- 31

- 0

**Schrodinger Equation in momentum space? ??**

I don't know if this makes any sense at all, but I'm studying QM and just trying to generalize some things I'm learning. Please let me know where I go wrong..

Basically by my understanding the most general form of the Schrodinger Equation can be written as follows:

[tex]i\hbar\frac{\partial\Psi}{\partial t}=\hat{H}\Psi[/tex]

Then we have that H is given by:

[tex]\hat{H}=\frac{\hat{p}^{2}}{2m}+V[/tex]

Then we just need to know what the 'p' operator is. My book shows that in configuration space it is:

[tex]\hat{p}=-i\hbar\nabla[/tex] (I generalized it a bit, the book just has an x derivative)

however, in momentum space it is just: [tex]\hat{p}=p[/tex]

Now.. if we take the momentum operator in configuration space and plug that into H and then plug that into the form of the Schrodinger Equation I wrote at the top we get the familiar form:

[tex]i\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi+V\Psi[/tex]

Ok... but what I am wondering now is what happens if we use the momentum operator in momentum space instead of configuration space, then plugging that into H and H into the first equation as before we would end up with:

[tex]i\hbar\frac{\partial\Psi}{\partial t}=\frac{p^{2}}{2m}\Psi+V\Psi[/tex]

This seems very odd though, I think it must be wrong but I don't know why... perhaps it is right but only for a wavefunction expressed in momentum space... ??

Can anyone help me understand what is going on here?? Perhaps there is no justification in plugging the momentum operator for momentum space into H like I did.. I just don't know?

Thanks for any insight.