P17.5 Another important uncertainty principle is encountered in time-dependent systems. It relates the lifetime of a state At with the measured spread in the photon energy AE associated with the decay of this state to a stationary state of the system. "Derive" the relation AE At ≥ h/2 in the following steps. a. Starting from E = p²/2m and AE show that AE = VxAPx. ΔΕ (dE/dpx) Apx, b. Using Vx Ax/At, show that AE At ApxAx ≥ 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. = =

P17.5 Another important uncertainty principle is encountered in time-dependent systems. It relates the lifetime of a state At with the measured spread in the photon energy AE associated with the decay of this state to a stationary state of the system. "Derive" the relation AE At ≥ h/2 in the following steps. a. Starting from E = p²/2m and AE show that AE = VxAPx. ΔΕ (dE/dpx) Apx, b. Using Vx Ax/At, show that AE At ApxAx ≥ 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. = =