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Solutions of the Schrodinger equation

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.


Time-independent Schrödinger equation for a free one dimensional particle

Time-independent Schrödinger equation for a free one dimensional particle

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Time-independent Schrödinger equation for a free one dimensional particle

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