An Interlude — Question 9

Question 9

Charge-to-mass comparisons constrain symmetry only with an extra assumption

A trap measures the magnitude ratio $R=(|q_{\bar p}|/m_{\bar p})/(|q_p|/m_p)=1+\delta$, with $\delta=(2.0\pm3.0)\times10^{-9}$. For a small fractional charge difference a and mass difference b, write $|q_{\bar p}|=|q_p|(1+a)$ and $m_{\bar p}=m_p(1+b)$.

1. Expand R to first order and identify the constrained combination.

2. Assuming a=0, infer b and its uncertainty; explain the limitation without that assumption.

Question 9 - Solution

Charge-to-mass comparisons constrain symmetry only with an extra assumption

(a) Method and calculation

$$R=\frac{1+a}{1+b}\simeq(1+a)(1-b)\simeq1+a-b.$$$$\boxed{a-b=(2.0\pm3.0)\times10^{-9}}.$$

(b) Interpretation and further calculation

$$a=0\Rightarrow\boxed{b=(-2.0\pm3.0)\times10^{-9}}.$$$$\boxed{(a,b)\to(a+\eta,b+\eta)}\quad\text{leaves the first-order observable unchanged}.$$

Physical check

This hypothetical measurement is consistent with zero difference. Measuring a ratio does not separately establish equality of charge and mass.