Problem 49 - College Physics

Question

Suppose that the uncertainty in position of an electron is equal to the radius of the \(n=1\) Bohr orbit, about \(0.5 \times 10^{-10} \mathrm{m} .\) Calculate the minimum uncertainty in the cor- responding momentum the minimum uncertainty in the cor- magnitude of the momentum of the electron in the \(n=1\) Bohr orbit.

Step-by-Step Solution

Answer

The minimum uncertainty in momentum is approximately \(1.05 \times 10^{-24} \text{ kg m/s}\).

Step 1: Understanding Uncertainty Principle

When dealing with uncertainties, we refer to Heisenberg's Uncertainty Principle, which states that the product of the uncertainty in position (\(\Delta x\)) and the uncertainty in momentum (\(\Delta p\)) is at least \(\hbar/2\), where \(\hbar\) is the reduced Planck's constant: \(\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\).

Step 2: Plugging Values into the Equation

Given that \(\Delta x = 0.5 \times 10^{-10} \text{ m}\), and knowing \(\hbar = 1.0545718 \times 10^{-34} \text{ Js}\), plug these values into the uncertainty equation: \(\Delta p \geq \frac{1.0545718 \times 10^{-34}}{2 \times 0.5 \times 10^{-10}}\).

Step 3: Calculating Minimum Uncertainty

Calculate the minimum uncertainty in momentum: \(\Delta p \geq \frac{1.0545718 \times 10^{-34}}{1 \times 10^{-10}} = 1.0545718 \times 10^{-24} \text{ kg m/s}\).

Step 4: Simplifying Result

The minimum uncertainty in the momentum, \(\Delta p\), simplifies to \(1.05 \times 10^{-24} \text{ kg m/s}\) when rounded to three significant figures.

Key Concepts

Bohr orbit

The concept of the Bohr orbit comes from Niels Bohr's model of the atom, introduced in the early 20th century. According to this model, electrons orbit the nucleus at specific distances, each corresponding to a particular energy level. This distance is known as the Bohr radius, particularly for the hydrogen atom. For the lowest energy level, known as the ground state or the first Bohr orbit, this radius is approximately \(0.5 \times 10^{-10} \text{ m}\).

This foundational model provides a simple explanation for atomic structure and spectral lines, by limiting electron orbits to quantized values. Bohr's model was a significant step forward from previous atomic theories, setting the stage for more complex quantum mechanical models that better describe electron behaviors in atoms other than hydrogen.

momentum

Momentum is a fundamental concept in physics and describes the amount of motion an object has. It is a vector quantity, which means it has both magnitude and direction. The momentum of an object is calculated as the product of its mass and velocity, represented by the formula \( p = mv \).

In the context of quantum mechanics, the concept of momentum becomes more nuanced. Quantum particles like electrons do not have definite positions or momenta simultaneously. This idea is encapsulated in Heisenberg's Uncertainty Principle, which relates the uncertainties in these quantities. For an electron in a Bohr orbit, the uncertainty in momentum is tied to the uncertainty in its position, highlighting the intrinsic quantum nature of subatomic particles.

Planck's constant

Planck's constant is a fundamental constant in quantum mechanics, symbolized by \( h \), and it sets the scale of quantum effects. The reduced Planck's constant, \( \hbar \), is \( h \) divided by \( 2\pi \), and it often appears in formulations involving quantum phenomena, such as the Heisenberg Uncertainty Principle.

Planck's constant is approximately \( 6.62607015 \times 10^{-34} \text{ Js} \). The reduced version, \( \hbar \), is about \( 1.0545718 \times 10^{-34} \text{ Js} \). In the uncertainty principle, \( \hbar \) plays a crucial role in defining the relationship between the uncertainties of position and momentum, showing the fundamental limits of precision for measurements in quantum mechanics. It underscores the discrete nature of quantum systems.

electron uncertainty

Electron uncertainty is central to the field of quantum mechanics, describing the unpredictable nature of an electron's position and momentum. According to Heisenberg's Uncertainty Principle, the precision in measuring these two properties is inherently limited. As one becomes more precise, the other becomes less certain.

This uncertainty is not due to measurement flaws, but a fundamental property of nature at the quantum level. It suggests that particles like electrons do not have precise positions and speeds but exist as a cloud of probabilities. For instance, knowing the position of an electron in a Bohr orbit with a high degree of certainty increases the uncertainty in its momentum. Such uncertainty poses challenges in predicting exact paths of particles, influencing modern physics and technology, such as the development of quantum computing.