In quantum mechanics, the wave function describes the quantum state of a particle. For a free particle in a void, where there are no external forces or potential energies acting on it, the wave function can be modeled as a plane wave.
The general form of a plane wave is given by:
ψ(x, t) = A * exp[i(kx - ωt)]
Where:
- ψ(x, t) represents the wave function at position x and time t.
- A is the amplitude of the wave.
- k is the wave vector, which determines the spatial frequency of the wave.
- ω is the angular frequency, which relates to the energy of the particle through the equation E = ℏω, where ℏ is the reduced Planck's constant.
For a free particle, the energy is given by the kinetic energy:
E = (p^2)/(2m)
Where:
- p is the momentum of the particle.
- m is the mass of the particle.
The momentum can be related to the wave vector through the de Broglie relation:
p = ℏk
Combining these equations, we can express the angular frequency as:
ω = (ℏk^2)/(2m)
Substituting this value of ω back into the equation for the plane wave, we get:
ψ(x, t) = A * exp[i(kx - (ℏk^2t)/(2m))]
This is the general form of the wave function for a free particle in a void. The modulus squared of the wave function, |ψ(x, t)|^2, gives the probability density of finding the particle at a particular position x at time t.