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

[qLinusq]

Hello,

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

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

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.) $$\frac{-\hbar^{2}}{2m} \frac{d^{2}\psi}{dx^{2}}=E \psi$$

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

The solution is:

$$\psi=sin(kx)$$
$$k=\frac{(2mE)^{2}}{\hbar}$$

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.

 

[torquil]

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

I think you can find it here:

http://www.cliffsnotes.com/study_guide/Constant-Coefficients.topicArticleId-19736,articleId-19720.html

 

[torquil]

i would like to derive it myself.

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

 

[qLinusq]

Lol, yes I can see how what I wrote is contradicting. That is the kind of help that I was looking for actually.

/Thank you torquil :)