Question

The decay energy of a short-lived particle has an uncertainty of 1.0 MeV due to its short lifetime. What is the smallest lifetime it can have?

a) 1.32 × 10^(-21) s
b) 5.27 × 10^(-21) s
c) 2.64 × 10^(-21) s
d) 8.42 × 10^(-34) s

Answer 1

The smallest possible lifetime for a particle with a decay energy uncertainty of 1.0 MeV is calculated using Heisenberg's uncertainty principle to be approximately 2.64 x 10^-21 s.

Step-by-step explanation:

The decay energy uncertainty of a short-lived particle and its lifetime are related by Heisenberg's uncertainty principle for energy and time, which states that the product of the uncertainty in energy (ΔE) and the uncertainty in time (Δt) is approximately equal to or greater than the reduced Planck constant (ħ) divided by 2: ΔEΔt ≥ ħ/2. Given that the decay energy uncertainty is 1.0 MeV (which is 1.0 x 106 eV), and using the value of the reduced Planck constant (ħ = 6.58 x 10-16 eV·s), we can calculate the smallest possible lifetime (Δt) of the particle.

Δt = ħ / (2ΔE) = (6.58 x 10-16 eV·s) / (2 x 1.0 x 106 eV) = 3.29 x 10-22 s, which is approximately option (c) 2.64 x 10-21 s after rounding to two significant figures.