Chapter 39, Problem 1FTD

To determine

The reason behind Yukawa’s conclusion about the nonzero mass of carrier particle of strong force.

Answer to Problem 1FTD

From the non-zero value of mass calculated from uncertainty principle, Yukawa concluded that the nonzero mass of carrier particle of strong force.

Explanation of Solution

Initially, Yukawa assumed that strong force is also mediated by some particles similar to case of electromagnetic interaction. Identification of non-zero value of mass of carrier particle of strong force by Yukawa is given below. The range of strong force in $1\text{fm}$ range.

Write the uncertainty principle in terms of energy of particle and time.

$t=\frac{\hslash }{m{c}^2}$

Here,

$t$ is the lifetime of unstable particle

$\hslash$ is the reduced planks constant

$m$ is the mass of particle

$c$ is the speed of light in vacuum

Write the equation for maximum distance that can be travelled by the particle using the energy $m{c}^2$.

$x=ct$

Here,

$x$ is the maximum distance travelled

Rewrite the above relation by substituting the previous relation for $t$.

$\begin{array}{c}x=c\frac{\hslash }{m{c}^2}\\ =\frac{\hslash }{mc}\end{array}$

Rewrite the above relation in terms of $m$.

$m=\frac{cx}{\hslash }$

For $1\text{fm}$ distance, the value of mass calculated from the above equation is $130\text{MeV}/{\text{c}}^{\text{2}}$. Thus, he concluded that field particles of strong force has non-zero mass.

Conclusion: Therefore, from the non-zero value of mass calculated from uncertainty principle, Yukawa concluded that the nonzero mass of carrier particle of strong force.