

where ∇² is the Laplacian operator.

The derivation of the above results makes use of the form of the vector operator ∇² in spherical coordinates. The derivation also make use of the fact that the time derivatives are all zero. For these derivations see Appendix I.

By the rules of quantum mechanics:

∂/∂x => (i/h)p_x
and similarly
for y and z
and for t

∂/∂t => -(i/h)W

where h is Planck's constant h divided by 2π, usually referred to as h-bar.

With these substitutions Yukawa's wave equation becomes:

-(1/h²)(p_>x² + p_y² + p_z²) - (1/c²)(-(1/h²)W² - λ²)U = 0
or
[(p_x² + p_y² + p_z²) - W²/c² + λ2h²]U = 0
or more succintly as
[p² - W²/c² + λ²h²]U = 0

Yukawa defines the mass of the particle associated with the field U as m_U such that

m_Uc = λh

This appears to be a definition rather than something derived from the analysis.

A similar relationship based upon Heisenberg's Uncertainty Principle was developed by G.C. Wick in Nature in 1938. The Uncertainty Principle in this case applies the canonical conjugate coordinates of time and energy:

ΔEΔt ≥ h/2π = h

In Wick's analysis the uncetainty in time Δt is the time required for light to traverse the range of the nuclear force r, which corresponds to 1/λ in Yukawa's analysis; i.e., Δt = r/c. The uncertainty of energy ΔE is the mass-energy of the particle, m_Uc². Thus, according to Wick's argument,

m_Uc²(r/c) = h
or m_Uc = (h/2π)/r = hλ

which is the same as Yukawa's relation.

Modification of Yukawa's Analysis

The potential energy function is defined such that the force of the field is equal to the negative of the gradient of the potential function. Although the potential Yukawa assumes involves an inverse function of distance with an exponential decay with distance it is not precisely the form that gives rise to the exponentially attenuated inverse distance squared form that has the proper form for a particle-based field. The potential function having that property is of the form:

V(r) = ∫^∞_r(exp(-λz)/z²)dz

where the term ±g² has been dropped. This will be called the true-form potential in the material which follows.

An integration by parts of this function yields the relationship:

V(r) = exp(-λr)/r - λ∫^∞_r(exp(-λz)/z)dz

The first term on the right is Yukawa's potential function.

This can be represented in the form:

V(r) = U(r) - λW(r)

where U(r) is Yukawa's potential and

W(r) = ±g²∫^∞_r(exp(-λz)/z)dz.
 

The term W(r) is closely related to what is known in mathematics as the exponential integral function, Ei(s) = ∫(exp(-s)/s)ds. Shown below are the values of U(r), V(r) and W(r) for the case λ=1 based upon numerical integration in the case of V(r). Alternatively the distance variable may be considered to be measured in units of 1/λ . The graph below goes up to r=20 where the potential is essentially zero as compared with the values in the vicinity of r=1. The potential variable plotted is the difference the potential at a particular value of r and the value at r=20.

Because the scale is so different at various parts of this graph it is more convenient to view the logarithms of the functions as shown below.

The parallel pattern for V(r) and U(r) for large distances indicates that for distances in such ranges the functions are roughly proportional.

In Appendix I it is shown that

∇²V = λ(∂V/∂r)
rather than as for the Yukawa potential
∇²U = λ²U.

This would appear to lead to a drastically different wave equation, but it is shown in Appendix II that the Laplacian for the true-form potential V(r) can be put into the form:

∇²V = λ²V + 2λX(r)
where
X(r) = ∫^∞_r(e^-λz/z³)/dz.

The integral involving the inverse cube of distance is extremely small compared to V for r large compared to unity. This means that the true-form potential implies the same things concerning mass as does the Yukawa potential, at least for distances greater than unity. The wave equation that would correspond to the true-form potential V(r) is:

(∇² - (1/c²)∂²/∂t² - λ²)V = 2λ∫^∞_r(e^-λz/z³)dz.

In particular the same relationship between λ and the mass of the particle would prevail.

m_Uc = hλ

However, the analysis indicates that the apparent mass of the particle for small distances would be different from that implied by the Yukawa potential.

The Half-Life and the
Mass of the Pi Mesons

If the decay of the nucleonic field potential with distance is due to the decay of the meson over time then there should be a relationship between the spatial rate of decay of the potential λ and the rate of time decay α; i.e., λ = α/v where v is the velocity of the meson.

The value of λ which corresponds to a mass 270 times the mass of an electron is 6.5x10¹²cm^-1. The reciprocal of λ is in units of length. The reciprocal of this λ is 1.54x10^-13 cm or 1.54 fermi. The half-lives of the positive and negative π mesons are equal and the value is 2.6x10^-8 seconds. The neutral π meson has an alternate mode of decay from the other π mesons and its half-life is equal to 9x10^-16 seconds. The reciprocal of the temporal rate of decay α is equal to the half-life divided by the natural logarithm of 2. Therefore the value of 1/α for the positive and negative π mesons is equal to 3.7x10^-8 seconds and for the neutral π meson it is 1.3x10^-15 seconds.

The ratio of the reciprocal of λ to the reciprocal of α, which corresponds to an apparent velocity, is about 4x10^-6 cm/sec for the positive and negative π mesons and to 1.2x10² or 120 cm/sec for the neutral π meson. This is an anomaly because some measurements indicate that the velocity of propagation of mesons is very close to the speed of light. When a particle moves at a speed close to the speed of light relativistic adjustments are required.

