Prove that a free electron cannot absorb a photon photoelectrically by showing that the conserva

step 1 Hi, students, welcome to the solution of problem number 15. We know that a free electron cannot

Prove that a free electron cannot absorb a photon...

Key Concepts

Photon Energy-Momentum Relation

Photons, being massless particles, have their energy and momentum related directly by the equation E = pc, where E is energy, p is momentum, and c is the speed of light. This relation provides a specific quantitative link between a photon’s energy and momentum, which becomes crucial when testing if these values can be reconciled with those of a massive particle like an electron.

Relativistic Kinematics

Relativistic kinematics involves the use of Einstein's energy-momentum relations for particles with mass, such as electrons. When a free electron attempts to absorb a photon, satisfying both the relativistic energy equation and momentum conservation simultaneously creates an inherent conflict, demonstrating that no reference frame can satisfy both requirements, and thereby proving that the process is forbidden.

Conservation of Energy

This principle asserts that the total energy in a closed system remains constant before and after an interaction. In the context of photon-electron interactions, any energy gained or lost by the electron must exactly match the energy provided or taken by the photon, making energy conservation an essential constraint in analyzing the process.

Conservation of Momentum

Momentum conservation requires that the total momentum of a closed system remains unchanged throughout an interaction. For a process where a photon is absorbed, the momentum carried by the photon must be fully accounted for by the electron's change in momentum, imposing a strict condition that must be simultaneously satisfied with energy conservation.

Transcript

00:01 Hi, students, welcome to the solution of problem number 15.

00:05 We know that a free electron cannot absorb a photon.

00:11 In this problem, we are going to prove that by showing the conservation of energy and conservation of momentum cannot be satisfied simultaneously in such a process.

00:28 Now, to start with, let us suppose that a free electron can absorb a photon.

00:42 Now according to the law of conservation of momentum if a free electron can absorb a photon the initial momentum the total initial momentum should be equal to the total initial the total final momentum which means p1 plus p new equals p2.

01:08 Here p1 and p2 are the initial and final momentums of the electron and p new is the momentum of the photon.

01:25 So let us mark this as our first equation.

01:35 So this is conservation of momentum.

01:43 The conservation of energy.

01:50 According to conservation of energy, the net initial energy should be equal to the net final energy.

02:00 So, the total energy before absorbing the photon should be equal to the total energy after the electron has absorbed the photon if such a process is happening.

02:16 According to law of conservation of energy we get the equation two where this is the initial total energy and this is the final total energy.

02:31 In the initial energy there are two terms because there are two particles then.

02:36 There is a photon.

02:38 There is an electron...

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