Solving the Schrodinger Equation for 1D Electron Motion

[irony of truth]

I am trying to find the Schrodinger's equation for the one-dimensional motion of an electron, not acted upon by any forces.

So.. should I begin using the time independent form of the Schrodinger's equation? What should I arrive at? Should I let my V(x) = 0?

Also, how do I show that the total energy of that particular schrodinger's equation is not quantized?

 

[dextercioby]

What's the general form of the SE's equation...?What's the free particle's Hamiltonian...?

Daniel.

 

[jtbell]

So.. should I begin using the time independent form of the Schrodinger's equation?

Yes.

Should I let my V(x) = 0?

Yes.

Also, how do I show that the total energy of that particular schrodinger's equation is not quantized?

Find the general solution of your Schrödinger equation, and show that any value of E leads to an acceptable solution for your boundary conditions. Of course, you don't really have any boundary conditions, which simplifies matters! (unlike the infinite square well a.k.a. "particle in a box" where the boundary conditions restrict the acceptable values of E to a discrete set)

 

[irony of truth]

Thank you for your helps... I can manage from here... E = (hbar)k^2 / (2m) >= 0

 

[dextercioby]

That's right.And the "k" wan take any real value...Making the energy spectrum the real positive semiaxis.

Daniel.

 

[BlackBaron]

Of course, you don't really have any boundary conditions

You do have boundary conditions, you have to require that your solution goes to zero at infinity, otherwise your solution is not normalizable.
For a free particle, that actually represents a big problem, since the free particle wavefunction ($$e^{ipx/\hbar$$) isn't nomalizable!
That's the reason one should really work with wavepackets in the cases where $$V(x) = 0$$, those are normalizable.
Almost any textbook in quantum mechanics has a discussion on that particular topic.