A massive particle with energy E0 and speed β0 c undergoes a head-on elastic collision with a ph

A massive particle with energy E0 and speed β0 c undergoes a...

A massive particle with energy $E_0$ and speed $\beta_0 c$ undergoes a head-on elastic collision with a photon with energy $E_{\gamma 0}$. Show that the final energy $E_\gamma$ of the photon is $$ E_\gamma=E_0 \frac{1+\beta_0}{2+\left(1-\beta_0\right) \frac{E_0}{E_{\gamma 0}}} . $$ Show that $E_\gamma<E_0$ but if $\beta_0 \rightarrow 1, \frac{E_\gamma}{E_0} \rightarrow 1$. That is, a high-energy particle loses most of its energy to the photon (in the ideal $\beta_0 \rightarrow 1$ limit, the particle retains energy $E_{\gamma 0}$ by the conservation of energy).

A massive particle with energy $E_0$ and speed $\beta_0 c$ undergoes a head-on elastic collision with a photon with energy $E_{\gamma 0}$. Show that the final energy $E_\gamma$ of the photon is
$$
E_\gamma=E_0 \frac{1+\beta_0}{2+\left(1-\beta_0\right) \frac{E_0}{E_{\gamma 0}}} .
$$
Show that $E_\gamma<E_0$ but if $\beta_0 \rightarrow 1, \frac{E_\gamma}{E_0} \rightarrow 1$. That is, a high-energy particle loses most of its energy to the photon (in the ideal $\beta_0 \rightarrow 1$ limit, the particle retains energy $E_{\gamma 0}$ by the conservation of energy).

Key Concepts

Relativistic Limit (? ? 1) Dynamics

In the regime where the velocity of the massive particle approaches the speed of light (? ? 1), relativistic effects become pronounced. This limit is particularly important as it demonstrates scenarios where the massive particle transfers most of its energy to the photon, showcasing the extreme energy redistribution that occurs under relativistic conditions.

Head-On Collision Geometry

Head-on collisions simplify the problem as the motion is confined to a single line, reducing the complexity of the momentum components. This direct opposition in movement allows for a more straightforward application of conservation laws, making it easier to derive relations between initial and final energies of the colliding particles.

Elastic Collision Dynamics

Elastic collisions are interactions in which the total kinetic energy of the system, as well as momentum, remain constant before and after the collision. In the relativistic context, this requires carefully equating the sums of the relativistic energies and momenta of all particles involved, leading to precise relationships such as the one derived for the final photon energy.

Photon Kinematics

Photons, being massless particles, always travel at the speed of light and have energy directly related to their frequency. Their energy-momentum relation is unique (E = pc) and plays a crucial role in collision calculations, especially in elastic scattering scenarios, as it alters the energy distribution when interacting with massive particles.

Relativistic Conservation Laws

In high-energy collisions, especially when the speeds involved are comparable to the speed of light, the traditional conservation laws of energy and momentum must be applied using their relativistic forms. This ensures that quantities such as the total energy (including rest mass energy) and momentum are correctly conserved across different inertial frames, taking into account Lorentz factors.

Transcript

00:01 In this problem, we have an atom at rest and an incoming photon, camera photon, and just a common notation is to draw a photon like this, even though we do consider it a particle, we do our paying homage to its electromagnetic wave nature of the, that it exhibits in different spots, different places.

00:24 It has energy, e .s .p.

00:27 That energy is such that it makes the atom go from, this state to an excited state.

00:36 And that excited state has an equivalent mass of m star.

00:39 And this excited atom will be moving.

00:42 Why? because the photon does have momentum.

00:45 So to conserve momentum, it must move.

00:48 So the first part of the problem wants us to write out the conservation laws.

00:54 So first for energy, we have the energy of the photon, plus the rest energy of the atom.

01:02 And this is equal to the energy, total energy, of the excited atom.

01:09 This has already got the kinetic energy in it.

01:11 This is the total energy.

01:12 I'll call this equation 1.

01:16 Momentum, the momentum of the photon plus 0 is equal to the momentum of the excited atom.

01:28 And this actually is e, i'll justify it in a second.

01:31 E p over c is equal to p m star.

01:35 This i'll call equation 2.

01:37 But this comes from, this comes from this expression that we have for all particle.

01:43 E squared is equal to p squared, c squared, plus m squared, c to the fourth.

01:53 That's for two of any.

01:55 We're doing it for the photon, the photon has zero rest mass.

01:58 So this term is gone.

01:59 So we just get to the momentum of the photon is e p over c.

02:03 That's what we have here.

02:06 So that was part a to write the conservation law.

02:08 So we got equation one, equation two.

02:11 Now let us use the same expression to make we write the momentum in equation 2.

02:18 So, e .p.

02:20 Over c is going to be em -star squared minus m -star squared, c -to -the -fourth over c -squared to the half -power.

02:33 So we just use the exact same expression, exact same basic expression to solve for the momentum of m -star.

02:41 So this time, though, it's not zero rest means.

02:45 And we got a c -squared to the half power in the bottom.

02:48 So this is going to go away.

02:51 And if we square both sides, we get the following...

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