What is the Meaning of Hilbert Space?

[plasma0073]

Please can anyone help me to display the meaning of Hilbert Space?

 

[mitchell porter]

I suggest that you start with two concepts: n-dimensional vector space, and analysis of a waveform into a sum of wave modes.

In a vector space, a vector is a sum of "basis vectors". For each basis vector, there is a coefficient which says how much it contributes to the sum.

In the same way, a waveform is a sum of "basis functions", and for each basis function, there is a coefficient in the sum.

The Hilbert space is a unifying concept which allows you to think of basis functions like basis vectors. The geometric ideas from vector space also apply in Hilbert space. For example, you change from one basis to another basis, and it does not change the total vector / total waveform, it just changes the coefficients in the sum.

In quantum mechanics, the coefficients can be complex numbers, the basis vectors / basis functions are the different physical states, and their probabilities are square of the absolute value of the complex number (Born rule).

 

[Fredrik]

The meaning of Hilbert space? I think you need to ask a more specific question. Do you want the definition? The reason why Hilbert spaces are useful in QM? Something else?

Which ones of the terms "vector space", "inner product", "norm", "metric space", "Cauchy sequence" and "complete" do you know already?

 

[BruceW]

Hilbert space is an abstract generalisation of Euclidean space. (Which is how it is explained to those who are studying it for the first time).
The Hilbert space has a mathematical definition, with several properties. And it happens to be useful in quantum mechanics. (Which is how I first heard of it).

EDIT: Maybe search it on wikipedia to get an introduction to it. Although, maybe you should get more familiar with the concepts of vectors, scalar products, etc before learning the more general Hilbert space. Can't hurt to look it up, though.