# Does derivative of wave function equal zero at infinity?

1. Sep 12, 2015

### Sturk200

I understand that ψ goes to zero as x goes to infinity. Is it also true that dψ/dx must go to zero as x goes to infinity?

2. Sep 12, 2015

### Dr. Courtney

I can't think of any counter examples.

3. Sep 12, 2015

### Staff: Mentor

It is true as long as $\psi$ and its first deriviative are well-behaved (differentiable, both limits exist, ...). This is a fun old calculus problem - you can prove it by contradiction from the mean value theorem.

4. Sep 12, 2015

### Heinera

Yes, so a counterexample would e.g. be let amplitude go to zero but frequency to infinity. This would be possible if potentials were introduced.

5. Sep 12, 2015

### Staff: Mentor

Yes. And that it goes to zero at infinity is a general assumption of physically realisable wavefunctions.

The correct formalism for QM is what's known as Rigged Hilbert Spaces and the restriction that's often imposed is somewhat stronger in the sense of being continuously differentiable and goes to zero fast enough. They are called good functions:
http://euclid.ucc.ie/pages/staff/thomas/AM2071/Notes/S3notes2011.pdf

It makes many things a lot simpler such as being able to rigorously define the Dirac Delta function and Fourier transforms become a snap with the usual issues of convergence etc a piece of cake.

Knowledge of this stuff really belongs in the toolkit of any applied mathematician in just about any area, not just QM. I stronly reccomend the following book:
https://www.amazon.com/The-Theory-Distributions-Nontechnical-Introduction/dp/0521558905

Thanks
Bill

Last edited by a moderator: May 7, 2017
6. Sep 12, 2015

### PeroK

A function based on $\frac{sin(x^2)}{x}$ would be a counter-example for the general case.