An ultrashort pulse has a duration of 9.009.00 fs and produces light at a wavelength of 556556 nm. What are the momentum and momentum uncertainty of a single photon in the pulse?

Ch 38: Photons: Light Waves Behaving as Particles

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 38: Photons: Light Waves Behaving as Particles

Problem 221

Chapter 37, Problem 221

An ultrashort pulse has a duration of $9.00$ fs and produces light at a wavelength of $556$ nm. What are the momentum and momentum uncertainty of a single photon in the pulse?

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Step 1: Calculate the energy of a single photon using the formula $ E = \frac{hc}{\lambda} $, where $ h $ is Planck's constant ($ 6.626 \times 10^{-34} \, \text{J·s} $), $ c $ is the speed of light ($ 3.00 \times 10^8 \, \text{m/s} $), and $ \lambda $ is the wavelength (556 nm, converted to meters as $ 556 \times 10^{-9} $).

Step 2: Use the relationship between energy and momentum for a photon, $ p = \frac{E}{c} $, to calculate the momentum of a single photon. Substitute the energy $ E $ from Step 1 and the speed of light $ c $.

Step 3: Determine the uncertainty in energy $ \Delta E $ using the time-energy uncertainty principle $ \Delta E \cdot \Delta t \geq \frac{\hbar}{2} $, where $ \Delta t $ is the pulse duration (9.00 fs, converted to seconds as $ 9.00 \times 10^{-15} $) and $ \hbar $ is the reduced Planck's constant ($ \hbar = \frac{h}{2\pi} $).

Step 4: Relate the uncertainty in energy $ \Delta E $ to the uncertainty in momentum $ \Delta p $ using the formula $ \Delta p = \frac{\Delta E}{c} $. Substitute $ \Delta E $ from Step 3 and the speed of light $ c $.

Step 5: Summarize the results: the momentum $ p $ of a single photon is calculated in Step 2, and the momentum uncertainty $ \Delta p $ is calculated in Step 4. These values depend on the given wavelength and pulse duration.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Photon Momentum

The momentum of a photon is given by the equation p = E/c, where E is the energy of the photon and c is the speed of light. The energy can be calculated using E = hc/λ, where h is Planck's constant and λ is the wavelength. For a photon with a wavelength of 556 nm, this relationship allows us to determine its momentum.

Uncertainty Principle

Heisenberg's Uncertainty Principle states that the more precisely the position of a particle is known, the less precisely its momentum can be known, and vice versa. This principle is crucial in quantum mechanics and can be expressed mathematically as ΔxΔp ≥ ħ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ is the reduced Planck's constant.

Pulse Duration and Frequency

The duration of an ultrashort pulse is related to its frequency spread, which can be understood through the Fourier transform. A shorter pulse duration corresponds to a broader range of frequencies, leading to greater uncertainty in the energy and, consequently, the momentum of the photons. This relationship is essential for calculating the momentum uncertainty of the photons in the pulse.

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