= Schrödinger equation for a free one dimensional particle
{c}
{{wiki=Free_particle#Quantum_free_particle}}
<Schrödinger equation for a one dimensional particle> with $V = 0$. The first step is to calculate the <time-independent Schrödinger equation for a free one dimensional particle>
Then, for each energy $E$, from the discussion at <solving the Schrodinger equation with the time-independent Schrödinger equation>{full}, the solution is:
$$
\psi(x) = \int_{E=-\infty}^{\infty} e^{\frac{\sqrt{2mE} i x}{\hbar}} e^{-iE t/\hbar} = e^{i\frac{\sqrt{2mE}x - E t}{\hbar}}
$$
Therefore, we see that the solution is made up of infinitely many <plane wave functions>.