The Schroedinger equation then becomes iħ∂Ψ(

Plane waves Ψ(

We have

∂Ψ(

ħωΨ(

using the de Broglie relations λ = h/p and f = E/h.

A plane wave represents a particle whose probability of presence is constant
throughout space.

|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 }a_{k }exp(i(**k∙r **- ω_{k}t)) will be a solution as long as for
each **k** we have ħω_{k }= ħ^{2}k^{2}/(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 **- ω_{k}t))
d^{3}k, d^{3}k = dk_{x}dk_{y}dk_{z}.

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.