Question

(a) An electron in the ground state of the hydrogen atom moves at an average speed of \(5 \times 10^{6} \mathrm{~m} / \mathrm{s}\). If the speed is known to an uncertainty of 20 percent, what is the minimum uncertainty in its position? Given that the radius of the hydrogen atom in the ground state is \(5.29 \times 10^{-11} \mathrm{~m},\) comment on your result. The mass of an electron is \(9.1094 \times 10^{-31} \mathrm{~kg} .\) (b) A \(0.15-\mathrm{kg}\) baseball thrown at 100 mph has a momentum of \(6.7 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\). If the uncertainty in measuring the momentum is \(1.0 \times 10^{-7}\) of the momentum, calculate the uncertainty in the baseball's position.

Short answer

(a) Minimum position uncertainty for electron is \(5.785 \times 10^{-10} \text{ m}\). (b) Minimum position uncertainty for baseball is \(7.87 \times 10^{-29} \text{ m}\).

Step-by-step solution

Calculate the Uncertainty in Velocity for the Electron

The uncertainty in velocity ( \( \Delta v \)) is given as 20% of the average velocity. The average velocity \( v \) is \( 5 \times 10^6 \text{ m/s} \). Thus, the uncertainty can be calculated as follows:\[ \Delta v = 0.2 \times 5 \times 10^6 \text{ m/s} = 1 \times 10^6 \text{ m/s}. \]

Use Heisenberg's Uncertainty Principle for the Electron

Heisenberg's Uncertainty Principle states that the product of the uncertainties in position and momentum is at least \( \frac{\hbar}{2} \), where \( \hbar = \frac{h}{2\pi} \approx 1.054 \times 10^{-34} \text{ m}^2 \text{ kg/s} \). The momentum \( p \) is \( mv \), where \( m \) is the electron's mass and \( v \) is its velocity.Therefore, for the electron:\[ \Delta p = m \times \Delta v = 9.1094 \times 10^{-31} \text{ kg} \times 1 \times 10^6 \text{ m/s} = 9.1094 \times 10^{-25} \text{ kg m/s}. \]Using the principle:\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \]\[ \Delta x \geq \frac{1.054 \times 10^{-34} \text{ m}^2 \text{ kg/s}}{2 \times 9.1094 \times 10^{-25} \text{ kg m/s}}. \]Solving gives:\[ \Delta x \geq 5.785 \times 10^{-10} \text{ m}. \]

Compare the Uncertainty in Position with the Hydrogen Atom Radius

The calculated uncertainty in position \( \Delta x \approx 5.785 \times 10^{-10} \text{ m} \) is substantially larger than the hydrogen atom's ground state radius of \( 5.29 \times 10^{-11} \text{ m} \). This indicates that the position is highly uncertain compared to the size of the atom.

Calculate the Uncertainty in Momentum for the Baseball

The uncertainty in momentum ( \( \Delta p \)) for the baseball is given as \(1.0 \times 10^{-7}\) of its momentum. The given momentum \( p \) is \(6.7 \text{ kg m/s}\). Thus, the uncertainty is:\[ \Delta p = 1.0 \times 10^{-7} \times 6.7 \text{ kg m/s} = 6.7 \times 10^{-7} \text{ kg m/s}. \]

Use Heisenberg's Uncertainty Principle for the Baseball

Use Heisenberg's Uncertainty Principle again:\[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \]\[ \Delta x \geq \frac{1.054 \times 10^{-34} \text{ m}^2 \text{ kg/s}}{2 \times 6.7 \times 10^{-7} \text{ kg m/s}} \]Solving gives:\[ \Delta x \geq 7.87 \times 10^{-29} \text{ m}. \]

Key concepts

Quantum Mechanics

Quantum mechanics is a fascinating area of physics that deals with the behavior of particles on a microscopic scale. Unlike classical physics, which provides precise predictions, quantum mechanics introduces the concept of probability and uncertainty.
This is where Heisenberg's Uncertainty Principle comes into play, stating that it's impossible to know both the position and momentum of a particle, like an electron, with absolute certainty at the same time.
- This principle is fundamental to understanding atomic and subatomic systems. - Quantum mechanics has led to revolutionary technologies such as semiconductors and lasers. The principle is represented mathematically and is vital in explaining phenomena within quantum mechanics, providing insights into atomic structures that are not possible with classical theories.

Electron Motion

The motion of electrons within an atom is both intriguing and complex. Due to their small mass and high speed, electrons do not follow traditional paths around the nucleus like planets around the sun.
Instead, they exist in areas of probability known as orbitals, where they are likely to be found at any given time.
- Electrons move rapidly, reaching speeds like those noted in the exercise, around \(5 \times 10^6 \text{ m/s}\).- Their motion affects how they bond with other atoms, influencing the chemical properties of the elements.Understanding electron motion is crucial, as it helps explain why elements have specific colors, odors, and reactivity levels. It further explains how light and energy interact with matter on a fundamental level.

Momentum Uncertainty

Momentum uncertainty is a crucial part of quantum mechanics. According to Heisenberg's Uncertainty Principle, the more precisely we know a particle's momentum, the less precisely we know its position, and vice versa. This principle fundamentally limits the precision of simultaneous measurements of key properties.
- For tiny particles like electrons, this uncertainty can significantly affect their behavior and measurement outcomes. - It has minimal impact on larger objects, such as a baseball, which operate under classic physics rules. This concept reveals the limitations of measurement at quantum levels and illustrates the underlying randomness of quantum systems. It also explains why observing or measuring a quantum system inevitably alters it.

Atomic Structure

Atomic structure refers to the composition and arrangement of particles, including protons, neutrons, and electrons, within an atom. The nucleus is made up of protons and neutrons, while electrons orbit this central core.
- The electron's role is critical because their arrangement dictates an atom's chemical properties and behavior. - Understanding atomic structure is crucial for comprehending how atoms bond to form molecules. In atomic theory, electrons are not fixed in one place but spread out over an area around the nucleus. This arrangement forms what we know as electron clouds or orbitals. Different elements have varying numbers of these particles, which accounts for their diverse attributes and interactions.

Hydrogen Atom

The hydrogen atom is the simplest and most fundamental atom in the universe, consisting of just one proton and one electron. This simplicity makes it an excellent model for understanding basic quantum mechanics and atomic physics.
- The electron in a hydrogen atom occupies orbitals that are spherical in the simplest model.- Its ground state radius, given as \(5.29 \times 10^{-11} \text{ m}\), is crucial to calculations in quantum mechanics.Learning about the hydrogen atom is pivotal because it serves as a foundation for understanding more complex atoms and molecules. Its study has provided substantial insights into atomic interaction, electromagnetic radiation, and the quantization of energy levels.