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.


Plane wave function

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{{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>.


Schrödinger equation for a free one dimensional particle

{tag=Schrödinger equation}
{wiki}

\Video[https://www.youtube.com/watch?v=8Iu74b5iCuQ]
{title=Why Relativity Breaks the Schrodinger Equation by Richard Behiel (2023)}
{description=Take a <plane wave function>, because we know its momentum perfectly. Apply a constant voltage to an electron. You can easily bring it beyond the speed of light at about 255.5 keV.}


Ciro Santilli @cirosantilli 35

 Incoming links: Plane wave function