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Chapter 21: Problem 9

(a) Assume that a force is carried by a virtual particle of mass \(m\). Assume that this particle can exist for a time \(h / 2 \pi m c^{2},\) and that it travels close to \(c .\) What is the approximate range of the force? (b) If the range of the strong force is \(10^{-13} \mathrm{cm},\) what is the mass of the particle carrying the force? How does it compare with the mass of the pion?

Short Answer

Expert verified
a) \(\approx \frac{h}{2\pi mc}\) b) \( \approx 120 \text{ MeV}/c^2 \), similar to pion mass

Step by step solution

01

- Calculate Time Duration

Use the formula for the time duration a virtual particle can exist: \[ t = \frac{h}{2\pi m c^2} \]This represents the time the virtual particle can exist with mass \(m\).
02

- Calculate the Distance Traveled

The approximate range of the force can be computed using the formula: \[ d = \frac{h}{2\pi m c} \]
03

- Given the Range of the Strong Force

Assume the range of the strong force is given by: \[ d = 10^{-13} \text{ cm} = 10^{-15} \text{ m} \]Using the formula from Step 2, solve for the mass \(m\): \[ 10^{-15} = \frac{h}{2\pi m c} \]Rearrange to find: \[ m = \frac{h}{2\pi d c} \]
04

- Calculate the Mass of the Particle

Plug in the values for constants to find the mass: \[ h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s} \]\[ c = 3 \times 10^8 \text{ m/s} \]\[ d = 10^{-15} \text{ m} \]Calculate: \[ m = \frac{6.626 \times 10^{-34}}{2\pi \times 10^{-15} \times 3 \times 10^8} \text{ kg} \]Simplify: \[ m = 1.1 \times 10^{-28} \text{ kg} \]
05

- Convert Mass to MeV/c² and Compare with Pion

Convert the mass from kg to \( \text{MeV}/c^2 \):\[ 1 \text{ kg} = 5.61 \times 10^{29} \text{ MeV}/c^2 \]\[ m = 1.1 \times 10^{-28} \times 5.61 \times 10^{29} = 120 \text{ MeV}/c^2 \]Compare this with the mass of the pion: \( 139.6 \text{ MeV}/c^2 \).The mass of the virtual particle is comparable to the mass of a pion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantum Mechanics
Quantum mechanics is a branch of physics that deals with the behavior of particles at extremely small scales. It describes how particles like electrons, protons, and even photons behave and interact. One of its essential principles is the uncertainty principle, which states that we cannot simultaneously know the precise position and momentum of a particle. This field also introduces the concept of wave-particle duality, where particles can exhibit both wave-like and particle-like properties. In essence, quantum mechanics provides a framework for understanding the probabilistic nature of the microscopic world.
Virtual Particles
Virtual particles are a fascinating aspect of particle physics. Unlike real particles, virtual particles cannot be directly observed. They exist temporarily and are used to explain the interactions between other particles in quantum field theory. These particles can 'borrow' energy from their surroundings, with the condition that they exist for a very short time. For example, in the context of the strong nuclear force, virtual particles like gluons and pions mediate the interactions between nucleons (protons and neutrons).
Virtual particles help to visualize how forces are transmitted at the quantum level, although they remain elusive in direct measurements.
Strong Nuclear Force
The strong nuclear force is one of the four fundamental forces of nature. It is responsible for holding the nucleus of an atom together, binding protons and neutrons tightly. This force is incredibly powerful but acts over very short distances, approximately the size of a nucleus. The strong force is mediated by particles called gluons, and in certain contexts, pions can also play a role. Without this force, atomic nuclei would not be stable, and matter as we know it would not exist.
The range of the strong force is around 10^{-15} meters (or 10^{-13} cm), which correlates to the distances over which virtual particles like pions can effectively mediate the force.
Pion Mass
Pions are subatomic particles that play an integral role in mediating the strong nuclear force. Specifically, pions are mesons, and they come in three varieties: positive, negative, and neutral. The mass of a pion is approximately 139.6 MeV/c². In the calculation provided, the virtual particle responsible for transmitting the strong force is compared to a pion. The calculated mass of about 120 MeV/c² is close to the pion's mass, suggesting a similar role in mediating the force. This comparison emphasizes the importance of pions in nuclear interactions and helps to understand the characteristics of these fundamental particles.
Particle Physics
Particle physics is the study of the fundamental particles of the universe and the forces they interact with. This field explores the smallest known building blocks of matter, such as quarks, leptons, and bosons. It also investigates how these particles interact through fundamental forces: gravitational, electromagnetic, weak, and strong forces. Particle accelerators are essential tools in this field, as they smash particles together at high speeds to observe the results of these interactions. Through particle physics, we uncover the underlying principles that govern the universe at the tiniest scales, leading to a deeper understanding of the fabric of reality.

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Most popular questions from this chapter

Assume that the energy density, integrated over all wavelengths, in the background radiation is given by the Stefan-Boltzmann law. Assume that we know the cosmological redshift, and show that these lead to the conclusion that the temperature must scale as \(1 / R\).

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(a) Suppose the scale factors at times \(t_{1}\) and \(t_{2}\) are \(R_{1}\) and \(R_{2},\) and the background radiation temperatures are \(T_{1}\) and \(T_{2}\). What is the relationship between \(T_{1}\) and \(T_{2}\) in terms of \(R_{1}\) and \(R_{2} ?(b)\) If we know that the back ground temperature now is \(3 \mathrm{K}\), and we let \(R=1,\) write an expression for the back ground temperature at any time when the scale factor is \(\mathrm{R}(t)\).

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If the electromagnetic force is carried by virtual photons, which can live for a time \(h / E\) explain how the first can be felt at any range. (Hint: Think about photons of different wavelength and energy.
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