Orthogonal basis in Hilbert Space

[Quartile]

Hilbert space is a tool used to mathematically identify properties of matter that are hidden to other types of analysis. In my understanding it is similar to a regular euclid space that includes all limits of all functions within its boundaries.

 

I can understand why an orthonormal basis is applicable to the notion of a Hilbert space, but I have a question. How can the relative position of two separate objects be determined, if the Hilbert space does not extend beyond the boundaries of the two objects themselves? That is, in a Hilbert space containing two separate objects, is the space between them required to accept an orthonormal basis as well?

 

Thanks!

[Bignose]

Hilbert space is a tool used to mathematically identify properties of matter that are hidden to other types of analysis.

 

I don't know quite where you got that interpretation of a Hilbert Space. A mathematical space doesn't really have anything to do with matter. A space is just some collection of members that obey certain rules. A Hilbert space is one where its collection of members have an inner product [math]<f,g> [/math] and they have a norm defined by [math] |f| = \sqrt{<f,f>} [/math] which forms a complete metric space.

 

That's pretty much it.

 

Nothing about "hidden properties of matter". So, I'm not sure where you got that idea from... maybe you could elaborate some?

[Quartile]

What I meant by the sentence you quoted is that the notion of Hilbert space is a mathematical tool which opens many doors for all types of analysis. It is a cornerstone of modern physics, which is the study of matter.

 

I know my understanding of Hilbert space may be somewhat weak compared to yours. Is the sentence that came after the one you quoted correct?

Edited July 4, 2008 by Quartile

[ajb]

Don't get too hung up on the idea of a Hilbert space. You should think of it as a generalisation of a vector space to an infinite dimensional space.

 

A Hilbert space also comes equipped with an inner product and is complete meaning all convergent series converge to something in the Hilbert space.

 

As this is posted in the physics section I assume you are interested in the Hilbert space [math]L^{2}[/math] as found in quantum mechanics. If so you don't really need to know much about the details of Hilbert space theory, all Hilbert spaces are essentially the same. It is the algebra of operators on that Hilbert space that are important.

[Quartile]

Rewording my poorly worded question, is there an operator that will allow the unoccupied area between two objects to be analyzed in a Hilbert space?

[ajb]

I really don't understand your question. Are you trying to count physical states or something?

[D H]

Go way back to simple vector spaces, way back to [math]\mathcal R^2[/math]: "Is there an operator that will allow the unoccupied area between two unit vectors to be analyzed in [math]\mathcal R^2[/math]?"

[Quartile]

Does including two separate objects on the same Hilbert space force them to become one system?

[ajb]

I think you need to be very careful with your language. By object on you mean a point? Or do you mean some geometric object? (e.g. Tensor)

 

In quantum mechanics the (usually infinite dimensional) Hilbert space is the space of all solutions to the Schrödinger equation. (In fact it is a little more than that at rays are physically identified). I don't quite follow what you mean by "two separate objects".

[Bignose]

Does including two separate objects on the same Hilbert space force them to become one system?

 

I still get the sense that there is a disconnect between the mathematical concept of a space and what is commonly thought of as space.

 

An "object" in a Hilbert space is a mathematical expression, one that obeys certain rules so that it can be a member of the Hilbert space. For example, something like [math]\sin{x}[/math] or the Hermite polynomials [math]H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n}e^{-x^2}[/math] or any other function/mathematical expression that the inner product exist for.

 

Nothing about a Hilbert space is connected to objects in space, like a baseball, a planet, or an electron. Now, physicists use the properties of a Hilbert space to mathematically describe some problem in "actual" space. But, the Hilbert space is just a collection of mathematical expressions. Nothing more. No connection to any actual physical object.

[nm-8]

Hi!

Studiing pysics somehow makes me understand what you want to ask. And to give an answer to your question: The space is defined over a property it has at each point: The scalar product.

Now, in physics, Hilbertspace is representing the possible states of a physical system. The actual state is represented by a coordinate in hilbertspace, and the probability of measuring this state is the scalar product between this coordinate in hilbertspace and the same coordinate in a dual-hilbertspace (which is almost exactly the same space). So Hilbertspace just indicates the posibillities of a system. SOmetimes the axes of Hilbertspace are continously (like the angle of a pendulum would make a cont. axis in HS) and sometimes the axis can be discrete (like the energyvalues of the hydrogen atom) building an axis of HS. Between the points of a discrete axis NO Measurement is possible. (because one hydrogen atom will never pop up in energystate 1.5. only in state 1 or in state 2)

 

For such discrete axis your question is relevant.

