Problem 66 - College Physics

Question

The measured energy width of the \(\phi\) meson is 4.0 \(\mathrm{MeV}\) and its mass is 1020 \(\mathrm{MeV} / c^{2} .\) Using the uncertainty principle (in the form \(\Delta E \Delta t \geq h / 2 \pi ),\) estimate the lifetime of the \(\phi\) meson.

Step-by-Step Solution

Answer

The estimated lifetime of the \(\phi\) meson is at least \(1.645 \times 10^{-22}\) seconds.

Step 1: Identify the variables

In this problem, the given values are the energy width of the \(\phi\) meson, \(\Delta E = 4.0\ \mathrm{MeV}\), and its mass, which is not directly needed for solving the problem. We need to estimate the lifetime (\(\Delta t\)) using the energy-time uncertainty principle.

Step 2: Recall the energy-time uncertainty principle

The energy-time uncertainty principle is given by the formula: \(\Delta E \Delta t \geq \frac{h}{2\pi}\). We need Planck's constant, \(h\), which is \(6.626 \times 10^{-34}\ \mathrm{Js}\) or approximately \(4.1357 \times 10^{-15}\ \mathrm{eV}\!\cdot\!\mathrm{s}\) when using eV for energy.

Step 3: Rearrange the formula

From the inequality \(\Delta E \Delta t \geq \frac{h}{2\pi}\), solve for \(\Delta t\) by rearranging the terms to give \(\Delta t \geq \frac{h}{2\pi \Delta E}\).

Step 4: Plug in the values

Substitute \(h = 4.1357 \times 10^{-15}\ \mathrm{eV}\cdot\mathrm{s}\), \(\Delta E = 4.0\ \mathrm{MeV} = 4.0 \times 10^6 \mathrm{eV}\), and \(2\pi \approx 6.283\). Calculate \(\Delta t \geq \frac{4.1357 \times 10^{-15}}{6.283 \times 4.0 \times 10^6}\).

Step 5: Calculate the result

Perform the calculation: \(\Delta t \geq \frac{4.1357 \times 10^{-15}}{2.5132 \times 10^7}\). This gives \(\Delta t \geq 1.645 \times 10^{-22}\ \mathrm{s}\).

Key Concepts

Phi Meson

The phi meson is a type of subatomic particle, classified as a meson. Mesons are particles made up of a quark and an antiquark. The phi meson specifically consists of a strange quark and its corresponding antiquark. It has a relatively short life, making it difficult to study. The phi meson plays a crucial role in the study of particle physics by helping scientists understand the strong force, one of the four fundamental forces in nature.

• The phi meson has a mass of approximately 1020 MeV/(\( c^2\)), which was given in the problem, but not needed for the lifetime estimation.
• It can decay into other types of particles, which is why knowing its lifetime is important.

Understanding its properties, such as energy width and lifetime, helps physicists explore and confirm theories related to the strong force and quantum chromodynamics.

Energy Width

Energy width refers to the range of energy possessed by a particle over its very short existence. For the phi meson, this energy width is measured as 4.0 MeV in this exercise.

• The energy width is an indication of how precise the energy level of the meson is.
• It also relates to the concept of decay, where a wider energy width implies a shorter lifetime.

By examining the energy width, scientists can better predict and understand the stability and behavior of subatomic particles. In quantum mechanics, it's crucial for determining how long a particle might exist before transitioning into other forms of energy or matter.

Planck's Constant

Planck's constant is a fundamental constant in physics that dictates the scale at which quantum effects become noticeable. It is a tiny number, specifically 6.626 x 10\(^{-34}\) Js, however it’s often used in its energy form as approximately 4.1357 x 10\(^{-15}\) (\( \text{eV} \, \cdot \, \text{s}\)).

• This constant is vital for various equations, including the energy-time uncertainty principle.
• It establishes a relationship between energy and frequency of a wave.
• Central to quantum mechanics, it serves as a bridge between the classical and quantum worlds.

In our exercise, Planck's constant is used to predict how precise our knowledge of time and energy can be, crucial for estimating the lifetime of particles like the phi meson. Without it, we couldn’t apply the uncertainty principle effectively.

Energy-Time Uncertainty

The energy-time uncertainty principle is a fundamental concept in quantum mechanics. It highlights that it is impossible to precisely measure both the energy and the time duration of an event simultaneously. This principle is mathematically expressed as:\[\Delta E \Delta t \geq \frac{h}{2\pi}\]

• \(\Delta E\) is the uncertainty in energy, in our context measured as the energy width of the phi meson.
• \(\Delta t\) is the uncertainty in time, which we want to find to estimate the particle's lifetime.

Essentially, the less certain we are of the energy, the more certain we can be about the time, and vice versa. This principle is not just theory; it has practical applications, like calculating the lifetime of rapidly decaying particles such as the phi meson.

Lifetime Estimation

Estimating the lifetime of a particle like the phi meson involves using the energy-time uncertainty principle. We know that there's always some level of uncertainty in our measurements due to quantum mechanical constraints.

• In the exercise, we start with the known energy width of 4.0 MeV.
• By rearranging the uncertainty principle, we solve for \(\Delta t\) to estimate the lifetime.
• The calculated lifetime provides insight into how long the meson exists before decaying into other particles.

These estimations are essential in particle physics for understanding unstable particles that don't last long enough to measure directly. By using theoretical principles and constants like Planck's constant, we infer the behavior of these particles, leading to advances in our understanding of the atomic and subatomic world.