In his 1934 Hideki Yukawa argued that the nuclear strong force is carried by a particle with a mass approximately 200 times that of an electron. This was based upon a relationship between the range of a force-carrying particle and its mass; i.e.,

m_U = h/(cρ_U)

where m_U is the mass of the force-carrying particle for a force field U, h is Planck's constant divided by 2π, c is the speed of light and ρ_U is the range of the force. Range here does not mean that the force is zero outside of that range; instead the range is a scale parameter.

Yukawa postulated that the potential fuction for the nuclear force was of the form

U=±g²e^-λr/r

where λ is a parameter and ρ_U=1/λ.

Yukawa used the very rough approximation of ρ_U being the scale of the nucleus; i.e., on the order of 10 fermi, 10×10⁻¹⁵ meters. This led him to predict that a particle with a mass 200 times that of an electron would be discovered. Ultimately such particles were discovered with a mass 270 times that of an electronl and that had the other required properties for a particle carrying the nuclear force. Hideki Yukawa was awarded the Nobel Prize in physics in 1949 for his work.

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

(∇² - (1/c²)∂²/∂t²)U = 0.

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

(∇² - (1/c²)∂²/∂t² - λ²)U = 0.

where ∇² is the Laplacian operator.

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

m_Uc = λh

This is the Yukawa relation.

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

ΔEΔt ≥ 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/r = hλ

which is the same as Yukawa's relation. What Wick really showed however is

m_Uc ≥ h

The constant in Heisenberg's Uncertainty Principle is usually given as h, as in Wick's derivation, but more correctly it is h/(4π)=h/2. That factor of (1/2) is not unimportant for empirical verification.

It is argued elsewhere that the exponential factor that Yukawa applied to the potential function should instead be applied to the force formula. This is justified on the basis that the particles carrying the nuclear force decay and the number of remaining particles is a negative exponential function of distance. Yukawa's potential function may be considered an approximation of the true potential which is derived from the exponentially weighted force. The approximation is better at greater distances than at small distances.

The formula for the force between two nucleons when the particles carrying the force decay is

F = − H*e^−λr/r²

where H* and λ are parmeters and r is the distance separating the nucleons.

Empirical estimates of H* and λ were derived from the binding energy of the deuteron being 2.22457 MeV. Those values are

H* = 1.92570×10⁻²⁵ kg m³/s²
and
λ = 1/(1.522 fermi)

There are two very interesting aspects to these estimates.

First, the value of H* is within 3 percent of the value of hc, the product of Planck's constant and the speed of light.

Second, consider the following calculations. The value of ρ₀ derived from λ, ρ₀=1/λ=1.522 fermi, would be a natural unit of length. This length divided by the speed of light would be a natural unit of time,

τ₀=ρ₀/c=1.522×10^-15/3×10⁸
=5.07×10^-24 seconds.

Planck's constant is a natural unit of action, energy×time, so h/τ₀ would be a natural unit of energy. Its value is

E₀ = 6.626×10⁻³⁴/5.07×10⁻²⁴
= 1.306×10⁻¹⁰ joules (kg*m²/s²)

Through the Einstein relation, E=mc², there would be a natural unit of mass equal to

m₀ = E₀/c² = 1.306×10⁻¹⁰/9×10¹⁶
= 1.45×10⁻²⁷ kg.

This is notably close to the masses of the proton and the neutron; i.e., 1.6726×10⁻²⁷ kg and 1.6749×10⁻²⁷ kg, respectively. There is only about a 15 percent difference betweeen the computed m₀ and the masses of the proton and neutron. This suggests that the value for ρ₀ may be in error by about 15 percent. If ρ₀ were 1.32 fermi instead of 1.522 fermi then the derived natural unit of mass would be equal to the mass of the proton.

The value of ρ_U deried from the Yukawa relation and the mass of the π mesons is about 1.522 fermi. Since this value is based upon h but the above "derivation" utilizes h rather than h a factor of 2π. If h had been used in the "derivation" the result would have been the mass of the π meson. Instead what resulted is 2π times the mass of the π meson. It just happens that the masses of the proton and neutron are approximately 2π times the mass of the π meson.

There is reason to believe that the binding energy of the deuteron is underestimated by a substantial amount. An estimate of H* based upon the separation distance of the proton and neutron in the deuteron being 3.2 fermi is

H* = 3.4×10⁻²⁶ kg m³/s²