Question

Suppose that photons of blue light \((430 \mathrm{nm})\) are used to locate the position of a 2.80 -g Ping-Pong ball in flight and that the uncertainty in the position is equal to one wavelength. What is the minimum uncertainty in the speed of the Ping-Pong ball? Comment on the magnitude of your result.

Short answer

The minimum uncertainty in speed is approximately \( 1.36 \times 10^{-23} \text{ m/s} \), which is negligible.

Step-by-step solution

Understand the Problem

We are given the wavelength of blue light \( \lambda = 430 \text{ nm} = 430 \times 10^{-9} \text{ m} \) and the mass of a Ping-Pong ball \( m = 2.80 \text{ g} = 0.00280 \text{ kg} \). The uncertainty in position \( \Delta x \) is equal to one wavelength, \( \Delta x = 430 \times 10^{-9} \text{ m} \). We want to find the minimum uncertainty in speed, \( \Delta v \).

Use Heisenberg's Uncertainty Principle

Heisenberg's uncertainty principle relates the uncertainties in position and momentum as \( \Delta x \Delta p \geq \frac{\hbar}{2} \), where \( \hbar = \frac{h}{2\pi} \) and \( h = 6.626 \times 10^{-34} \text{ Js} \) is Planck's constant. First, find \( \Delta p \), the uncertainty in momentum, which is \( m \Delta v \). We use \( \Delta x (m \Delta v) \geq \frac{\hbar}{2} \).

Solve for Uncertainty in Speed

Rearrange the inequality: \( \Delta v \geq \frac{\hbar}{2m\Delta x} \). Calculate \( \hbar = \frac{6.626 \times 10^{-34}}{2\pi} \approx 1.055 \times 10^{-34} \text{ Js} \). Substitute the values: \( \Delta v \geq \frac{1.055 \times 10^{-34}}{2 \times 0.00280 \times 430 \times 10^{-9}} \approx 1.36 \times 10^{-23} \text{ m/s} \).

Comment on the Result

The calculated minimum uncertainty in speed, \( 1.36 \times 10^{-23} \text{ m/s} \), is an extremely small and practically negligible value for a macroscopic object like a Ping-Pong ball, as its actual speed would be many orders of magnitude larger. This demonstrates how quantum uncertainty has a negligible effect on macroscopic objects.

Key concepts

uncertainty in position and momentum

Heisenberg's Uncertainty Principle provides a fundamental limit to the precision with which we can know the position and momentum of a particle simultaneously. When we refer to uncertainty in position and momentum, what we mean is that there is a trade-off in the accuracy with which these two properties can be known. The principle is expressed mathematically as:

• \( \Delta x \Delta p \geq \frac{\hbar}{2} \)

Here, \( \Delta x \) is the uncertainty in the position, and \( \Delta p \) is the uncertainty in the momentum of the particle.
When one is measured with high precision, the other becomes less certain.
The constant \( \hbar \) is derived from Planck's constant, emphasizing the quantum nature of this principle, which is more noticeable at the microscopic scale.
In the problem, the uncertainty in position is equal to one wavelength of the blue light used to observe a Ping-Pong ball. The incredibly small resulting uncertainty in the ball's speed highlights that Heisenberg's principle is not significant for macroscopic objects like a Ping-Pong ball.

Planck's constant

Planck's constant is fundamental to quantum mechanics and it shows up in various forms throughout the field. Represented by the symbol \( h \), its value is approximately \( 6.626 \times 10^{-34} \text{ Js} \). This very small constant is part of equations that describe how energy is quantized in the form of quantum levels or packets.
For Heisenberg's Uncertainty Principle, we use a reduced form of Planck's constant known as \( \hbar \), which is given by:

• \( \hbar = \frac{h}{2\pi} \approx 1.055 \times 10^{-34} \text{ Js} \)

This simplified version, \( \hbar \), is used in calculations involving angular momentum and quantum uncertainties.
Its existence reflects the intrinsic properties of particles at quantum scales, supporting how phenomena at atomic and subatomic levels differ vastly from our macroscopic experiences.
Planck's constant reminds us how minute and discreet the quantum world really is.

quantum mechanics

Quantum mechanics is a branch of physics that fundamentally changed our understanding of nature. It describes the behavior of particles at atomic and subatomic scales, a level at which classical physics cannot accurately predict outcomes.
Quantum mechanics introduces principles such as wave-particle duality, uncertainty, and quantization of energy levels.
One of the key aspects of quantum mechanics is the notion of probability, rather than certainty, leading to the statistical nature of particle properties.

• Heisenberg's Uncertainty Principle is a cornerstone of this field, illustrating the inherent limitations in measuring quantum systems precisely.
• Particles, such as electrons, do not have definite positions or velocities until measured.

This necessitates a probabilistic approach in their study, as events on this scale do not align with deterministic classical physics.
Quantum mechanics challenges us to think beyond the Newtonian physics of everyday objects.

wavelength of light

The wavelength of light is a crucial characteristic of electromagnetic waves, determining its color and energy. It is the distance between successive crests of a wave and is measured in nanometers (nm).
In the exercise provided, blue light with a wavelength of \( 430 \text{ nm} \) is used. This wavelength corresponds to a higher energy than red light and is part of the visible spectrum.
Using light to measure position introduces uncertainties as defined by the Uncertainty Principle, where the act of measuring interferes with what is being measured because of the wave nature of light.

• Shorter wavelengths like blue light provide higher precision in measurements as they are smaller than longer wavelengths.
• Yet, regardless of the wavelength, the inherent limit in accuracy due to quantum effects is unavoidable.

This exercise underscores how such quantum characteristics apply even when using classical tools like light to measure position, although the effects are more significant at microscopic scales.