Problem 24 - University Physics with Modern Physics

Question

A horizontal beam of laser light of wavelength 585 nm passes through a narrow slit that has width 0.0620 mm. The intensity of the light is measured on a vertical screen that is 2.00 m from the slit. (a) What is the minimum uncertainty in the vertical component of the momentum of each photon in the beam after the photon has passed through the slit? (b) Use the result of part (a) to estimate the width of the central diffraction maximum that is observed on the screen.

Step-by-Step Solution

Answer

(a) Minimum momentum uncertainty is approximately \( 8.49 \times 10^{-31} \text{ kg m/s} \). (b) Central diffraction maximum width is about 37.7 mm.

Step 1: Understanding the Concept

We need to apply the Heisenberg Uncertainty Principle to find the minimum uncertainty in the momentum of the photons passing through the slit and use this information to estimate the width of the central diffraction maximum on a screen. The Heisenberg Uncertainty Principle is given by: \[ \Delta y \cdot \Delta p_y \geq \frac{h}{4\pi} \]where \( \Delta y \) is the width of the slit and \( \Delta p_y \) is the uncertainty in the momentum of the photons.

Step 2: Calculating Minimum Momentum Uncertainty

Given \( \Delta y = 0.0620 \) mm or \( 0.0620 \times 10^{-3} \) m, use the Heisenberg Uncertainty Principle:\[ \Delta p_y = \frac{h}{4\pi \cdot \Delta y} \]where \( h = 6.626 \times 10^{-34} \) J·s is Planck's constant. Calculate \( \Delta p_y \) using the given slit width.

Step 3: Plug in the Values

Substitute the known values into the equation:\[ \Delta p_y = \frac{6.626 \times 10^{-34}}{4\pi \times 0.0620 \times 10^{-3}} \approx 8.49 \times 10^{-31} \text{ kg m/s} \]Thus, the minimum uncertainty in the vertical component of momentum is approximately \( 8.49 \times 10^{-31} \text{ kg m/s} \).

Step 4: Estimate Width of Central Diffraction Maximum

The uncertainty in momentum \( \Delta p_y \) affects the spreading of the light in the form of a diffraction pattern. For a small angle, use \( \theta \approx \frac{\Delta p_y}{p} \). The width of the central maximum \( \Delta Y \) can be estimated by\[ \Delta Y = \frac{2\lambda L}{\Delta y} \]Where \( L = 2.00 \) m is the distance to the screen and \( \lambda = 585 \) nm is the wavelength.

Step 5: Calculate the Width of the Central Diffraction Maximum

Convert the wavelength into meters: \( \lambda = 585 \times 10^{-9} \) m. Then substitute the values into the equation:\[ \Delta Y = \frac{2 \times 585 \times 10^{-9} \times 2.00}{0.0620 \times 10^{-3}} \approx 0.0377 \text{ m or } 37.7 \text{ mm} \]The estimated width of the central diffraction maximum is 37.7 mm.

Key Concepts

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is a fundamental concept in quantum mechanics. It states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision simultaneously. This principle is especially crucial when dealing with very small particles, like photons. In our problem, it is used to determine the uncertainty in a photon’s momentum after it passes through a narrow slit.
For the vertical component of momentum, the principle is expressed as: \[ \Delta y \cdot \Delta p_y \geq \frac{h}{4\pi} \]where:

• \( \Delta y \) is the width of the slit.
• \( \Delta p_y \) is the uncertainty in the momentum.
• \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ J·s}) \).

This formula implies that a smaller slit (smaller \( \Delta y \)) causes a greater uncertainty in momentum (larger \( \Delta p_y \)), which links closely to the diffraction pattern we observe.

Photon Momentum

Photon momentum might seem like a strange concept, as we usually associate momentum with mass. However, photons are particles of light that, despite having no mass, still exhibit momentum. This momentum is linked to the energy and wavelength of the photons. If you recall from physics, the momentum \( p \) of a photon is given by:\[ p = \frac{h}{\lambda} \]where:

• \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{ J·s}\).
• \( \lambda \) is the wavelength of the light \( (in \text{ m}) \).

This relationship indicates that photons of shorter wavelengths (higher energy) have more momentum. When photons pass through a slit, their momentum is distributed differently, leading to a spread in their paths on the other side—a phenomenon described by the diffraction pattern.

Diffraction Pattern

A diffraction pattern is a series of light and dark bands resulting from the bending of light waves around an obstacle or through a slit. In our problem, the horizontal laser beam passes through a narrow vertical slit, creating a characteristic pattern on a screen placed some distance away.When light encounters a slit, the light waves spread out due to diffraction. The degree of this spreading depends on the wavelength of the light and the width of the slit. The central diffraction maximum we calculated marks the brightest band at the center of the pattern, and its width is determined by the formula:\[ \Delta Y = \frac{2\lambda L}{\Delta y} \]where:

• \( \lambda \) is the wavelength of the light \( (in \text{ m}) \).
• \( L \) is the distance from the slit to the screen.
• \( \Delta y \) is the slit width \( (in \text{ m}) \).

Understanding diffraction patterns is crucial for applications in optics and experimental physics, where precision in measuring wave behavior is necessary.

Laser Light Wavelength

The wavelength of laser light is essential in determining the characteristics of the light's interaction with various mediums, including narrow slits. Wavelength is the distance between consecutive peaks of a wave. For laser light, which is highly monochromatic, this value is very precise.In our specific problem, the wavelength of the laser light is given as 585 nm. This precise wavelength affects how the light will diffract when it passes through the slit. Converting this into meters for calculations, we get:\[ \lambda = 585 \times 10^{-9} \text{ m} \]This wavelength is used in the equations to find both the photon momentum and the resulting diffraction pattern. In practice, knowing the wavelength allows scientists to predict how light will behave in different scenarios, including how and where light intensity falls on a screen when performing experiments.