# University of Wisconsin Green Bay

A photon with an initial energy of 14 keV scatters off of a free electron and changes direction by 550. What is the wavelength of the scattered photon? What is the recoil speed of the electron?

• In introduction to quantum mechanics units, you are introduced to a variety of experimental results involving light, atoms, or constituents of atoms. In most cases, when you look at interactions on these tiny scales, you have a foundations of quantum mechanics problem.

In this case, the interaction of interest is that of a photon interacting with a free electron. This is known as the Compton Effect. At root, the Compton Effect is understood through Conservation of Momentum and Conservation of Energy, and so if you recognized this as a momentum problem that shows a good understanding of basic physics applied to a new situation. As always, problems should be initially approached through the core physics concepts regardless of what is requested in the solution.

• A photon interacting with a free electron can be treated as any other collision in which the interacting objects are free to move. Therefore, the picture looks like that for any other two-dimensional collision problem. There is no need, however, to draw a second picture in which the velocities are divided into components.

• Whenever a photon interacts with a free electron, it is asking you to think about the Compton Effect. Your book has probably provided you with this partially-solved version of the Conservation of Momentumand Conservation of Energy equations as they apply to photon-electron scattering. One caution with partially-solved or situation specific equations: Make sure that you understand both when the equation applies and what each symbol represents.

In this problem, you are asked first for the photon wavelength and then for the electron energy. As always, begin with the core physics behind the problem and let that understanding lead you to the answer to secondary questions. You do not need to picture the full solution before you begin.

• The Compton Effect equation directly gives the wavelength of the scattered photon. Scroll down in order to find the final speed of the electron.

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Step 2

In the second question, you are asked to find the speed of the electron after the interaction with the photon. As has been discussed throughout this problem, the Compton Effect reflects a conservation of energy and momentum between the electron and the photon. It is always easier to use energy than momentum if you can, and so I began by tracking what happened to the initial energy of the photon. No further solution in necessary in this problem.

• Once you recognize this problem as the Compton Effect, you can answer the first question mathematically merely by plugging into* the Compton equation. However, to understand the problem you need to recognize that this equation tracks momentum and energy.

Remember that in order to understand the experimental results of the Compton Effect, we treat light as both a particle rather and as a wave. In other words, we understand the interaction of light with the electron as an elastic collision between a photon (light particle) and the electron. In this interaction, the photon gives some of its energy to the electron. In understanding light as a wave, however, we recognize that lower energy light has a longer wavelength (as given by E = hf = hc/λ) than high energy light. This is in agreement with the experimental results that show the scattered light has a longer wavelength than the incident light.

*Note that the energy of a photon is related to its wavelength and so the definition of photon energy was also used in order to relate the given quantities to those used in the equation.