The photoelectric effect, discovered by Heinrich Hertz in 1887, can be mathematically interpreted using the principles of quantum theory. One of the key aspects of the photoelectric effect is the emission of electrons from a material surface when it is exposed to light of a certain frequency or higher.

According to Max Planck, the energy of light is quantized into discrete packets called photons. The energy (E) of a photon can be calculated using the equation E = hf, where h is Planck's constant (approximately 6.626 × 10^-34 Joule-seconds) and f is the frequency of the incident light. In the photoelectric effect, an electron in the material absorbs the energy of a photon and gains enough energy to overcome the binding forces and be emitted from the surface.

In order for electrons to be emitted in the photoelectric effect, the frequency of the incident light must be greater than or equal to a certain threshold frequency (f_threshold), unique to each material. This threshold frequency corresponds to the minimum amount of energy required to liberate an electron from the material. If the frequency of the incident light is lower than f_threshold, no electrons are emitted regardless of the intensity of the light.

Einstein further expanded on the mathematical interpretation of the photoelectric effect by proposing that the kinetic energy (KE) of the emitted electron is given by the equation KE = hf - W, where W is the work function of the material. The work function represents the minimum energy required for an electron to escape the surface and is specific to the material. If the energy of the incident light (hf) is greater than the work function (W), the excess energy is converted into the kinetic energy of the emitted electron.

These mathematical equations provide a quantitative understanding of the photoelectric effect and allow us to predict and analyze the behavior of electrons in different materials. They explain why the photoelectric effect is independent of the intensity of the incident light, but dependent on its frequency and the properties of the material. Moreover, these equations successfully explain the experimental observations and provide a solid foundation for the development of quantum theory.