Answer: No, between every two neighbouring "objects" (="coordinates") in a discrete Hilbertspace no measurement can be made inbetween.

 

this all is just in case that you are making a single measurement on a discrete system. as soon as you talk of mean values, and do a lot of measurements, also values between the discrete values can be resulting. Energystate of a hydrogen cloud can be 1.5, if one half is state1 and the others are state2...

 

As you see, Hilberspace really is just a mathematical tool, to categorize your system. though it has it's well defined role in physics, it remains a tool, not a space that is somehow existing in our world...

hope that helped

[ajb]

I don't think that is quite right nm-8.

 

Usually you give the Hilbert space the natural topology using open balls defined using the norm. I think (usually) Hilbert spaces are considered not to be discrete, (unless you give it the discrete topology).

[nm-8]

Most examples of HS assume infinite measurements, which is what you are talking about. But this extremely changes the statistics of states, and has nothing to do with a real quantum-experiment-discription (which needs a discrete topology).

Otherwise tell me, how shall I ever measure an action of PI*h, without infinite measurements. With continouus space this would be possible, but in our world (and it's corresponding HS) its not.

 

what do you mean with "open balls"?

 

p.s: thanks for your reply, it made me think

[ajb]

I'll give you some links to reputable sites.

 

As for the topology on a metric space see metric topology.

 

For the definition of a Hilbert space see Hilbert Space.

 

The important thing here is that the idea of being discrete or not is a question of topology. Given the metric topology, the Hilbert space is "continuous", meaning we can think about points being arbitrary close.

 

This property is separate from the completeness, which is a property of the norm (metric) and not that of the topology chosen.

[prometeu]

orthonormal, in Hilbert and any other spaces, bsis is sometimes better only for simplifying the calculations and perhaps to give physical meaning.

I dont understand the other part of the letter.

[Quartile]

An "object" in a Hilbert space is a mathematical expression, one that obeys certain rules so that it can be a member of the Hilbert space... But, the Hilbert space is just a collection of mathematical expressions. Nothing more. No connection to any actual physical object.

 

But there is a connection between physics and reality, I hope? Aren't the qualities of Hilbert space chosen so that physicists are able to mathematically model some aspect of quantum reality?

[D H]

It's the other war around. Physicists chose to use the concept of Hilbert spaces (note: Not Hilbert space; this is a very generic concept) because the concept is useful in modeling aspects of quantum reality.

[booker]

But there is a connection between physics and reality, I hope? Aren't the qualities of Hilbert space chosen so that physicists are able to mathematically model some aspect of quantum reality?

 

Quartile,

 

Does your original question have anything to do with two particles like electrons or photons measured at two distant places in space, usually by Alice and Bob?

[Quartile]

It's the other war around. Physicists chose to use the concept of Hilbert spaces (note: Not Hilbert space; this is a very generic concept) because the concept is useful in modeling aspects of quantum reality.

 

Right thats what I meant lol thanks

 

Does your original question have anything to do with two particles like electrons or photons measured at two distant places in space, usually by Alice and Bob?

 

Yes. Two photons measured in different positions.

 

The notion of a discrete topology actually answered most of my question. One last thing: Considering that orthogonality is a necessary quality of the space on which quantum-scale matter is analyzed, is it too far a stretch to say that it is a necessary quality of any relevant geometry/space?

[booker]

Yes. Two photons measured in different positions.

 

The notion of a discrete topology actually answered most of my question. One last thing: Considering that orthogonality is a necessary quality of the space on which quantum-scale matter is analyzed, is it too far a stretch to say that it is a necessary quality of any relevant geometry/space?

 

You have some confusion about hilbert space and space, or spacetime.

Hilbert space doen't span space. It's an abstract space of probability amplitudes. The magnitude of the vector squared is the propbability that the particle will have some particular value when measured.

 

I think DH could expain it to you better.

[Quartile]

It's an abstract space of probability amplitudes.

 

Like you say, the probability amplitude describes the possible states that a wave function might collapse to if it were measured. In the case of an electron the probability amplitude describe the space that the electron "probably" occupies at some time, before measurement. So it does describe space, doesn't it?