Find the kinetic energy of an initially stationary electron after a 0.10 -nm \(\mathrm{X}\) -ray photon scatters from it at \(90^{\circ}\)

Kinetic energy is the energy that an object possesses by virtue of its motion. In physics, the kinetic energy of a particle, like an electron, can be calculated once we know its mass and velocity. The relationship is given by the formula:
\[ KE = \frac{1}{2}mv^2 \]
where \(KE\) represents kinetic energy, \(m\) is the mass of the electron, and \(v\) is its velocity. In the context of Compton scattering, an initially stationary electron gains kinetic energy after being hit by a photon. The photon's energy is partly transferred to the electron, propelling it into motion. The change in energy of the photon is what gives rise to the kinetic energy of the electron, thus fulfilling the law of conservation of energy.
To find the kinetic energy imparted to the electron in Compton scattering, we calculate the energy difference between the incident and scattered photon and assume that this difference is equal to the kinetic energy gained by the electron.

The energy of a photon, the elementary particle of light, is directly correlated with its electromagnetic wave frequency (or inversely with its wavelength). This energy can be calculated using Planck's relation:
\[ E = \frac{hc}{\lambda} \]
where \(E\) is the energy of the photon, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength of the photon. When a photon scatters off an electron, as in Compton scattering, its energy changes because the scattering process affects its wavelength—increasing it if the photon loses energy and decreasing it if it gains energy. In the steps to solving the original exercise, the energy of the photon before and after scattering is critical to determining the kinetic energy transferred to the electron.

The electron mass, denoted by \(m_e\), is a fundamental physical constant representing the rest mass of an electron. It is a key parameter in calculating Compton scattering effects. The rest mass of an electron is essential because it remains constant regardless of the electron's energy state. The Compton scattering formula incorporates the electron mass to find the change in wavelength of the photon that scatters from the electron. The change in wavelength (and hence the energy) of the photon is inversely proportional to the mass of the electron; a more massive object would exhibit a smaller change in wavelength for the same photon energy. This explains why electrons, with their relatively small mass, are significantly affected by the energy of photons like X-rays.

Planck's constant, symbolized as \(h\), is a fundamental quantity in quantum mechanics that has units of energy multiplied by time (often joules times seconds). It relates the energy of a photon to the frequency of its electromagnetic wave. Planck’s constant is a cornerstone in the theory of quantum mechanics and is particularly important in quantifying the interactions between electromagnetic radiation and matter.
In Compton scattering, Planck's constant is used to calculate the change in wavelength of the photon after the scattering process. It appears in the Compton scattering formula and is vital in determining the energies of both the initial and scattered photons. Since the value of Planck's constant is very small, the effects described by it, such as Compton scattering, are most notable at the microscopic level, like the interactions between photons and electrons.