# Wave formula question

asdf1
why is the wave formula equal to zero when x equals postive/negative infinity?

Homework Helper
Gold Member
Hi,

I think is a requirement that follows from the probabilistic interpretation. If the integral

$$\int_{-\infty}^{\infty}|\Psi|^2dx$$

is to converge, psi is to go to zero at infinity

Last edited:
Generally. it's not zero. Yes for bound states. But, scattering particles obey the Sommerfeld radiation conditions, outgoing waves go as exp(ikx), incoming go as exp(-ikx). To see this in action in a relatively simple way, see scattering from a square well potential in any QM text.
Regards,
Reilly Atkinson

asdf1
thank you very much! :)

Homework Helper
Scattering states are typically eigenstates of unbounded operators (we usually call them "plane waves" and "spherical waves"), like the free particle's Hamiltonian and momentum operators. Ortonormalizing such eigenstates (which are NOT in $L^{2}\left(\mathbb{R}^{3}\right)$ ) is a tricky problem. See Bogoliubov's text on axiomatical QFT.

Daniel.