Question regarding a Free particle and Hilbert space (QM)

[CGandC]

TL;DR Summary

In quantum mechanics, is the Eigenfunction resulting from the Hamiltonian of a free particle in 1D system, belongs to Hilbert Space?

In quantum mechanics, the Eigenfunction resulting from the Hamiltonian of a free particle in 1D system is $$ \phi = \frac{e^{ikx} }{\sqrt{2\pi} } $$
We know that a function $$ f(x) $$ belongs to Hilbert space if it satisfies $$ \int_{-\infty}^{+\infty} |f(x)|^2 dx < \infty $$

But since the Eigenfunction $$ \phi(x) $$

doesn't satisfy the above condition to belong in Hilbert space:
$$ \int_{-\infty}^{+\infty} |\phi(x)|^2 dx= \infty $$

Therefore, I say that $$ \phi(x) = \frac{e^{ikx} }{\sqrt{2\pi} } $$ does not belong to Hilbert space.

Am I right in my saying? if not, why?

 

[dextercioby]

Yes, you are right, if the particle is completely free, i.e. unconstrained to move in 1 dimension from minus infinity to plus infinity.

 

[PeroK]

Summary: In quantum mechanics, is the Eigenfunction resulting from the Hamiltonian of a free particle in 1D system, belongs to Hilbert Space?

In quantum mechanics, the Eigenfunction resulting from the Hamiltonian of a free particle in 1D system is $$ \phi = \frac{e^{ikx} }{\sqrt{2\pi} } $$
We know that a function $$ f(x) $$ belongs to Hilbert space if it satisfies $$ \int_{-\infty}^{+\infty} |f(x)|^2 dx < \infty $$

But since the Eigenfunction $$ \phi(x) $$

doesn't satisfy the above condition to belong in Hilbert space:
$$ \int_{-\infty}^{+\infty} |\phi(x)|^2 dx= \infty $$

Therefore, I say that $$ \phi(x) = \frac{e^{ikx} }{\sqrt{2\pi} } $$ does not belong to Hilbert space.

Am I right in my saying? if not, why?

That function is not in the Hilbert space of square-integrable functions - because it is not square-integrable.

 

[martinbn]

This is just a starting point, but it might be helpful, the motivation is exactly your question.

https://en.wikipedia.org/wiki/Rigged_Hilbert_space

 

[CGandC]

I understand.
Thanks for the help!

 

[HomogenousCow]

There are some short sections in Ballentine on this. However I personally did not find the whole concept of rigged hilbert spaces to be that physically important, it seemed to me just a mathematical rubber stamp to proceed as we did.