Problem 29 - College Physics

Question

An electron and a proton each have a momentum of \(3.28470 \times 10^{-30} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s} \pm 0.00025 \times 10^{-30} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\) (a) The minimum uncertainty in the position of the electron compared with that of the proton will be (1) larger, (2) the same, (3) smaller. Why? (b) Calculate the minimum uncertainty in the position for each.

Step-by-Step Solution

Answer

(a) Larger for electron due to mass difference. (b) Both uncertainties calculate to \(2.11 \times 10^{-5} \mathrm{~m}\).

Step 1: Understand the Problem

We are given the momentum of an electron and a proton, along with its uncertainty. Our task is to compare the minimum uncertainty in position between the electron and the proton, and then calculate these uncertainties.

Step 2: Review Heisenberg's Uncertainty Principle

Heisenberg's Uncertainty Principle states that the product of the uncertainties in position \((\Delta x)\) and momentum \((\Delta p)\) of a particle is at least \(\frac{\hbar}{2}\). Formally: \(\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\) where \(\hbar = \frac{h}{2\pi} = 1.0545718 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\).

Step 3: Analyze Mass Differences

The electron and proton have significantly different masses. The electron's mass \((m_e)\) is approximately \(9.10938356 \times 10^{-31} \mathrm{~kg}\) and the proton's mass \((m_p)\) is approximately \(1.6726219 \times 10^{-27} \mathrm{~kg}\). The lighter mass of the electron means its velocity for the same momentum is higher, affecting its position uncertainty.

Step 4: Determine Which Particle has Greater Position Uncertainty

Using the formula \(\Delta x \geq \frac{\hbar}{2\Delta p}\), note that despite the same momentum uncertainty \(\Delta p\), the electron (being much lighter) will have a greater velocity and therefore a greater uncertainty in position than the proton. This is due to the larger spread of potential velocities for a fixed momentum uncertainty.

Step 5: Calculate Minimum Uncertainty in Position for Electron

For the electron, \(\Delta p_e = 0.00025 \times 10^{-30} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\). The uncertainty in position, \(\Delta x_e\), is given by \(\Delta x_e \cdot \Delta p_e \geq \frac{\hbar}{2}\), so \(\Delta x_e \geq \frac{1.0545718 \times 10^{-34}}{2 \times 0.00025 \times 10^{-30}}\). Calculate \(\Delta x_e\).

Step 6: Calculate Minimum Uncertainty in Position for Proton

For the proton, \(\Delta p_p = 0.00025 \times 10^{-30} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\). The uncertainty in position, \(\Delta x_p\), is identical in calculation to the electron: \(\Delta x_p \geq \frac{1.0545718 \times 10^{-34}}{2 \times 0.00025 \times 10^{-30}}\). Calculate \(\Delta x_p\).

Step 7: Compare the Results

Since \(\Delta x_e\) and \(\Delta x_p\) are determined by the same formula from \(\Delta p\), their numerical values will be equivalent. However, conceptually, the effect on position due to mass difference means the larger actual spread in potential positions belongs to the electron as established before.

Key Concepts

Momentum Uncertainty

Heisenberg's Uncertainty Principle tells us that we can never know both the position and momentum of a particle with absolute precision. Momentum uncertainty, represented by \( \Delta p \), plays a crucial role in this principle. The given problem involves both an electron and a proton with a specific momentum of \(3.28470 \times 10^{-30} \mathrm{~kg} \cdot \mathrm{m}/\mathrm{s}\), and a momentum uncertainty of \(0.00025 \times 10^{-30} \mathrm{~kg} \cdot \mathrm{m}/\mathrm{s}\).

This tiny uncertainty might seem insignificant, but it's fundamental in quantum mechanics. It affects how precisely we can pinpoint where a particle is located at any given time. The greater the momentum uncertainty, the less precisely we can know the particle's position. This interdependence is central to understanding the behavior of subatomic particles under quantum mechanics.

Position Uncertainty

Position uncertainty, noted as \( \Delta x \), is directly linked with momentum uncertainty through Heisenberg's principle. This principle can be expressed with the formula \( \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \), where \( \hbar \) is the reduced Planck's constant. This means that as \( \Delta p \) increases, \( \Delta x \) must decrease and vice versa.

For the electron and proton with the same momentum uncertainty, the position uncertainty is theoretically the same when calculated mathematically. However, the electron's smaller mass means it has a higher velocity for the same momentum. This results in the electron having a larger practical range of position uncertainty compared to the proton, due to larger variations in its trajectory. Understanding this aspect helps us envision the unpredictability and fluidity of positions for subatomic particles.

Electron versus Proton

The markedly different masses of electrons and protons are essential in understanding their behavior under quantum mechanics. The mass of an electron is approximately \(9.10938356 \times 10^{-31} \mathrm{~kg}\), while a proton is significantly heavier at about \(1.6726219 \times 10^{-27} \mathrm{~kg}\).

This significant mass difference means that for the same amount of momentum, an electron must travel at a much quicker velocity compared to a proton. Consequently, this affects the electron's range of position, making its location prediction more uncertain than that of a proton.

• The electron has smaller mass, leading to higher speeds for the same momentum.
• The proton's greater mass leads to slower speeds and reduced position uncertainty.
• The conceptual understanding shows that electrons exhibit greater positional spread.

These properties are key to understanding why, even though both particles might share the same calculated momentum uncertainty, the electron practically displays a greater position uncertainty.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of very small particles, such as electrons and protons. The principles of quantum mechanics provide insight into the peculiar nature of the subatomic world, where traditional physics sometimes falls short.

At the core of quantum mechanics is the idea that particles do not have definite states until they are measured. Instead, they exist in superpositions of states, leading to phenomena such as wave-particle duality.

• Particles behave like waves, making their exact position inherently uncertain.
• Probability plays a significant role, rather than deterministic outcomes.
• Quantum mechanics explains the fundamental limitations in measuring precise values of position and momentum.

Understanding these principles is pivotal for grasping why concepts like Heisenberg's Uncertainty Principle are integral to describing and predicting the behavior of particles at such small scales.