Problem 34 - Physics for Scientists and Engineers with Modern Physics

Question

(1) The measured width of the \(\psi(3686)\) meson is about 300 \(\mathrm{keV}\) . Estimate its mean life.

Step-by-Step Solution

Answer

The mean life of the \( \psi(3686) \) meson is approximately \( 2.194 \times 10^{-21} \text{ seconds}.\)

Step 1: Understanding the relationship

The mean life of a particle is related to its decay width via the uncertainty principle. Specifically, this is expressed as the relationship between the lifetime ( \( \tau \)) and the decay width ( \( \Gamma \)): \[ \tau = \frac{\hbar}{\Gamma} \]where \( \hbar \) is the reduced Planck's constant (\( \hbar \approx 6.582 \times 10^{-22} \text{ MeV} \cdot \text{s}\)). In this problem, the decay width \( \Gamma \) is given as 300 keV.

Step 2: Converting units for consistency

We need to ensure that the units are consistent to calculate the mean life. So, we convert the decay width from keV to MeV since \( \hbar\) is in MeV \( \cdot \) s:\[ 300 \text{ keV} = 0.300 \text{ MeV} \]

Step 3: Substitute values and calculate mean life

Using the relationship \( \tau = \frac{\hbar}{\Gamma} \), substitute \( \hbar \approx 6.582 \times 10^{-22} \text{ MeV} \cdot \text{s} \) and \( \Gamma = 0.300 \text{ MeV} \):\[ \tau = \frac{6.582 \times 10^{-22}}{0.300} \approx 2.194 \times 10^{-21} \text{ s} \]Thus, the mean life of \( \psi(3686) \) is approximately \( 2.194 \times 10^{-21} \text{ seconds} \).

Key Concepts

Uncertainty Principle

The Uncertainty Principle is a fundamental concept in quantum mechanics. It states that there is a limit to how precisely we can know certain pairs of properties of a particle, such as position and momentum.

In the context of particle physics, this principle is crucial because it establishes a relationship between the lifetime of a particle and its decay width. This principle is beautifully captured in the formula:

• \( \tau = \frac{\hbar}{\Gamma} \)

Here, \( \tau \) represents the mean life (or lifetime) of a particle, and \( \Gamma \) denotes the decay width.

This relationship indicates that particles with a very short lifetime must have a higher decay width, and vice versa. It suggests a fundamental trade-off: the more accurately we know a particle's energy level (related to \( \Gamma \)), the less accurately we can know its lifetime.

Decay Width

Decay width, denoted by \( \Gamma \), is an important concept in studying unstable particles. It describes the range or spread in the energy levels of the particle due to its instability.

Think of decay width as a measure of how quickly a particle decays or transitions to a lower energy state. More simply, it's like measuring how 'wide' the particle's potential to decay is.

In our exercise, the decay width of the \( \psi(3686) \) meson is given as 300 keV, which is a small energy value indicating a very short but significant measure of lifetime. When particles have a larger decay width, they tend to decay faster compared to those with a smaller decay width.

Mean Life

The mean life, also called lifetime, is the average time a particle will exist before it decays. It’s often represented by \( \tau \) in equations.

The mean life is directly related to the decay width: the larger the lifetime, the smaller the decay width, indicating a stable particle. Conversely, particles that decay rapidly have shorter lifetimes and larger corresponding decay widths.

In terms of calculation, as seen in the exercise, when we know the decay width \( \Gamma \), we can compute the mean life \( \tau \) using the formula:

• \( \tau = \frac{\hbar}{\Gamma} \)

The mean life is an essential concept as it helps physicists understand how long certain subatomic particles can exist before experiencing decay.

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics, and it's often denoted by \( h \). However, in many calculations, especially in particle physics, we use the reduced Planck's constant \( \hbar \), which is \( h/2\pi \).

This constant is extremely small, with a value of approximately \( 6.626 \times 10^{-34} \) Js in its full form. The reduced Planck’s constant is \( 6.582 \times 10^{-22} \text{ MeV} \cdot \text{s} \).

Planck’s constant appears in the uncertainty principle formula, providing a bridge between quantum mechanics and classical physics.

Understanding and using Planck's constant helps us to quantify the highly constrained and sensitive measurements of energy and time in the quantum realm. Its significance underlies many fundamental concepts and constants within the field of physics.