Schrödinger Wave Function Of A Free Particle

For a free particle the potential energy is zero everywhere and the Hamiltonian eigenfunctions are:

Ae^ik·r

with corresponding eigenvalues:

ħ² k² / (2m)

for A and k where r is the position vector, m is the mass and ħ is the reduced Planck constant. The Schrödinger wave function is an integral, over all k, of functions of the following form:

C(r, k) e^ik·r

where the C(r, k) functions are chosen to give the correct initial Schrödinger wave function.