Source: cirosantilli/time-independent-schrodinger-equation-for-a-free-one-dimensional-particle

= Time-independent Schrödinger equation for a free one dimensional particle

$$
\pdv{^2}{x^2} \psi(x) = -\frac{2mE}{\hbar^2} \psi(x)
$$
so the solution is:
$$
\psi(x) = e^{\frac{\sqrt{2mE} i x}{\hbar}}
$$
We notice that the solution has <continuous spectrum (functional analysis)>, since any value of $E$ can provide a solution.

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