Free particle solutions to the Schroedinger equation
A plane wave represents a particle whose probability of presence is constant
|Ψ(r,t)|2 = |A|2 = constant.
But a plane wave is not square-integrable, it is not a proper solution.
The Schroedinger equation is a linear equation, the principle of superposition applies. A linear combination of plane wave solutions is also a solution.
Ψ(r,t) = ∑k ak exp(i(k∙r - ωkt)) will be a solution as long as for each k we have ħωk = ħ2k2/(2m).
Since k is a continuous variable, the most general solution is not a sum, but an integral.
Ψ(r,t) = (2π)-3/2∫g(k) exp(i(k∙r - ωkt)) d3k, d3k = dkxdkydkz.
Such a wave function is called a three-dimensional wave packet and can represent any non-pathological square-integrable function.
(g(k) can be complex: g(k) = |g(k)|exp(iα(k)) ; α(k) changes the phase of the plane wave.)
Proper wave functions of free particles are wave packets.