Problem 8 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A 12-g bullet leaves a rifle horizontally at a speed of \(180 \mathrm{~m} / \mathrm{s}\). ( \(a\) ) What is the wavelength of this bullet? \((b)\) If the position of the bullet is known to a precision of \(0.65 \mathrm{~cm}\) (radius of the barrel), what is the minimum uncertainty in its vertical momentum?

Step-by-Step Solution

Answer

(a) Wavelength is approximately \(3.07 \times 10^{-34}\) m. (b) Minimum uncertainty in vertical momentum is \(8.10 \times 10^{-33}\) kg m/s.

Step 1: Understand the Problem

We need to find two things: (a) the wavelength of a bullet with a given speed and mass, and (b) the minimum uncertainty in its vertical momentum, given the position's precision.

Step 2: Apply the de Broglie Wavelength Formula

To find the wavelength of the bullet, we use the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant \(6.63 \times 10^{-34} \, \text{m}^2 \text{kg} / \text{s}\), \( m \) is the mass of the bullet \(0.012 \, \text{kg}\), and \( v \) is the velocity \(180 \, \text{m/s}\).

Step 3: Calculate the Wavelength

Substitute the known values into the de Broglie equation: \( \lambda = \frac{6.63 \times 10^{-34}}{0.012 \times 180} \). Calculating this gives \( \lambda \approx 3.07 \times 10^{-34} \, \text{m} \).

Step 4: Use the Uncertainty Principle for Vertical Momentum

The uncertainty principle is given by \( \Delta x \Delta p \geq \frac{\hbar}{2} \), where \( \Delta x \) is the uncertainty in position (converted from cm to meters as 0.0065 m), and \( \Delta p \) is the uncertainty in momentum. \( \hbar = \frac{h}{2\pi} \).

Step 5: Calculate the Minimum Uncertainty in Vertical Momentum

Substitute into the uncertainty equation: \( \Delta p \geq \frac{6.63 \times 10^{-34} / (2 \pi)}{2 \times 0.0065} \). Calculate this to find \( \Delta p \approx 8.10 \times 10^{-33} \, \text{kg m/s} \).

Key Concepts

Uncertainty Principle

The uncertainty principle is a fundamental concept in quantum mechanics, stating there's a limit to how precisely we can know both the position and momentum of a particle. Mathematically, it is expressed as:

• \( \Delta x \Delta p \geq \frac{\hbar}{2} \), where
  • \( \Delta x \) is the uncertainty in position
  • \( \Delta p \) is the uncertainty in momentum
  • \( \hbar \) is the reduced Planck's constant

The principle highlights the inherent limitations when measuring quantum systems. It doesn't result from technical imperfections but is a fundamental property of nature. This means that the more precisely we know a particle's position, the less precisely we can know its momentum, and vice versa. This relationship is particularly significant at microscopic scales.
In the given exercise, knowing the bullet's position to a specific degree introduces an unavoidable uncertainty in its momentum. We need to use the principle to compute this uncertainty.

Momentum

Momentum is an essential concept in physics related to motion, defined as the product of an object's mass and velocity. Described by the formula:

• \( p = mv \), where
  • \( p \) is momentum
  • \( m \) is mass
  • \( v \) is velocity

In classical physics, momentum helps predict results like collision outcomes and energy transfers. In the quantum realm, it's also a crucial aspect, especially when discussed alongside the uncertainty principle. Knowing the precise momentum of particles can be challenging because of their wave-like behavior.
For the bullet in our exercise, it has a mass of 0.012 kg and a horizontal speed of 180 m/s. Yet, even with these precise parameters, there's an uncertainty in the vertical component of its momentum, influenced significantly by the precision of its measured position, as explained by the uncertainty principle.

Planck's Constant

Planck's constant is a critical figure in quantum mechanics, denoted as \( h \). Its value is approximately \( 6.63 \times 10^{-34} \, \text{Js} \). This constant acts as a bridge between the wave and particle properties of matter.
Planck's constant appears in the de Broglie equation for wavelength, making it crucial for determining the wave characteristics of particles with mass and speed. In the equation:

• \( \lambda = \frac{h}{mv} \)

\( h \) represents how particles with different masses and velocities can exhibit wave-like behavior.
This constant also appears in the uncertainty principle as part of the reduced Planck's constant, \( \hbar = \frac{h}{2\pi} \). Both contexts illustrate how \( h \) is fundamental to describing the quantum mechanics landscapes.
In the exercise, the de Broglie equation utilizes Planck's constant to explore the wave nature of the bullet. Despite its larger scale where quantum effects are nearly negligible, Planck's constant helps model the theoretical limits of these characteristics.