The wave function of a free electron, meaning an electron not subject to any external forces or interactions, is a plane wave. Mathematically, it is represented as:
ψ(r, t) = Ae^(i(k·r - ωt))
In this equation, ψ represents the wave function, r is the position vector, t is time, A is a normalization constant, k is the wave vector, and ω is the angular frequency.
The wave vector k is related to the momentum of the electron through the de Broglie relation: k = (2π/λ)p, where λ is the wavelength and p is the momentum. For a free electron, the momentum can be described by its magnitude and direction, so k specifies both the direction and the magnitude of the electron's momentum.
The angular frequency ω is related to the energy of the electron through the equation ω = E/ħ, where E is the energy and ħ is the reduced Planck's constant.
The wave function of a free electron is a complex exponential that oscillates in space and time. The exponential term contains the wave vector and the position vector, determining the spatial distribution of the wave function, while the time dependence is given by the term e^(-iωt).
It's important to note that this wave function describes a free electron in the context of non-relativistic quantum mechanics. In relativistic quantum mechanics or quantum field theory, the treatment of electrons becomes more complex, involving spin and additional mathematical formalisms.