Question

In the Compton effect, we choose the electron to be at the origin and the initial photon's direction of motion to be in the+x-direction.

(a) We may also choose the xy-plane so that it contains the velocities of the outgoing electron and photon. Why? (b) The incoming photon's wavelengthλis assumed to be known. The unknowns after the collision are the outgoing photon's wavelength and direction,λ′, and θ,and the speed and direction of theelectron,u_e,andϕ.With only three equationstwocomponents of momentum conservation and one of energy, we can't find all four. Equation(3−8)givesλ′in terms ofθ.Our lack of knowledge of θθ after the collision (without an experiment) is directly related to a lack of knowledge of something before the collision. What is it? (imagine the two objects are hard spheres.) (c) Is it reasonable to suppose that we could know this? Explain.

Short answer

a) Restricting the motion in the$xy$plane is necessary, to conserve the linear momentum.

b) The lack of knowledge of the impact parameter is responsible for the lack of knowledge of the angle after the collision$\left({\theta }_{\text{after}}\right)$.

c) The impact parameter can't be determined due to the Heisenberg uncertainty relation.

Step-by-step solution

Given data

The incoming photons and the stationary electrons are considered hard spheres. The angle of approach of the incoming photon upon collision is not inthe positive x-direction.

Concept used

Electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, propagating through space, carrying electromagnetic radiant energy.

Heisenberg’s uncertainty principle:

$\begin{array}{l}\Delta p\Delta x\ge \frac{h}{4\pi }\\ \Delta t\Delta E\ge \frac{h}{4\pi }\end{array}$

where$\Delta$ refers to the uncertainty in that variable and$\text{h}$ is Planck's constant.

Determine the nature of photons and stationary electrons

(a)

Well, we are free to choose the coordinate system to be oriented in an arbitrary direction, however, the particular configuration chosen here simplifies our analysis a bit. If we restrict the initial photon momentum to be in the$+\mathbf{x}$ direction and restrict the outgoing electron to be in the$xy$plane, which we can always do i.e. rotate your coordinate system around the$\mathbf{x}$-axis, that means we are restricting the outgoing photon to be in the$\mathbf{xy}$plane as well.

The previous conclusion is based on the fact that the linear momentum is conserved, hence, if the initial momentum is only in the x-direction, we should expect the final momentum to have a net component in the x-direction as well.

Therefore, the outgoing photon can't have momentum in the $z$ direction otherwise, the total momentum won't be conserved, it's only allowed to have its momentum in the $xy$ plane. Nevertheless, the outgoing photon will have its momentum in such a way as to cancel the electron momentum in the $\mathbf{y}$-direction.

Use the Heisenberg’s Uncertainty Principle

(b)

Although the motion of the incident photon is confined to be in the positive$x$-direction, we still can't figure out the angle of approach$\left({\theta }_{\text{before}}\right)$, in other words, the angle between the direction of motion and the line joining the center of the two spheres. Using two hard spheres to model both the photon and the electron will reduce the problem of the undetermined angle to the missing impact parameter, which refers to the vertical distance between the two centers.

Hence, we could say that the lack of knowledge of the impact parameter is responsible for the lack of knowledge of the angle after the collision $\left({\theta }_{\text{before}}\right)$e.g. for $a>R_1+R_2$ we won't have a collision at all.

Determine the nature of the impact parameter

(c)

Could we know the exact value of the impact parameter? The answer is no, we can't. The reason behind this has its root in the Heisenberg uncertainty principle, where we can't determine the exact position of a quantum particle i.e the photon and the electron with its momentum simultaneously.

Therefore, the hard-sphere model isn't a real model, it's an approximation that we use to simplify our analysis there is always uncertainty associated with their position. However, we need to emphasize two points here,

The uncertainty principle isn't related to the precision of our measurement, it's a fundamental principle of nature, even a measurement with great precision won't be able to monitor the exact position e.g. for a quantum particle.

The uncertainty principle doesn't contradict with de Broglie relation that we have encountered earlier in this chapter. As de Broglie relation is concerned with the average values of the wavelength and the momentum.

However, experimentally, the wavelength measurement is always associated with a certain band that represents the uncertainty in its measurement.