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CGandC

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- 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?

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?