Question regarding a Free particle and Hilbert space (QM)

• I
• CGandC
In summary: So I would not say that you are "wrong in your saying," but I would say that I don't know why rigged hilbert spaces are so important.
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?

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

CGandC said:
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.

I understand.
Thanks for the help!

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.

1. What is a free particle in quantum mechanics?

A free particle in quantum mechanics refers to a particle that is not subject to any external forces or interactions. This means that its motion is only influenced by its own momentum and energy. In other words, it is a particle that is not bound to any potential energy well.

2. How is a free particle described in Hilbert space?

In Hilbert space, a free particle is described by a wave function, which is a mathematical representation of the probability amplitude of the particle's position and momentum. The wave function is a complex-valued function that evolves over time according to the Schrödinger equation.

3. What is the significance of the momentum operator in the Hilbert space representation of a free particle?

The momentum operator in Hilbert space is a mathematical operator that represents the momentum of a free particle. It is a key component in the Schrödinger equation and is used to calculate the evolution of the wave function over time. It is also used to determine the uncertainty in the particle's position and momentum.

4. How does the concept of superposition apply to a free particle in Hilbert space?

In Hilbert space, a free particle can exist in a state of superposition, meaning it can have multiple possible values for its position and momentum at the same time. This is because the wave function of a free particle is a complex-valued function that can have multiple components with different amplitudes and phases. This allows for the particle to have a range of possible positions and momentums until it is observed and its wave function collapses to a specific state.

5. Can a free particle be in a bound state in Hilbert space?

No, a free particle cannot be in a bound state in Hilbert space. A bound state refers to a particle that is confined to a specific region due to external forces or interactions. However, a free particle is not subject to any external forces or interactions, so it is not possible for it to be in a bound state. Its wave function will always spread out over all possible positions and momentums, rather than being confined to a specific region.

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