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Chapter 27: Problem 56

A positron and an electron that were at rest suddenly vanish and two photons of identical frequency appear. What is the wavelength of each of these photons?

Short Answer

Expert verified
Answer: The wavelength of each photon can be calculated using the formula 位 = \frac{4h}{(mp + me)c}, where h is the Planck constant, mp is the mass of the positron, me is the mass of the electron, and c is the speed of light.

Step by step solution

01

Understanding the given information

A positron (mass mp) and an electron (mass me) at rest suddenly vanish, and two photons of equal frequency appear. Since the particles are at rest, their total energy and momentum are conserved. We are to find the wavelengths of the two photons.
02

Conservation of energy and momentum

Since energy and momentum are conserved, the total initial energy (Ein) and momentum (Pin) equal the total final energy (Eout) and momentum (Pout). Ein = Eout and Pin = Pout. Before vanishing, the combined energy of the positron and electron at rest is given by the sum of their rest mass energies: Ein = (mp + me)c^2.
03

Energy and momentum of photons

After the particles vanish, the two photons move in opposite directions. Let's denote the energy of one photon as E and its momentum as p. Since the two photons are identical and moving in opposite directions, the total energy will be Eout = 2E. The total momentum is given by Pout = 2p (in opposite directions, so they cancel when calculating the total momentum).
04

Planck's relation

We use the Planck's relation to find the energy of one photon: E = h * f , where h is the Planck constant and f is the frequency of the photon. We also know that the speed of light, c, is equal to the product of frequency and wavelength: c=f位 . Solving for the frequency, we get: f = c/位. Plugging this into the formula for energy, we get: E = h * (c/位).
05

Finding the wavelength of the photons

Now we have the equation for Ein (Ein = (mp + me)c^2) and Eout (Eout = 2h * (c/位)). Setting Ein equal to Eout, we get: (mp + me)c^2 = 2h * (c/位). Solving for 位, we obtain the wavelength of each photon as: 位 = \frac{4h}{(mp + me)c}. Now we have a formula to find the wavelength of the photons given their mass and the speed of light.

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