Problem 52 - Fundamentals of Physics

Question

A laser emits at \(424 \mathrm{~nm}\) in a single pulse that lasts \(0.500 \mu \mathrm{s}\). The power of the pulse is \(2.80 \mathrm{MW}\). If we assume that the atoms contributing to the pulse underwent stimulated emission only once during the \(0.500 \mu \mathrm{s}\), how many atoms contributed?

Step-by-Step Solution

Answer

Approximately \( 2.98 \times 10^{18} \) atoms contributed to the pulse.

Step 1: Calculate the energy of the pulse

First, we need to find out the total energy delivered by the pulse. We can use the formula \( E = P \times t \), where \( E \) is energy, \( P \) is power, and \( t \) is time. Here, \( P = 2.80 \mathrm{~MW} = 2.80 \times 10^6 \mathrm{~W} \) and \( t = 0.500 \times 10^{-6} \mathrm{~s} \). Substituting these values, we get: \[ E = (2.80 \times 10^6)(0.500 \times 10^{-6}) = 1.40 \mathrm{~J} \].

Step 2: Calculate the energy of a single photon

The energy of a single photon can be calculated using the formula \( E_{photon} = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s} \), \( c \) is the speed of light \( 3 \times 10^8 \mathrm{~m/s} \), and \( \lambda = 424 \mathrm{~nm} = 424 \times 10^{-9} \mathrm{~m} \). Substituting these values, we get: \[ E_{photon} = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{424 \times 10^{-9}} = 4.69 \times 10^{-19} \mathrm{~J} \].

Step 3: Calculate the number of photons in the pulse

To find the number of photons, we divide the total energy of the pulse by the energy of a single photon. Using \( N = \frac{E}{E_{photon}} \), where \( E = 1.40 \mathrm{~J} \) and \( E_{photon} = 4.69 \times 10^{-19} \mathrm{~J} \), we have: \[ N = \frac{1.40}{4.69 \times 10^{-19}} = 2.98 \times 10^{18} \] photons.

Step 4: Calculate the number of atoms

Since each atom contributes one photon per stimulated emission event, the number of atoms contributing to the pulse is equal to the number of photons. Therefore, there are \( 2.98 \times 10^{18} \) atoms that contributed to the pulse.

Key Concepts

Photon Energy Calculation

To understand photon energy calculation, we first need to explore what a photon is. A photon is a fundamental unit of light. Each photon carries a specific amount of energy, determined by its wavelength. The energy of a photon can be calculated with the formula: \[ E_{photon} = \frac{hc}{\lambda} \]Here is where each component of the formula matters:

• \( h \) is Planck's constant, which is \( 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \). This is a fundamental constant of nature, connecting the energy of a photon to its frequency.
• \( c \) is the speed of light, valued approximately at \( 3 \times 10^8 \text{ m/s} \).
• \( \lambda \) is the wavelength of the light, which needs to be in meters for this calculation. In the exercise, the wavelength is given as 424 nm, which converts to \( 424 \times 10^{-9} \text{ m} \).

By substituting these constants and the given wavelength into the formula, the energy of a single photon can be determined. This step is crucial in calculating the number of photons involved in a laser pulse.

Stimulated Emission

Stimulated emission is a process where an incoming photon prompts an excited atom to release a photon of the same frequency, phase, polarization, and direction. This concept is fundamental to how lasers operate. In a laser, many atoms undergo stimulated emission, which results in the emission of coherent light. Key characteristics of stimulated emission include:

• Coherency: Photons produced are identical in phase and direction to the stimulating photon, leading to a consistent beam.
• Amplification: The emitted photon stimulates more emissions, amplifying the light intensity.

In the given exercise, we assume each atom in the laser contributes just one time through stimulated emission during the pulse duration. This means the number of contributing atoms matches the number of photons emitted. Understanding this process helps explain how lasers can emit powerful, uniform light beams.

Planck's Constant

Planck's constant is denoted as \( h \) and is a pivotal constant in quantum mechanics. Its value is \( 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \). Max Planck introduced this constant in the context of black-body radiation, and it has since become central to describing the behavior of particles at the quantum level.Planck's constant's importance includes:

• Linking the energy of a photon to its frequency, directly represented as \( E = hf \).
• Serving as a bridge between the macroscopic and microscopic worlds, aiding in the transition from classical to quantum physics.
• Having broad implications in technologies such as lasers, photonics, and other fields that rely on quantum mechanics.

Understanding Planck's constant is crucial for executing precise photon energy calculations, such as determining the energy carried by light in quantum physics applications. In the presented exercise, it enables us to accurately calculate the energy of photons emitted by a laser.