In quantum mechanics, the wavefunction of a single photon, which is a quantum of electromagnetic radiation, is typically described by a quantum state known as a photon wavepacket. The wavepacket represents the probability distribution of finding the photon at different positions and times.
The mathematical equation that describes the wavefunction of a single photon depends on the specific context and the properties of the photon. However, a common and simplified representation of a photon wavepacket can be described by a Gaussian wavefunction.
In one dimension, the Gaussian wavefunction for a single photon can be written as:
ψ(x, t) = A * exp[-((x - x₀)² / (4σ²)) + (iωt)]
Here, ψ(x, t) represents the wavefunction of the photon as a function of position (x) and time (t). A is the amplitude of the wavefunction, x₀ represents the central position of the wavepacket, σ is the width of the wavepacket (related to the spatial extent), ω is the angular frequency, and i is the imaginary unit (√(-1)).
It's important to note that this equation provides a simplified description and does not capture all the intricacies of the behavior of a photon, especially in more complex scenarios. The complete description of a photon in quantum mechanics requires the use of quantum field theory and the concept of photon states in the quantum field.
Additionally, it's worth mentioning that the wavefunction of a photon is typically associated with its spatial properties rather than its momentum, as photons are massless particles and have a fixed speed (the speed of light) in vacuum.