# I Question regarding a Free particle and Hilbert space (QM)

#### CGandC

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?

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#### dextercioby

Homework Helper
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

Homework Helper
Gold Member
2018 Award
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.

• Demystifier

#### 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.