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## Q. 14.1

Use the Heisenberg uncertainty principle to estimate the mass of the meson if the range of the nuclear force $R_{N}$ is about 1 fm.

Strategy The uncertainty principle allows energy conservation to be violated during a short time Δt. We will let energy conservation be violated to the extent of the excess energy ΔE required to create the meson. The fastest speed possible for the meson is c, so the distance the meson travels in time Δt is c Δt. This distance c Δt must be about the range of the nuclear force$R_{N}$.

## Verified Solution

The energy ΔE and the time Δt are related by the uncertainty principle, $\Delta E \Delta t \approx \hbar / 2, \text { so } \Delta t \text { is given by } \Delta t \approx \hbar /(2 \Delta E)$. The value of ΔE must be at least as large as is needed to create the mass particle, the meson in this case, so $\Delta E=m_{\pi} c^{2}$. We combine these results to give

$\Delta t \approx \frac{\hbar}{2 \Delta E}=\frac{\hbar}{2 m_{\pi} c^{2}}$ (14.2)

We solve Equation (14.2) for the meson mass and use $R_{N}=c \Delta t$:

$m_{\pi} c^{2}=\frac{\hbar}{2 \Delta t}=\frac{\hbar c}{2 R_{N}}$ (14.3)

If the mean range of the nuclear force is about 1 fm, we have

$m_{\pi} c^{2}=\frac{1.973 \times 10^{2} eV \cdot nm }{2 \times 10^{-15} m } \approx 100 MeV$

Equation (14.3) is useful to relate the effective length of any force R and the mass m that mediates the force. We write R as

$R=\frac{\hbar}{2 m c}=\frac{\hbar c}{2 m c^{2}}$ (14.4)