Postponing the matter of relativistic correction, consider for now the implications of a meson velocity essentially equal to the speed of light. If the velocity of the mesons is equal to the speed of light then

λ = α/c.

Since

m_Uc = hλ
and
ατ = ln(2),
 

where τ is the half-life, it follows that the relationship between the mass of the particle and its half-life would be given by:

m_U = (h/c)λ = (h/c)(β/c) = (h/c²)(ln(2)/τ
or
m_Uc² = hln(2)/τ
or
m_Uc²τ = hln(2) = constant.

Since particle masses are often expressed relative to the mass of the electron the above relationship is more conveniently expressed as:

(m_U/m_e)τ = ln(2)(h/m_ec²) = constant.

The term h/m_ec² is the reciprocal of the de Broglie frequency of an electron which is the de Broglie wavelength of an electron divided by the speed of light.

The estimate of mass based upon the above relation could be in error by many orders of magnitude because the half-life of the meson is vastly longer than the time required for light to traverse a distance equal to the range of the nuclear force, r=1/λ. The vastly extended life-time of the mesons can be accounted for by relativistic effects. According to the Special Theory of Relativity time appears to be dilated in a coordinate system moving at uniform speed with respect to another coordinate system. If the true half-life of the meson is τ₀ in a meson not moving with respect to the coordinate system in which measurement are being made then the half-life appears to be τ₀/(1-β²)^1/2 (where β=v/c) when the mesons are moving. The half-life of mesons is measured with respect to the nucleus and the laboratory coordinates system and can be vastly extended. In the coordinate system moving with the meson the half-life would be τ₀ but the distance the meson would appear to travel away from the nucleus before decay would appear to be contracted according to the Special Theory of Relativity. The distance in the coordinate system of the meson would be:

vτ₀(1-β²)^1/2

The determination of the true half-life of a positive or negative π meson and the velocity of propagation of the the mesons is a matter of finding the simultaneous solution to the two equations:

τ₀/(1-(v/c)²)^1/2 = 2.6x10^-8 seconds
vτ₀(1-(v/c)²)^1/2 = 1.54x10^-13 cm

Dividing the second equation by the first and then dividing by c gives the equation:

(v/c)(1-(v/c)²) = (1.54x10^-13)/((2.6x10^-8)(3x10¹⁰)
= 1.97x10^-16

which has a real solution of approximately (v/c)=1-1x10^-16

Thus τ₀ = 2.6x10^-8(1.4x10^-16) = 3.65x10^-24 sec, which is approximately the time required for light to traverse a distance equal to the range of the nuclear force. This value of τ₀ corresponds to a mean life of τ₀/ln(2) or 5.27x10^-24 seconds and this is the time required for light to traverse the range of the nuclear force.

Appendix I: Derivation of Laplacian in Spherical
Coordinates of Two Potential Functions

The Laplacian operator in spherical coordinates when applied to a spherically symmetrical function P(r) reduces to the evaluation of:

∇²V = (1/r²)(∂(r²∂P/∂r)/∂r))

When P(r) is the Yukawa potential U(r)=e^-λr/r the Laplacian reduces to:

∂(e^-λr/r)/∂r = -λ(e^-λr/r) - (e^-λr/r²)
so
r²∂V/∂r = -λr(e^-λr) - e^-λr
and thus
∂(r²∂V/∂r)/∂r = λ²r(e^-λr) -λ(e^-λr) + λe^-λr
= λ²r(e^-λr)
and finally
∇²U = (1/r²)(∂(r²(∂U/∂r)/∂r)) = λ²(e^-λr/r)
or
∇²U = λ²U
 

When P(r) is the true-form potential V(r) the Laplacian reduces to:

∂V/∂r = -e^-λr/r²
and thus
r²∂V/∂r = -e^-λr
hence
∂(r²∂V/∂r)/∂r = λe^-λr
and therefore
∇²V = λ(e^-λr/r²)
and finally
∇²V = λ(∂V/∂r)
 

Appendix II: A Second Form for the
Laplacian of theTrue-Form Potential

The true-form potential

V(r) = ∫^∞_r(exp(-λz)/z²)dz

may be integrated by parts using u=1/z² and dv=e^-λzdz to give

V(r) = (1/λ)e^-λr/r² - (2/λ)∫^∞_r(exp(-λz)/z³)dz.

But e^-λr/r² = -dV/dr so

V(r) = -(1/λ)dV/dr - (2/λ)∫^∞_r(e^-λz/z³)dz
which can be solved for dV/dr to give
dV/dr = -λV(r) - 2∫^∞_r(e^-λz/z³)dz

When this expression for dV/dr is substituted into the expression for the Laplacian

∇²V(r) = -λ(∂V/∂r)

the result is:

∇²V(r) = λ²V(r) +2λX(r)
where
X(r) = ∫^∞_r(e^-λz/z³)dz.

For large values of r, X(r) is small. Also λ is a large number so λ, the coefficient of the second term, is small in comparison with λ², the coefficient in the first term. Therefore the true-form potential is essentially the same as the Yukawa potential for large values of r.

The second term on the right in the equation for V(r) involves a factor of λ. Since λ has dimensions of inverse length it is impossible to apply perturbation techniques based upon the magnitude of this coefficient. With the equations in dimensionless form it might be appropriate to use perturbation techniques for analyzing the relationship between the implications associated with Yukawa's potential and the potential function V(r).