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Time-independent Schrödinger equation for a free one dimensional particle

\frac{\partial ^{2}}{\partial x ^{2}} ψ (x) = - \frac{2 m E}{ℏ ^{2}} ψ (x)

so the solution is:

ψ (x) = e^{\frac{2 m E i x}{ℏ}}

We notice that the solution has continuous spectrum, since any value of

E

can provide a solution.

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Schrödinger equation for a free one dimensional particle