Problem 46 - College Physics

Question

\(\bullet\) (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

(a) Minimum velocity uncertainty is \(1.58 \times 10^4 \text{ m/s}\). (b) Minimum position uncertainty is \(2.32 \times 10^{-4} \text{ m}\).

Step 1: Understand the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that the product of the uncertainties of position and momentum of a particle cannot be smaller than a particular constant: \[\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\] where \( \Delta x \) is the uncertainty in position, \( \Delta p \) is the uncertainty in momentum, and \( h \) is Planck's constant (\( h = 6.626 \times 10^{-34} \text{J⋅s} \)). For momentum, \( p = m \cdot v \) where \( m \) is mass and \( v \) is velocity, allowing us to express the uncertainty in velocity as: \[\Delta x \cdot m \cdot \Delta v \geq \frac{h}{4\pi}\] Solve for \( \Delta v \): \[\Delta v \geq \frac{h}{4\pi m \Delta x}\]

Step 2: Calculate the Minimum Uncertainty in Velocity of the Proton

For the proton, given: \( \Delta x = 2.0 \times 10^{-12} \text{ m} \), and the mass of a proton \( m_p = 1.67 \times 10^{-27} \text{ kg} \). Substitute these values into the formula for velocity uncertainty: \[\Delta v \geq \frac{6.626 \times 10^{-34}}{4\pi \times 1.67 \times 10^{-27} \times 2.0 \times 10^{-12}}\] Calculate \( \Delta v \): \[\Delta v \geq 1.58 \times 10^{4} \text{ m/s}\]

Step 3: Calculate the Minimum Uncertainty in Position of the Electron

For the electron, given: \( \Delta v = 0.250 \text{ m/s} \), and the mass of an electron \( m_e = 9.11 \times 10^{-31} \text{ kg} \). Rearranging the formula for \( \Delta x \): \[\Delta x \geq \frac{h}{4\pi m_e \Delta v}\] Substitute these values into the rearranged formula: \[\Delta x \geq \frac{6.626 \times 10^{-34}}{4\pi \times 9.11 \times 10^{-31} \times 0.250}\]Calculate \( \Delta x \): \[\Delta x \geq 2.32 \times 10^{-4} \text{ m}\]

Key Concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the nature at the smallest scales of atoms and subatomic particles. It offers a framework for understanding the physical universe under these conditions, where classical theories like Newton's laws of motion fail. This field of study explores the interactions between energy and matter, providing a detailed picture of the behavior of particles at quantum levels.
Some of the core elements of quantum mechanics include:

• Wave-Particle Duality: Particles such as electrons exhibit properties of both waves and particles.
• Quantum Entanglement: Particles may become entangled, indicating that the state of one is directly connected to the state of another, regardless of distance.
• Probability and Uncertainty: Exact states of particles are often described using probabilities rather than certainties.

This uncertainty is precisely what the Heisenberg Uncertainty Principle captures, a pivotal part of quantum mechanics, stating that certain pairs of physical properties cannot both be known to arbitrary precision. This leads us directly into the concepts of momentum and position uncertainty.

Momentum Uncertainty

Momentum uncertainty refers to the indeterminacy in measuring both the momentum and position of a particle simultaneously. According to the Heisenberg Uncertainty Principle, the product of uncertainties in position (\( \Delta x \)) and momentum (\( \Delta p \)) of a particle cannot be less than a specific value determined by Planck's constant:\[\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\]Momentum itself is a measure of the quantity of motion of a particle, which depends on its mass and velocity (\( p = m \cdot v \)).
This principle implies that increasing the precision in measuring a particle's position will proportionally increase the uncertainty in its momentum, and vice versa. For a proton or an electron, as examined in the step-by-step solution, if position is known precisely, the minimum uncertainty in velocity or momentum becomes larger, illustrating the practical applications of quantum mechanics at a microscopic level.

Position Uncertainty

Position uncertainty is a critical aspect of the quantum behavior of particles. It asserts that there is a fundamental limit to the precision with which the position of a particle can be measured, especially when its momentum is measured with high precision. This is embodied in Heisenberg's precision equation for position uncertainty, which links the confidence of a position measurement to that of momentum (\( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \)).
When applied to actual particles like electrons, small uncertainties in momentum can culminate in relatively large uncertainties in their position. This is particularly significant due to electrons' smaller mass, thereby amplifying the position uncertainty. Understanding this can help elucidate not only theoretical aspects but also lead to practical understanding in fields like quantum computing and quantum cryptography where precise state measurements of particles are pivotal.

Planck's Constant

Planck's constant (\( h \)) is one of the fundamental constants in physics, pivotal to quantum mechanics. It serves as the proportionality constant between the energy of a photon and the frequency of its electromagnetic wave, expressed as:\[E = h \cdot f\]where \( E \)is energy and \( f \)is frequency.
Planck's constant is also integral to Heisenberg's Uncertainty Principle, signifying the limit to precision in simultaneous measurements of particles' complementary properties, such as position and momentum. Heisenberg's inequality uses \( h \)to set this boundary:\[\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\]Planck's constant has an approximate value of \( 6.626 \times 10^{-34} \text{ J⋅s} \), extraordinarily small, highlighting how uncertainty becomes significant mainly at quantum scales where such tiny constants influence particle behavior in ways not seen in the macroscopic world. This underscores the quantum realm's unique nature, distinct from classical physics.