Ch 44: Particle Physics and Cosmology

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경

Young & Freedman Calc 14th Edition

Ch 44: Particle Physics and Cosmology

문제 4a

44장, 문제 4a

A proton and an antiproton annihilate, producing two photons. Find the energy, frequency, and wavelength of each photon if the $p$ and $\overline{p}$ are initially at rest.

검증된 단계별 안내

Start by understanding the annihilation process: When a proton and an antiproton annihilate, their entire rest energy is converted into the energy of the two photons. The rest energy of a particle is given by Einstein's equation:

E_{rest} = mc

², where

m

is the mass of the particle and

c

is the speed of light.

Calculate the total rest energy of the proton and antiproton system. Since the proton and antiproton have the same mass, the total rest energy is:

E_{total} = 2mc

², where

m

is the mass of a proton.

Since the proton and antiproton are initially at rest, their total momentum is zero. By conservation of momentum, the two photons produced must have equal and opposite momenta. This ensures that the system's total momentum remains zero. Therefore, the energy of each photon is equal to half of the total rest energy:

E_{photon} = mc

².

Relate the energy of each photon to its frequency using the Planck-Einstein relation:

E_{photon} = hν

, where

h

is Planck's constant and

ν

is the frequency of the photon. Solve for the frequency:

ν = E_{photon} / h

Relate the frequency of each photon to its wavelength using the wave equation:

c = λν

, where

λ

is the wavelength and

c

is the speed of light. Solve for the wavelength:

λ = c / ν

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주요 개념

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Mass-Energy Equivalence

Mass-energy equivalence, expressed by Einstein's equation E=mc², states that mass can be converted into energy and vice versa. In the context of particle annihilation, the rest mass of the proton and antiproton is converted into energy, which is then emitted as photons. This principle is fundamental for calculating the total energy produced in the annihilation process.

Photon Properties

Photons are massless particles of light that carry energy and momentum. The energy of a photon is directly related to its frequency (E=hf) and inversely related to its wavelength (λ=c/f), where h is Planck's constant and c is the speed of light. Understanding these relationships is crucial for determining the energy, frequency, and wavelength of the photons produced in the annihilation.

Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the case of proton-antiproton annihilation, the total energy before the event (the rest mass energy of the particles) equals the total energy after the event (the energy of the resulting photons). This principle allows us to calculate the energy of each photon produced in the annihilation.