Traveling wave equation derivation from free space electron? Following my textbook I came across this derivation. I can make the assumption in 3.11 that solving ODE will give you $e^{jkx}$. But I get lost in the substitutions and assumptions being made in 3.12. Can anyone explain those steps.

 A: Given
$$\psi(x) = Ce^{jkx}$$
if we take the second derivative we get
$$\frac{d^2 \psi(x)}{d x^2} = -k^2Ce^{jkx}$$
So substituting into equation 3.11 on both sides, we get
$$\frac{\hbar^2k^2}{2m}Ce^{jkx} = ECe^{jkx}$$
which gives
$$k= \frac{\sqrt{2mE}}{\hbar}$$
A: Equation 3.11, which is also called time-independent Schroedinger equation,  is derived from the original time dependent Schroedinger equation, which is a partial differential equation, by separation of variable, assuming that the solutions can be written as a product of a function that depends only on t and another function that depends only on x. This gives two ordinary differential equations, one for the time dependent function having the solution exp(-jt) with E= h, and one, which is equation 3.11, for the x-coordinate dependent function which has the exp(jkx) solutions. Therefore, in order to obtain the complete solution of the time dependent Schroedinger equation you have to multiply the t-dependent and the x-dependent functions giving the exp j(kz-t) travelling wave solutions. 
