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Chapter 6: Problem 48

Among the elementary subatomic particles of physics is the muon, which decays within a few microseconds 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

Expert verified
The de Broglie wavelength associated with a muon traveling at \(8.85 \times 10^{5} \mathrm{cm} / \mathrm{s}\) is calculated as \(λ = \dfrac{6.626 \times 10^{-27}}{206.8 \times (9.11 \times 10^{-28}) \times (8.85 \times 10^5)} \) cm.

Step by step solution

01

Calculate the mass of the muon

Given that the muon has a rest mass 206.8 times that of an electron, we will calculate the mass of the muon by multiplying the mass of an electron by this factor. Mass of electron, \(m_e = 9.11 \times 10^{-28}\) g Mass of muon, \(m_\mu = 206.8 \times m_e = 206.8 \times (9.11 \times 10^{-28})\) g
02

Calculate the momentum of the muon

Using the formula for momentum \(p = mv\) and the given velocity \(v = 8.85 \times 10^5 \) cm/s, we will calculate the momentum: \(p = m_\mu v = 206.8 \times (9.11 \times 10^{-28}) \times (8.85 \times 10^5) \) g cm/s
03

Calculate the de Broglie wavelength

Now that we have the momentum of the muon, we can use the de Broglie wavelength formula to calculate the wavelength: \(λ = \dfrac{h}{p}\), Where Planck's constant, \(h = 6.626 \times 10^{-27} \) erg s. \(λ = \dfrac{6.626 \times 10^{-27}}{206.8 \times (9.11 \times 10^{-28}) \times (8.85 \times 10^5)} \) cm Now that we have all the values plugged in, we can compute the de Broglie wavelength of the muon traveling at \(8.85 \times 10^5 \mathrm{cm} / \mathrm{s} \).

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Most popular questions from this chapter

Determine which of the following statements are false and correct them. (a) The frequency of radiation increases as the wavelength increases. (b) Electromagnetic radiation travels through a vacuum at a constant speed, regardless of wavelength. (c) Infrared light has higher frequencies than visible light. (d) The glow from a fireplace, the energy within a microwave oven, and a foghorn blast are all forms of electromagnetic radiation.

Consider a fictitious one-dimensional system with one electron. The wave function for the electron, drawn below, is \(\psi(x)=\sin x\) from \(x=0\) to \(x=2 \pi\) . (a) Sketch the probability density, \(\psi^{2}(x),\) from \(x=0\) to \(x=2 \pi .(\mathbf{b})\) At value or values of \(x\) will there be the greatest probability of finding the electron? (c) What is the probability that the electron will be found at \(x=\pi ?\) What is such a point in a wave function called? [Section 6.5\(]\)

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