# Schrodinger equation

1. Mar 10, 2005

### 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?

2. Mar 10, 2005

### dextercioby

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

Daniel.

3. Mar 10, 2005

### Staff: Mentor

Yes.

Yes.

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)

4. Mar 10, 2005

### irony of truth

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

5. Mar 10, 2005

### dextercioby

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

Daniel.

6. Mar 11, 2005

### BlackBaron

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.

Last edited: Mar 11, 2005