{c}
{{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

{{wiki=Schrödinger_equation#Time-independent_equation}}

The <time-independent Schrödinger equation> is a variant of the <Schrödinger equation> defined as:
$$
\hat{H}[\psi_x(E, \vv{x})] = E \psi_x{\vv{x}}
$$
{title=Time-independent Schrodinger equation}

So we see that for any <Schrödinger equation>, which is fully defined by the <Hamiltonian> $\hat{H}$, there is a corresponding <time-independent Schrödinger equation>, which is also uniquely defined by the same <Hamiltonian>.

The cool thing about the <Time-independent Schrödinger equation> is that we can always reduce solving the full <Schrödinger equation> to solving this slightly simpler time-independent version, as described at: <Solving the Schrodinger equation with the time-independent Schrödinger equation>{full}.

Because this method is fully general, and it simplifies the initial time-dependent problem to a time independent one, it is the approach that we will always take <solutions of the Schrodinger equation>[when solving the Schrodinger equation], see e.g. <quantum harmonic oscillator>.


Ciro Santilli @cirosantilli 34

 Incoming links: Solving the Schrodinger equation with the time-independent Schrödinger equation