Problem 33 - Physics for Scientists and Engineers with Modern Physics

Question

(1) The mean life of the \(\Sigma^{0}\) particle is \(7 \times 10^{-20}\) s. What is the uncertainty in its rest energy? Express your answer in MeV.

Step-by-Step Solution

Answer

4.70 MeV

Step 1: Identify the given information

We are given the mean life (average lifetime) of the \(\Sigma^{0}\) particle, \(\tau = 7 \times 10^{-20}\) seconds. We need to find the uncertainty in its rest energy.

Step 2: Use the Energy-Time Uncertainty Principle

The Energy-Time Uncertainty Principle is given by the relationship \(\Delta E \cdot \Delta t \geq \frac{\hbar}{2}\), where \(\Delta E\) is the uncertainty in energy, \(\Delta t = \tau\) is the lifetime or time uncertainty, and \(\hbar = 1.0545718 \times 10^{-34}\, \text{Js}\) is the reduced Planck's constant.

Step 3: Rearrange to solve for uncertainty in energy

Rearrange the formula to solve for \(\Delta E\):\[ \Delta E \geq \frac{\hbar}{2 \Delta t} \]This will give us the minimum possible uncertainty in energy for the \(\Sigma^{0}\) particle.

Step 4: Substitute the values into the formula

Substitute the values into the equation:\[\Delta E \geq \frac{1.0545718 \times 10^{-34} \text{ Js}}{2 \times 7 \times 10^{-20}\text{ s}}\]

Step 5: Calculate the uncertainty in energy

Calculate the right side of the equation to find \(\Delta E\):\[\Delta E \geq \frac{1.0545718 \times 10^{-34}}{1.4 \times 10^{-19}} \approx 7.53 \times 10^{-16} \text{ J}\]

Step 6: Convert energy from Joules to MeV

To convert the energy from Joules to Mega-electronvolts (MeV), use the conversion factor \(1 \text{ eV} = 1.602176634 \times 10^{-19} \text{ J}\). Therefore, \(1 \text{ MeV} = 1.602176634 \times 10^{-13} \text{ J}\).\[\Delta E_{\text{MeV}} = \frac{7.53 \times 10^{-16}}{1.602176634 \times 10^{-13}} \approx 4.70 \text{ MeV}\]

Step 7: Conclusion

Thus, the uncertainty in the rest energy of the \(\Sigma^{0}\) particle is approximately \(4.70 \text{ MeV}\).

Key Concepts

mean life

The mean life of a particle, often symbolized as \(\tau\), is a fundamental concept in physics that describes the average lifespan of an unstable particle before it decays. Specifically, it provides a measure of how long a particle, like the \(\Sigma^0\), exists before transitioning into other particles.

To calculate the mean life, scientists often examine a large number of the same type of particle and average the times they take to decay. Mathematically, it can be related to the decay constant and the half-life of the particle. However, mean life is different because it considers the statistical properties of all possible decay times.

• The mean life is longer than the half-life due to the exponential nature of decay.
• Understanding the mean life helps physicists predict the behavior of particles and is crucial in fields like particle physics and cosmology.
• In our problem, the mean life of the \(\Sigma^0\) particle was given as \(7 \times 10^{-20}\) seconds.

reduced Planck's constant

The reduced Planck's constant, denoted as \(\hbar\), is a fundamental quantity in quantum mechanics. It equals the standard Planck's constant \(h\) divided by \(2\pi\), simplifying many equations in quantum physics.
\[\hbar = \frac{h}{2\pi} = 1.0545718 \times 10^{-34} \text{ Js}\]

The reduced Planck’s constant frequently appears in quantum mechanics formulas, such as the Energy-Time Uncertainty Principle, which we use to determine the energy uncertainty.

• Provides a way to quantify the uncertainty and behavior at quantum scales.
• Plays a crucial role in the Heisenberg Uncertainty Principle, linking inseparability of certain pairs of physical properties like position and momentum, and energy and time.
• In our exercise, using \(\hbar\) enables us to calculate the minimum energy uncertainty of the \(\Sigma^0\) particle.

uncertainty in energy

Uncertainty in energy, symbolized as \(\Delta E\), is a key concept derived from the Energy-Time Uncertainty Principle, which sets the limits on the precision with which we can know complementary variables. In our case, it's the energy and the particle's time of existence.
The Energy-Time Uncertainty Principle is given by:
\[\Delta E \cdot \Delta t \geq \frac{\hbar}{2}\]

This principle implies that there is a trade-off between how precisely we can know a particle’s energy and how precisely we understand its lifetime. If a particle lives a very short time (like the \(\Sigma^0\)), its energy becomes very uncertain.

• Here, knowing the mean life offers a way to estimate this uncertainty.
• Using the principle, we rearranged to find uncertainty as \(\Delta E \geq \frac{\hbar}{2\Delta t}\).
• This uncertainty plays a crucial role in explaining phenomena at quantum levels, including particle decay and stability.

conversion from Joules to MeV

Converting energy from Joules to Mega-electronvolts (MeV) is essential in particle physics because it's the preferred unit of energy. MeV allows scientists to express extremely small energy values more conveniently, given that particles at atomic and subatomic levels utilize such energies.

The conversion is straightforward using the factor:
\[1 \text{ eV} = 1.602176634 \times 10^{-19} \text{ J}\]
Therefore,
\[1 \text{ MeV} = 1.602176634 \times 10^{-13} \text{ J}\]
To convert Joules to MeV, divide the energy in Joules by the conversion factor:
\[\Delta E_{\text{MeV}} = \frac{\Delta E_{\text{J}}}{1.602176634 \times 10^{-13}}\]

For the \(\Sigma^0\) particle, after calculating the energy uncertainty in Joules, using this conversion provides the energy in the more user-friendly MeV unit. This aids in understanding and comparing with known particle data in scientific literature.

• Essential for expressing particle physics energy levels.
• Utilized in our problem to make the energy uncertainty relatable and comparable to known physics constants and measurements.