Problem 43 - College Physics

Question

(a) The uncertainty in the \(x\) component of the position of a proton is \(2.0 \times 10^{-12} \mathrm{~m} .\) What is the minimum uncertainty in the \(x\) component of the velocity of the proton? (b) The uncertainty in the \(x\) component of the velocity of an electron is \(0.250 \mathrm{~m} / \mathrm{s}\). What is the minimum uncertainty in the \(x\) coordinate of the electron?

Step-by-Step Solution

Answer

Minimum velocity uncertainty of proton: \( 1.58 \times 10^4 \) m/s. Minimum position uncertainty of electron: \( 2.32 \times 10^{-3} \) m.

Step 1: Understand the Problem and Given Data

We need to use the Heisenberg Uncertainty Principle for both parts. The principle states that the product of the uncertainties in position and momentum is at least on the order of Planck's constant: \[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]Furthermore, since the momentum \( p = mv \), the uncertainty in momentum can be written as \( \Delta p = m \cdot \Delta v \), where \( m \) is the mass and \( \Delta v \) is the uncertainty in velocity.

Step 2: Calculate Minimum Uncertainty in Velocity of the Proton

For part (a), rearrange the formula to solve for \( \Delta v \):\[ \Delta v \geq \frac{h}{4\pi m \Delta x} \]We are given \( \Delta x = 2.0 \times 10^{-12} \) m. The mass of a proton is approximately \( 1.67 \times 10^{-27} \) kg. Planck’s constant \( h = 6.626 \times 10^{-34} \text{ Js} \).Substitute these into the formula:\[ \Delta v \geq \frac{6.626 \times 10^{-34}}{4\pi \times 1.67 \times 10^{-27} \times 2.0 \times 10^{-12}} \]Simplify to find \( \Delta v \).

Step 3: Solve the Proton Calculation

After substituting the values:\[ \Delta v \geq \frac{6.626 \times 10^{-34}}{4\pi \times 1.67 \times 10^{-27} \times 2.0 \times 10^{-12}} \approx 1.58 \times 10^4 \, \text{m/s} \]So, the minimum uncertainty in the velocity of the proton is approximately \( 1.58 \times 10^4 \) m/s.

Step 4: Calculate Minimum Uncertainty in Position of the Electron

For part (b), rearrange the formula to solve for \( \Delta x \):\[ \Delta x \geq \frac{h}{4\pi m \Delta v} \]We are given \( \Delta v = 0.250 \) m/s. The mass of an electron is approximately \( 9.11 \times 10^{-31} \) kg.Substitute these values into the formula:\[ \Delta x \geq \frac{6.626 \times 10^{-34}}{4\pi \times 9.11 \times 10^{-31} \times 0.250} \]Simplify to find \( \Delta x \).

Step 5: Solve the Electron Calculation

After substituting the values:\[ \Delta x \geq \frac{6.626 \times 10^{-34}}{4\pi \times 9.11 \times 10^{-31} \times 0.250} \approx 2.32 \times 10^{-3} \text{ m} \]So, the minimum uncertainty in the position of the electron is approximately \( 2.32 \times 10^{-3} \) m.

Key Concepts

Quantum Mechanics

Quantum mechanics is the fascinating branch of physics that studies the smallest particles in the universe. It challenges our understanding of how nature behaves at fundamental levels, principles defying intuition from classical physics. One of its core ideas is that particles, like protons and electrons, do not have definite positions or velocities at any given point.
Instead, these properties are described by probabilities, which is a significant departure from the deterministic view of classical mechanics.

• This field accounts for phenomena that classical physics cannot explain, such as the behavior of particles at the atomic and subatomic scales.
• It includes concepts like wave-particle duality, which suggests particles exhibit both wave-like and particle-like properties.
• The Heisenberg Uncertainty Principle, a cornerstone of quantum mechanics, illustrates these inherent uncertainties.

Understanding quantum mechanics requires a shift in thinking, embracing probability and uncertainty as fundamental attributes of physical systems at the microscopic level.

Proton Position Uncertainty

In quantum mechanics, the position of a proton, like any particle, is not precisely known—it comes with an intrinsic uncertainty. This uncertainty is not due to imperfect measurement tools but is a fundamental aspect of nature known as the Heisenberg Uncertainty Principle.
In the original exercise, the uncertainty in the proton's position \(\Delta x\) is stated to be \(2.0 \times 10^{-12}\) meters.

• This means the proton's true position could be anywhere within this range.
• This uncertainty impacts calculations involving the proton's velocity, as exact values cannot be pinned down simultaneously with certainty.
• Such uncertainties are crucial when studying atomic interactions and nuclear reactions where precise behavior of protons plays a role.

Understanding position uncertainty helps us grapple with how subatomic particles behave and ensures our models align with reality at quantum scales.

Electron Velocity Uncertainty

Similar to position, the velocity of an electron in quantum mechanics is also characterized by uncertainty. Instead of being able to precisely define electron velocity, we recognize a range within which it might fall.
For the electron in the exercise, its velocity uncertainty \(\Delta v\) is given as \(0.250\) m/s.

• This indicates that the electron's exact movement is uncertain within this velocity range.
• The smaller a particle’s mass, the more significantly uncertainty affects observable properties like velocity.
• Electrons, being much lighter than protons, often exhibit larger disparities in velocity uncertainty.

Such understanding is central to predicting and interpreting behaviors in quantum systems, including electron configurations in atoms and chemical bonding processes.

Planck's Constant

Planck's constant, denoted by \(h\), is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. Its value is approximately \(6.626 \times 10^{-34}\) Joule-seconds.

• It signifies the scale at which quantum mechanical effects become significant.
• The Heisenberg Uncertainty Principle uses Planck's constant to quantify how precisely position and momentum can be known simultaneously.
• In calculations, like those in the exercise, Planck’s constant sets the baseline for determining limits of precision due to quantum effects.

Planck's constant is integral in formulas used in quantum physics and chemistry, highlighting the quantized nature of energy, momentum, and other properties at microscopic levels. Its role underscores quantum mechanics' departure from classical concepts, shaping our modern understanding of particle physics.