Can This Integral Be Simplified?

[Erik 05]

Homework Statement

∫^∞_-∞|∂Ψ/∂x|²dx

Homework Equations

Can this be simplified at all

The Attempt at a Solution

 

[PeroK]

What have you tried?

 

[Erik 05]

I haven't tried anything. The question, you are given E = (1/2m) <pΨ|pΨ> and asked to express it in ∂Ψ/∂x, so it may be fine just to leave it as the above. I was just thinking that maybe there is a 'standard' simplification for that integral. For example if Ψ is normalized, does that imply ∂Ψ/∂x also is? Well probably not, but would it simplify the integral somehow?

 

[PeroK]

If you are talking about a general wavefunction, then your original expression is as simple as anything else. You can,of course, try integration by parts - which should lead back to an alternative version of the inner product as well.

For example if Ψ is normalized, does that imply ∂Ψ/∂x also is?

That's definitely something you should explore for yourself. A Gaussian is often a good example to start.

 

[Erik 05]

Ok, thanks.