Meson mass and Heisenberg principle

To explain the nuclear forces, Yukawa (Japanese physicist) postulated the existence of the mesons. Derive the relation between the range r of nuclear forces and mass m of the meson, using the Heisenberg incertitude principle. Evaluate the mass of a meson (in electron mass units – $m_e$), if we admit that the range of nuclear forces is $r =1.4*10^{^{15}} m$


The incertitude principles says that  the imprecision in the simultaneous determination of both position and momentum is:

$\Delta(X)*\Delta(P) \geq \hbar$

Suppose that the mesons of mass m are the exchange quanta of nuclear forces and such all quanta their speed is c (speed of light). When a meson is emitted by a nucleon and absorbed by another, it has to travel the average distance between two interacting nucleons.

Therefore the imprecision in the meson position is $\Delta(X) =r$

Since $\Delta(P) =m c$ we can write

$r*(m c) \geq \hbar$

$m \geq \hbar/(r c) =276*m_e$

Thus, the smallest estimated mass of a meson should be about 276 electron masses.

(the experimental measured smallest mass of a meson is 278 electron masses).