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In the book physical chemistry (P. Atkins & Julio de Paula, 2009, 5 ED) the authors derive a justification of the Schrödinger equation.

1.) [tex]\frac{-\hbar^{2}}{2m} \frac{d^{2}\psi}{dx^{2}}+V(x)\psi=E \psi[/tex]

The derivation goes as follows:

Derivation:

We can justify the form of the Schrödinger equation to a certain extent by showing that it implies the de Broglie relation for a freely moving particle.

By free motion we mean motion in a region where the potential energy is zero (V=0 everywhere).

If V=0, equation 1 simplifies to:

2.) [tex]\frac{-\hbar^{2}}{2m} \frac{d^{2}\psi}{dx^{2}}=E \psi[/tex]

So far all good, however they then present a solution to equation 2. without showing how they obtained it.

The solution is:

[tex]\psi=sin(kx)[/tex]

[tex]k=\frac{(2mE)^{2}}{\hbar}[/tex]

I have no problem understanding that this is a valid solution however i would like to derive it myself.

Could you provide me with the derivation to the solution of equation 2?

/Thanks in advance,

Linus.

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# Showing that the Schrödinger equation implies the de Broglie relation when PE=0

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