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The neutral pion π0 is an unstable particle (mass m=135 MeV/c2 ) that can decay into two photons, π0γ+γ. (a) If the pion is at rest, what is the energy of each photon? (b) Suppose instead that the pion is traveling along the x axis and that the photons are observed also traveling along the x axis, one forward and one backward. If the first photon has three times the energy of the second, what was the pion's original speed v?

Short Answer

Expert verified
(a) 67.5 MeV per photon. (b) Pion's speed was c2.

Step by step solution

01

Analyze the Decay Process at Rest

For part (a), when the neutral pion π0 is at rest and decays into two photons, the total energy of the pion is its rest mass energy E=mc2, where m=135 MeV/c2. Each photon carries half of this energy since the initial system is symmetric and both photons share equal energy.
02

Calculate Photon Energy at Rest

The energy of each photon is given by Eγ=E2=mc22. Substituting the given mass, we find Eγ=135 MeV2=67.5 MeV. Thus, each photon has an energy of 67.5 MeV.
03

Analyze the Moving Pion Scenario

For part (b), when the pion is moving along the x-axis, let's consider the conservation of energy and momentum. If one photon moves in the forward direction with energy E1 and the other in the opposite direction with energy E2, and E1=3E2, then the initial energy Eπ of the pion equals E1+E2.
04

Set Up Equations for Conservation of Momentum

Similarly, conservation of momentum dictates that p1p2=pπ, where p1 and p2 are the magnitudes of momentum for the photons. Since energy E=pc for photons, we have p1p2=Eπc.
05

Substitute Energies and Solve for Velocity

Given that E1=3E2, substitute E1=3E2 into the energy equation Eπ=E1+E2=4E2 and momentum equation Eπc=3E2cE2c=2E2c. Solving these equations, we find Eπ=4E2 and pπc=2E2, which gives v=pπc2Eπ=2E2c2/c4E2=12c. Thus, the original velocity of the pion was v=c2.

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Key Concepts

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

Photon Energy
When a neutral pion (π0) decays while at rest, it transforms into two photons. The energy of each photon can be found using the principle of energy conservation. Initially, the pion has an energy equal to its rest mass energy, given by E=mc2. For a pion with mass m=135MeV/c2, this energy is 135 \mathrm{MeV}.Since the pion is at rest and decays evenly into two photons, the energy is shared equally between the two photons. Thus, each photon obtains half of the original energy, calculated as:
  • Eγ=mc22=135 MeV2=67.5 MeV
Consequently, each photon has an energy of 67.5 MeV. This simple yet vital calculation demonstrates the symmetry and balanced energy sharing in particle decay when at rest. Understanding energy distribution is crucial in particle physics, especially in scenarios involving conservation laws.
Conservation of Momentum
In scenarios where the pion (π0) is moving, it's key to consider both energy and momentum conservation to understand how the particles interact post-decay.Momentum conservation plays a critical role. For instance, when the pion is originally moving along the x-axis, it decays into two photons, one moving forward and the other backward. The energies of these photons are different, where one photon has three times the energy of the other. Let these energies be E1=3E2 and E2. The total original energy Eπ of the pion is thus:
  • Eπ=E1+E2=4E2
For momentum conservation, since photons have energy E=pc, the relation is:
  • p1p2=Eπc=pπ
These equations capture the entirety of energy and momentum before and after the decay, maintaining the conservation laws integral in physics that hold true in all inertial frames. Recognizing these concepts allows for deeper analysis and understanding of particle interactions and transformations.
Relativistic Velocity Calculation
To find the original velocity of a moving pion (π0) before it decays, we delve into relativistic physics.Given the scenario where the energy of the first photon is three times the energy of the second (E1=3E2), we need to use the relationship between energy and momentum. The equations drawn from energy and momentum conservation provide the foundation for calculating the velocity.The equations are:
  • Eπ=4E2
  • Eπc=2E2c=pπ
By solving for velocity, we equate momentum :
  • v=pπc2Eπ=2E2c2/c4E2=12c
This reveals that the pion was originally moving at half the speed of light v=c2. Relativistic calculations are essential when dealing with high-speed particles, ensuring our theoretical predictions match observed behaviors.

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Most popular questions from this chapter

If one defines a variable mass mvar =γm, then the relativistic momentum p=γmv becomes mvar v which looks more like the classical definition. Show, however, that the relativistic kinetic energy is not equal to 12mvarv2

Show that any two zero-mass particles have a CM frame, provided their three- momenta are not parallel. [Hint: As you should explain, this is equivalent to showing that the sum of two forward light-like vectors is forward time-like, unless the spatial parts are parallel.]

A space explorer A sets off at a steady 0.95c to a distant star. After exploring the star for a short time, he returns at the same speed and gets home after a total absence of 80 years (as measured by earth-bound observers). How long do A 's clocks say that he was gone, and by how much has he aged as compared to his twin B who stayed behind on earth? [Note: This is the famous "twin paradox." It is fairly easy to get the right answer by judicious insertion of a factor of γ in the right place, but to understand it, you need to recognize that it involves three inertial frames: the earth-bound frame S, the frame S of the outbound rocket, and the frame S of the returning rocket. Write down the time dilation formula for the two halves of the journey and then add. Notice that the experiment is not symmetrical between the two twins: A stays at rest in the single inertial frame δ, but B occupies at least two different frames. This is what allows the result to be unsymmetrical.]

Two particles a and b with masses ma=0 and mb>0 approach one another. Prove that they have a CM frame (that is, a frame in which their total three-momentum is zero). [Hint: As you should explain, this is equivalent to showing that the sum of two four-vectors, one of which is forward light-like and one forward time-like, is itself forward time-like.]

A rocket traveling at speed 12c relative to frame S shoots forward bullets traveling at speed 34c relative to the rocket. What is the speed of the bullets relative to S?

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