Another important uncertainty principle is encountered in time-dependent systems. It relates the lifetime of a state t

Another important uncertainty principle is encountered in time-dependent systems. It relates the lifetime of a state Δt with the measured spread in the photon energy ΔE associated with the decay of this state to a stationary state of the system. “Derive” the relation ΔE Δt ≥ h/2 in the following steps 

a. Starting from E = p²_x/2m and ∆E = (dE/dp_x) Δp_x show that ΔE = v_x Δp_x.

b. Using v_x = Δx/Δt, show that ΔE Δt = Δp_xΔx ≥ h/2.

c. Estimate the width of a spectral line originating from the decay of a state of lifetime 1.0 × 10⁻⁹ s and 1.0 × 10⁻¹¹ s in inverse seconds and inverse centimeters.