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 $8.85\times {10}^5\mathrm{cm}/\mathrm{s}$.

Short answer

The de Broglie wavelength associated with a muon traveling at $8.85\times {10}^5\mathrm{cm}/\mathrm{s}$ is approximately $4.00\times {10}^{-8}\mathrm{cm}$.

Step-by-step solution

Calculate the mass of the muon

We are given that the mass of the muon is $206.8$ times that of an electron. The mass of an electron is approximately $9.11\times {10}^{-28}\mathrm{g}$. To calculate the mass of the muon, multiply the mass of the electron by the factor: $$m_{\text{muon}}=206.8\times m_{\text{electron}}=206.8\times (9.11\times {10}^{-28}\mathrm{g})\approx 1.88\times {10}^{-25}\mathrm{g}$$

Calculate the momentum of the muon

Now that we have the muon's mass, we can calculate its momentum using the formula $p=m\cdot v$. We are given the muon's speed, $v=8.85\times {10}^5\mathrm{cm}/\mathrm{s}$: $$p=m_{\text{muon}}\cdot v=(1.88\times {10}^{-25}\mathrm{g})\cdot (8.85\times {10}^5\mathrm{cm}/\mathrm{s})\approx 1.66\times {10}^{-19}\mathrm{g}\cdot \mathrm{cm}/\mathrm{s}$$

Calculate the de Broglie wavelength

Finally, we can use the de Broglie wavelength formula to find the wavelength associated with the muon. The Planck constant is approximately $6.63\times {10}^{-27}\mathrm{erg}\cdot \mathrm{s}$: $$\lambda =\frac{h}{p}=\frac{6.63\times {10}^{-27}\mathrm{erg}\cdot \mathrm{s}}{1.66\times {10}^{-19}\mathrm{g}\cdot \mathrm{cm}/\mathrm{s}}\approx 4.00\times {10}^{-8}\mathrm{cm}$$ The de Broglie wavelength associated with a muon traveling at $8.85\times {10}^5\mathrm{cm}/\mathrm{s}$ is approximately $4.00\times {10}^{-8}\mathrm{cm}$.

Key concepts

Subatomic Particles

Subatomic particles are the fundamental constituents of matter, smaller than an atom and generally not divisible into smaller units. They include protons, neutrons, electrons, and a variety of particles under the category of leptons and quarks. In our exercise, the focus is on the muon, a type of lepton. Unlike the more familiar electron, muons are much more massive, as the calculation indicates they are over 200 times the mass of an electron. Understanding these particles is crucial in the world of particle physics, as their properties, interactions, and behaviors are fundamental to the laws governing the universe.

Muons, similar to electrons, have a negative electric charge, but they possess a greater mass and are unstable, leading to their decay into other particles. Their unique characteristics make muons incredibly useful in experimental physics, including applications like muon tomography, which can be used to study the internal structure of large objects such as volcanoes or pyramids.

Muon Decay

Muons are not stable particles and typically decay into a set of other particles, including electrons and neutrinos. This decay process is governed by the weak nuclear force, one of the four fundamental forces of nature. Specifically, a muon will decay into an electron, a muon neutrino, and an electron antineutrino.

Understanding muon decay is essential for several reasons. Firstly, it helps physicists to test the principles of conservation, such as that of charge and lepton number. Secondly, it provides insight into the behavior of weak interactions, which are responsible for processes that occur in the sun and other stars. Finally, it has practical implications in areas like particle detectors, where muons are used to validate experimental setups and calibrate instruments.

Relativistic Mass

The concept of relativistic mass arises from Einstein's theory of relativity, which suggests that an object's mass increases as it approaches the speed of light. This should not be confused with the object's 'rest mass', which is the inherent mass of a stationary particle. For subatomic particles that move at high velocities, such as our muon example, relativistic effects must be considered when calculating kinetic energies and dynamics.

The exercise provided does not require a relativistic consideration due to the muon's velocity being much less than the speed of light. However, it's important to note that in cases where the velocities are close to that of light, using the relativistic mass is crucial in making accurate calculations related to momentum and energy.

Momentum Calculation

Momentum is a fundamental concept in physics, representing the quantity of motion an object possesses. It's the product of an object's mass and its velocity, described by the equation p=mv. In classical mechanics, this calculation is straightforward, but when dealing with subatomic particles, particularly those moving at high speeds, we sometimes need to take relativistic mechanics into account for greater precision.
For the muon in the exercise, we used its rest mass and given velocity to calculate its classical momentum. This is an essential step before applying the de Broglie hypothesis, which relates the momentum of a particle to its wavelength. In more relativistic scenarios, the formula for momentum would need to include factors of the Lorentz transformation, ensuring that the increased mass at high velocities is appropriately reflected in the particle's momentum.