6.3 The compton effect

Two of Einstein’s influential ideas introduced in 1905 were the theory of special relativity and the concept of a light quantum, which we now call a photon. Beyond 1905, Einstein went further to suggest that freely propagating electromagnetic waves consisted of photons that are particles of light in the same sense that electrons or other massive particles are particles of matter. A beam of monochromatic light of wavelength $\lambda$ (or equivalently, of frequency f ) can be seen either as a classical wave or as a collection of photons that travel in a vacuum with one speed, c (the speed of light), and all carrying the same energy, $E_f=hf.$ This idea proved useful for explaining the interactions of light with particles of matter.

Momentum of a photon

Unlike a particle of matter that is characterized by its rest mass $m_0,$ a photon is massless. In a vacuum, unlike a particle of matter that may vary its speed but cannot reach the speed of light, a photon travels at only one speed, which is exactly the speed of light. From the point of view of Newtonian classical mechanics, these two characteristics imply that a photon should not exist at all. For example, how can we find the linear momentum or kinetic energy of a body whose mass is zero? This apparent paradox vanishes if we describe a photon as a relativistic particle. According to the theory of special relativity, any particle in nature obeys the relativistic energy equation

This relation can also be applied to a photon. In [link] , E is the total energy of a particle, p is its linear momentum, and $m_0$ is its rest mass. For a photon, we simply set $m_0=0$ in this equation. This leads to the expression for the momentum $p_f$ of a photon

Here the photon’s energy $E_f$ is the same as that of a light quantum of frequency f , which we introduced to explain the photoelectric effect:

The wave relation that connects frequency f with wavelength $\lambda$ and speed c also holds for photons:

Therefore, a photon can be equivalently characterized by either its energy and wavelength, or its frequency and momentum. [link] and [link] can be combined into the explicit relation between a photon’s momentum and its wavelength:

Notice that this equation gives us only the magnitude of the photon’s momentum and contains no information about the direction in which the photon is moving. To include the direction, it is customary to write the photon’s momentum as a vector:

In [link] , $\hslash =h\text{/}2\pi$ is the reduced Planck’s constant    (pronounced “h-bar”), which is just Planck’s constant divided by the factor $2\pi .$ Vector $\overrightarrow{k}$ is called the “wave vector” or propagation vector (the direction in which a photon is moving). The propagation vector    shows the direction of the photon’s linear momentum vector. The magnitude of the wave vector is $k=\left|\overrightarrow{k}\right|=2\pi \text{/}\lambda$ and is called the wave number    . Notice that this equation does not introduce any new physics. We can verify that the magnitude of the vector in [link] is the same as that given by [link] .