When an electron and a positron interact, they can annihilate each other, resulting in the emission of one or more photons. The behavior of the emitted photons is described by quantum electrodynamics (QED), which is a quantum field theory that incorporates both quantum mechanics and special relativity.
In the context of QED, virtual particles such as virtual photons are mathematical entities that represent the exchange of energy and momentum during interactions. They do not have the same properties as real, observable particles. The virtual photon is said to mediate the electromagnetic interaction between the electron and positron.
The wave function of the virtual photon is not directly affected by the wave function of the positron or the electron. Instead, the wave function of the entire system, including the electron, positron, and the virtual photon, is described by a joint wave function that takes into account the quantum state of all the particles involved. The combined wave function describes the probabilities and amplitudes of the various possible outcomes of the interaction, including the emission of photons.
Regarding the speed of the wave function, it is important to note that the wave function itself is not a physical entity but rather a mathematical description of a quantum system. It does not have a physical location or travel through space. However, the collapse or change of the wave function can propagate at a maximum speed limited by the speed of light (c). This means that the effects of an interaction, including the detection or measurement of a particle, can only be observed after a finite time interval dictated by the distance between the interacting particles.
In summary, the wave function of a virtual photon emitted by an electron is influenced by the joint wave function of the entire system, including the positron and other particles involved. The wave function itself does not have a speed or physical location, but the effects of the wave function's collapse or change propagate at a maximum speed limited by the speed of light.