Problem 11 - Physics for Scientists and Engineers with Modern Physics

Question

(1I) A free neutron \(\left(m=1.67 \times 10^{-27} \mathrm{kg}\right)\) has a mean life of 900 s. What is the uncertainty in its mass (in kg)?

Step-by-Step Solution

Answer

The mass uncertainty is approximately \( 3.9 \times 10^{-40} \text{ kg} \).

Step 1: Understand the Problem

We are given the mean life of a neutron and asked to find the uncertainty in its mass. This problem relates to the quantum physics Heisenberg uncertainty principle, which in this context is written in terms of energy and time.

Step 2: Apply Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that \( \Delta E \cdot \Delta t \geq \frac{\hbar}{2} \), where \( \Delta E \) is the uncertainty in energy, \( \Delta t \) is the mean lifetime (900 s), and \( \hbar \) is the reduced Planck constant \( \approx 1.05 \times 10^{-34} \text{Js} \). Here, \( \Delta E = \Delta m \cdot c^2 \), with \( \Delta m \) as the uncertainty in mass.

Step 3: Rearrange and Solve for Mass Uncertainty

Rearrange the formula for \( \Delta E \) to find \( \Delta m \):\[ \Delta m = \frac{\hbar}{2 \cdot c^2 \cdot \Delta t} \]Insert the given values:\[ \Delta m = \frac{1.05 \times 10^{-34} \, \text{Js}}{2 \cdot (3 \times 10^8)^2 \, \text{m/s}^2 \cdot 900 \, \text{s}} \]

Step 4: Calculate the Result

Using the values in the formula, we can solve for \( \Delta m \):\[ \Delta m = \frac{1.05 \times 10^{-34}}{2 \times 9 \times 10^{16} \times 900} \approx 3.9 \times 10^{-40} \, \text{kg} \]The uncertainty in the mass of the neutron is approximately \( 3.9 \times 10^{-40} \text{ kg} \).

Key Concepts

Quantum Physics

Quantum physics is the science that explores the very small particles, like atoms and subatomic particles, located at the core of everything. Unlike classical physics, quantum physics reveals a universe that is unpredictable and strange. One of its fundamental principles is the Heisenberg Uncertainty Principle. This principle implies that certain paired properties, such as position and momentum, or energy and time, cannot both be precisely measured simultaneously.
For instance, if we know very accurately where an electron is, we lose precision about how fast it is moving, and vice versa. This limitation is not due to imperfections in our instruments, but rather a fundamental property of nature.
### Key Characteristics of Quantum Physics

• Wave-Particle Duality: Particles at the quantum level display characteristics of both particles and waves, depending on how they are observed.
• Probability and Uncertainty: Outcomes in quantum physics are often probabilistic rather than deterministic.
• Quantization: Many properties such as energy are quantized, meaning they can only take discrete values.

Understanding quantum physics requires a shift from classical intuition. It reveals the intricate workings of the universe at microscopic scales.

Neutron Decay

Neutron decay is a crucial process in understanding particle physics and nuclear reactions. A free neutron, which is not part of an atomic nucleus, is unstable, and over time it decays into three particles: a proton, an electron, and an antineutrino. This process is termed beta decay and it increases the stability of the atom by turning a neutron into a proton, which is more stable outside an atomic nucleus.
### Importance of Neutron Decay

• Energy Release: Neutron decay releases energy, which is essential for understanding processes in nuclear reactors and stellar evolution.
• Neutrino Production: Neutrinos produced during decay help physicists study fundamental forces and contribute to the solar neutrino problem understanding.

The mean life of a neutron is about 900 seconds, which means half of a group of free neutrons will decay in this time. This period is relevant when considering the uncertainty in properties like mass, as calculated using principles from quantum physics.

Planck Constant

The Planck constant (\(h\)) is one of the most important numbers in physics, marking the scale at which quantum effects become significant. It represents the smallest action that can be taken within the universe, showcasing the limits below which classical physics loses its validity.
The reduced form, known as the reduced Planck constant (\(\hbar\)) is the Planck constant divided by \(2\pi\), often appearing in quantum mechanics equations like the Heisenberg Uncertainty Principle. The value for \(\hbar\) is approximately \(1.05 \times 10^{-34}\) Js.
### Why is the Planck Constant Significant?

• Defines Quantum Action: Sets the scale for quantum phenomena, determining when classical physics will no longer work.
• Heisenberg Uncertainty: Central to understanding limitations in measuring quantum systems, particularly in energy and time calculations.

The Planck constant is a bridge between the macroscopic classical world and the microscopic quantum realm, governing the actions of particles and waves at their smallest scales.