Question

The omega particle \((\Omega)\) decays on average about \(0.1 \mathrm{ns}\) after it is created. Its rest energy is 1672 MeV. Estimate the fractional uncertainty in the \(\Omega\) 's rest energy \(\left(\Delta E_{0} / E_{0}\right)\) [Hint: Use the energy-time uncertainty principle, Eq. \((28-3) .]\)

Short answer

Answer: The fractional uncertainty in the Omega particle's rest energy is approximately \(1.97 \times 10^{-16}\).

Step-by-step solution

Determine relevant quantities and constants

We are given the average decay time, which we will consider as the time uncertainty in this calculation: \(\Delta t = 0.1\, \mathrm{ns}\). Additionally, we have the rest energy: \(E_0 = 1672\, \mathrm{MeV}\). We also need the reduced Planck constant: \(\hbar = 6.582 \times 10^{-16}\, \mathrm{eV}\,\mathrm{s}\).

Solve for energy uncertainty using the energy-time uncertainty principle

Using the energy-time uncertainty principle formula (\(\Delta E \cdot \Delta t \gtrsim \hbar / 2\)), we can solve for the energy uncertainty: \(\Delta E = \frac{\hbar}{2\Delta t}\) Note that we will consider the equality for simplicity. Now, we can plug in the known values: \(\Delta E = \frac{6.582 \times 10^{-16}\, \mathrm{eV}\,\mathrm{s}}{2(0.1 \times 10^{-9}\, \mathrm{s})}\)

Calculate the energy uncertainty

Now we can calculate the energy uncertainty: \(\Delta E = \frac{6.582 \times 10^{-16}\, \mathrm{eV}\,\mathrm{s}}{2\times0.1 \times 10^{-9}\, \mathrm{s}} = 3.291\times 10^{-7}\, \mathrm{eV}\) Note that to compare this with the rest energy in MeV, we need to convert it to MeV: \(\Delta E = 3.291 \times 10^{-7}\, \mathrm{eV} \times \frac{1\, \mathrm{MeV}}{10^6\, \mathrm{eV}} = 3.291 \times 10^{-13}\, \mathrm{MeV}\)

Determine the fractional uncertainty

Finally, we can determine the fractional uncertainty using the energy uncertainty and the rest energy: \(\frac{\Delta E_{0}}{E_{0}} = \frac{3.291 \times 10^{-13}\, \mathrm{MeV}}{1672\, \mathrm{MeV}}\)

Calculate the fractional uncertainty

Now we can calculate the fractional uncertainty: \(\frac{\Delta E_{0}}{E_{0}} = \frac{3.291 \times 10^{-13}\,\mathrm{MeV}}{1672\, \mathrm{MeV}} \approx 1.97 \times 10^{-16}\) The fractional uncertainty in the Omega particle's rest energy is approximately \(1.97 \times 10^{-16}\).