Question

Among the elementary subatomic particles of physics is the muon, which decays within a few nanoseconds after formation. The muon has a rest mass 206.8 times that of an electron. Calculate the de Broglie wavelength associated with a muon traveling at a velocity of \(8.85 \times 10^{5} \mathrm{~cm} / \mathrm{s}\)

Short answer

The de Broglie wavelength associated with a muon traveling at a velocity of \( 8.85 \times 10^{5} \mathrm{~cm} / \mathrm{s} \) is approximately \( 2.05 \times 10^{-11} \mathrm{cm} \).

Step-by-step solution

Calculate the muon's mass

First, we need to determine the actual mass of the muon. We're given the information that its rest mass is 206.8 times the mass of an electron. We can find the mass of the muon (m) by multiplying this factor with the mass of an electron (me), which is approximately \( 9.11 \times 10^{-28} \mathrm{g} \): \[ m_\mu = 206.8 \times m_e = 206.8 \times (9.11 \times 10^{-28} \mathrm{g}) \]

Calculate the muon's momentum

Now that we have the mass of the muon, we can calculate its momentum by multiplying its mass with its given velocity: \[ p_\mu = m_\mu \cdot v_\mu = (206.8 \times (9.11 \times 10^{-28} \mathrm{g})) \times (8.85 \times 10^{5} \mathrm{~cm} / \mathrm{s}) \]

Calculate the de Broglie wavelength of the muon

With the momentum calculated, we can now find the de Broglie wavelength using the formula mentioned earlier: \[ \lambda_\mu = \frac{h}{p_\mu} \] where h is the Planck's constant (\( 6.63 \times 10^{-27} \mathrm{~erg} \cdot \mathrm{s}\)) \[ \lambda_\mu = \frac{6.63 \times 10^{-27} \mathrm{~erg} \cdot \mathrm{s}}{ (206.8 \times (9.11 \times 10^{-28} \mathrm{g})) \times (8.85 \times 10^{5} \mathrm{~cm} / \mathrm{s})} \] Now, simply solve for the de Broglie wavelength: \[ \lambda_\mu \approx 2.05 \times 10^{-11} \mathrm{cm} \] So, the de Broglie wavelength associated with a muon traveling at a velocity of \( 8.85 \times 10^{5} \mathrm{~cm} / \mathrm{s} \) is approximately \( 2.05 \times 10^{-11} \mathrm{cm} \).

Key concepts

Muon

The muon is a fascinating elementary particle belonging to the lepton family, similar to the electron but much heavier. It plays a crucial role in particle physics experiments and naturally occurs in cosmic rays. A muon has a rest mass that is about 206.8 times that of an electron, making it significantly more massive. Due to this mass difference, muons can behave differently compared to electrons, especially when traveling at high speeds. Importantly, muons are unstable. They tend to decay into electrons and neutrinos within a few microseconds after their formation. Despite their short lifespan, their presence and behavior offer valuable insights into the fundamental forces in the universe.

Momentum

Momentum is a fundamental concept in physics related to an object's mass and velocity. It defines the quantity of motion an object possesses. The momentum (\( p \)) of an object is found using the formula:

• \( p = m \times v \)

where \( m \) is the object's mass and \( v \) is its velocity.
For subatomic particles like muons, knowing their momentum is crucial, particularly when calculating properties like the de Broglie wavelength. Higher momentum usually indicates a more pronounced particle-like characteristic, affecting interactions and trajectories in experiments.

Subatomic Particles

Subatomic particles are smaller than atoms and are the building blocks of matter. They include particles like electrons, protons, neutrons, and exotic particles such as muons. These particles are crucial to understanding the composition and behavior of matter at the smallest scale.
The study of subatomic particles is integral to fields like quantum mechanics and particle physics. Each particle has unique characteristics, such as mass, charge, and spin, influencing their interactions. For instance, muons are part of the lepton family and behave differently from protons, which belong to the baryon family due to their distinct properties.

Planck's Constant

Planck's constant (\( h \)) is a fundamental quantity in physics that relates energy and frequency. Its value is approximately:

• \( 6.63 \times 10^{-27} \mathrm{~erg \cdot s} \)

This constant is central to understanding quantum mechanics and is pivotal in calculations involving the de Broglie wavelength. The de Broglie wavelength is found using the formula:

• \( \lambda = \frac{h}{p} \)

where \( p \) is momentum.
Planck's constant ensures that we can determine the wave-like nature of particles. It is a bridge between the particle and quantum worlds, helping to predict how particles behave at different scales.