Particle-antiparticle pairs are occasionally created out of empty space. Looking at energy-time uncertainty, how long would such particles be expected to exist if they are: a) an electron/positron pair? b) a proton/antiproton pair?

First, we need to find the rest mass energy for an electron/positron pair and a proton/antiproton pair. The masses of electron and proton are given as: m_electron = 9.10938356 × 10^-31 kg m_proton = 1.67262192 × 10^-27 kg Now, calculate the rest mass energy for each pair using the formula E = mc^2, where c = 3 × 10^8 m/s is the speed of light: E_electron = m_electron * c^2 ≈ 8.19 × 10^-14 J E_proton = m_proton * c^2 ≈ 1.50 × 10^-10 J

Now that we have the rest mass energy for each particle-antiparticle pair, we can use the energy-time uncertainty principle to find the time uncertainty for each pair: Δt ≥ ħ / (2 * ΔE) Here, ħ = h /(2 * π) is the reduced Planck constant, and h ≈ 6.62607015 × 10^-34 Js is the Planck constant. Firstly, calculate the value of ħ: ħ = h /(2 * π) ≈ 1.054571817 × 10^-34 Js Now, calculate the time uncertainty for an electron/positron pair: Δt_electron ≥ (1.054571817 × 10^-34 Js) / (2 * 8.19 × 10^-14 J) Δt_electron ≥ 6.42 × 10^-22 s And for a proton/antiproton pair: Δt_proton ≥ (1.054571817 × 10^-34 Js) / (2 * 1.50 × 10^-10 J) Δt_proton ≥ 3.52 × 10^-25 s In conclusion, the time uncertainties for a particle-antiparticle pair created out of empty space are: a) electron/positron pair: Δt_electron ≳ 6.42 × 10^-22 s b) proton/antiproton pair: Δt_proton ≳ 3.52 × 10^-25 s