Find the initial wavelength of a photon that loses half its energy when it Compton-scatters from an electron and emerges at \(90^{\circ}\) to its initial direction.

The Compton Shift is fundamental to understanding how light interacts with matter at a microscopic level. During Compton Scattering, a photon – a quantum of light – strikes an electron and bounces off in a different direction. This interaction causes a change in both the photon's direction and its energy, which corresponds to a change in wavelength known as the Compton Shift. The amount of this shift can be quantified using the equation:

\[ \Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta) \]

Here, \( \Delta \lambda \) represents the Compton Shift, while \( \lambda \) and \( \lambda' \) denote the initial and final wavelengths of the photon, respectively. The constants are Planck's constant (\( h \)), the mass of the electron (\( m_e \)), and the speed of light (\( c \)). The angle \( \theta \) is the angle at which the photon scatters relative to its original path, and the cosine function reflects the directional change's influence on the shift.

In our exercise, the photon loses half its energy and scatters at a right angle (\(90^\circ\)), a specific case that provides insight into how energy transfer during scattering affects the photon's wavelength.

Photon energy is a critical concept in quantum mechanics that describes the amount of energy carried by a photon. The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength. This relationship is captured by the equation:

\[ E = \frac{h c}{\lambda} \]

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 the case with Compton Scattering, energy is transferred to the electron, leading to a decrease in the photon's energy and an increase in its wavelength.

In the context of the exercise at hand, the photon in question loses half its energy upon scattering. As the energy and wavelength are inversely related, this loss of energy means the photon's wavelength after the encounter is doubled when compared to its initial wavelength. Understanding this inverse relationship is key to solving problems involving energy and wavelength changes due to photon interactions.

Calculating the wavelength of a photon before or after a scattering event requires understanding the relationship between energy, frequency, and wavelength, as well as the specific details of the interaction. The initial exercise involves calculating the initial wavelength of a photon that loses half its energy. The final wavelength is double the initial one because the energy of the photon is halved as a result of the scattering.

By using the Compton Shift formula with given values and the knowledge that the scattering angle is \(90^\circ\), we can determine the initial wavelength with the calculation:

\[ \lambda = \Delta \lambda = \frac{h}{m_e c} \]

The simplicity of the calculation in this case arises from the complete transfer of half the photon's energy to the electron at a right angle, eliminating the need for complex trigonometric evaluations. Plugging in the values for Planck's constant, the mass of the electron, and the speed of light allows us to arrive at the numerical value for the initial wavelength. Through this exercise, students can grasp how scattering parameters affect the photon's properties, particularly its wavelength, and harness equations to predict outcomes quantitatively.