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. = =
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![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 = p/2m and AE = (dE/dpx)^px,
Vx Apx.
show that AE =
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 × 10s and 1.0 × 10-¹¹ s
in inverse seconds and inverse centimeters.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78fbc2f9-5ce5-4aa9-99e1-1139be15f6b7%2F03e6c659-4cca-4560-8372-f2ac2e23859e%2Fheam5bn_processed.png&w=3840&q=75)
Transcribed Image Text: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 = p/2m and AE = (dE/dpx)^px,
Vx Apx.
show that AE =
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 × 10s and 1.0 × 10-¹¹ s
in inverse seconds and inverse centimeters.
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