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?

Verified step by step guidance

1. Calculate the momentum of a single photon using the formula $p = \frac{h}{\lambda}$, where $h$ is Planck's constant ($6.626 \times 10^{-34}$ Js) and $\lambda$ is the wavelength of the photon.
2. Convert the pulse duration from femtoseconds to seconds to use in further calculations. Recall that 1 fs = $10^{-15}$ s.
3. Use the energy-time uncertainty principle, $\Delta E \Delta t \geq \frac{\hbar}{2}$, where $\Delta t$ is the pulse duration and $\hbar = \frac{h}{2\pi}$. Solve for the energy uncertainty $\Delta E$.
4. Relate the energy uncertainty $\Delta E$ to the momentum uncertainty $\Delta p$ using the relation $\Delta E = c \Delta p$, where $c$ is the speed of light ($3.00 \times 10^8$ m/s). Solve for $\Delta p$.
5. Summarize the calculated values of the momentum $p$ and the momentum uncertainty $\Delta p$ for the photon.

Key Concepts

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 = hf, where h is Planck's constant and f is the frequency of the light. For a photon with a wavelength of 556 nm, the frequency can be found using the relationship f = c/λ.

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 can be applied to photons, where the uncertainty in momentum (Δp) can be related to the uncertainty in time (Δt) and energy (ΔE) through the relation ΔEΔt ≥ ħ/2, where ħ is the reduced Planck's constant.

Wave-Particle Duality

Wave-particle duality is a fundamental concept in quantum mechanics that describes how particles, such as photons, exhibit both wave-like and particle-like properties. This duality is crucial for understanding phenomena like interference and diffraction, as well as the behavior of photons in ultrashort pulses, which can be analyzed in terms of both their wave characteristics and their quantized energy levels.