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 Section "Solving the Schrodinger equation with the time-independent Schrödinger equation", the solution is:
Therefore, we see that the solution is made up of infinitely many plane wave functions.

$ψ(x)=∫_{E=−∞}e_{ℏ2mEix}e_{−iEt/ℏ}=e_{iℏ2mEx−Et}$

In this solution of the Schrödinger equation, by the uncertainty principle, position is completely unknown (the particle could be anywhere in space), and momentum (and therefore, energy) is perfectly known.

The plane wave function appears for example in the solution of the Schrödinger equation for a free one dimensional particle. This makes sense, because when solving with the time-independent Schrödinger equation, we do separation of variable on fixed energy levels explicitly, and the plane wave solutions are exactly fixed energy level ones.

$∂x_{2}∂_{2} ψ(x)=−ℏ_{2}2mE ψ(x)$

$ψ(x)=e_{ℏ2mEix}$

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