Hilbert Spaces And Quantum Mechanics

[schumi1991`]

While going through the hermitian nature of quantum operators i came upon the term hilbert spaces... i have no idea what so ever what does this means and would like to know that what are Hilbert spaces and what are they doing in quantum mechanics...

 

[netheril96]

For such a general question, Wikipedia will be the best source.
In a nutshell, Hilbert space is a complete inner product space.

 

[Matterwave]

It is the space of all normalizable wave-functions (not normalized, mind you). It is a vector space which means it obeys all of the rules required for vector spaces (closed under addition, scalar multiplication, etc). For more on this, you can also look at linear algebra and what it means to be a "vector space".

 

[netheril96]

It is the space of all normalizable wave-functions (not normalized, mind you). It is a vector space which means it obeys all of the rules required for vector spaces (closed under addition, scalar multiplication, etc). For more on this, you can also look at linear algebra and what it means to be a "vector space".

Hibert space is much more than a vector space. It has an inner product and is complete under the metric induced from said inner product.

 

[Matterwave]

Well, I don't know too rigorously the exact definitions, but I always thought that Hilbert space could be considered as some kind of a vector space.

 

[San K]

While going through the hermitian nature of quantum operators i came upon the term hilbert spaces... i have no idea what so ever what does this means and would like to know that what are Hilbert spaces and what are they doing in quantum mechanics...

http://en.wikipedia.org/wiki/Hilbert_space

btw...just my (unsolicited) opinion...

all the things we encounter in maths must have something corresponding to it that exists in reality...

hilbert space assumes infinite dimensions and that sounds reasonable to me, because the three axis (dimensions) we define are within one dimension...i.e. time and space ...thus there is either only one dimension (space-time) or infinite dimensions because the three axis describe things only in the time-space dimension...and even within time and space...you could have more than 3 axis...

for example besides space time you need another dimension for instantaneous transmission of law of conservation of momentum...and (perhaps...as a hypothesis...) for explaining the select properties of gyroscopes...in short if maths comes up with things like complex numbers, wave functions to describe something...it tells us that ...there is something in reality that we are missing/don't understand.

 

[netheril96]

http://en.wikipedia.org/wiki/Hilbert_space

btw...just my (unsolicited) opinion...

all the things we encounter in maths must have something corresponding to it that exists in reality...

hilbert space assumes infinite dimensions and that sounds reasonable to me, because the three axis (dimensions) we define are within one dimension...i.e. time and space ...thus there is either only one dimension (space-time) or infinite dimensions because the three axis describe things only in the time-space dimension...and even within time and space...you could have more than 3 axis...

for example besides space time you need another dimension for instantaneous transmission of law of conservation of momentum...and (perhaps...as a hypothesis...) for explaining the select properties of gyroscopes...


in short if maths comes up with things like complex numbers, wave functions to describe something...it tells us that ...there is something in reality that we are missing/don't understand.

No, Hilbert space doesn't imply any dimension. It can finite or infinite. For example, all Euclidean spaces are Hilbert spaces.

 

[schumi1991`]

what I could gather from wikipedia was that hilbert spaces are the generalization of all the spaces(whether with infinite or finite dimensions...)
But still what are their applications in quantum was pretty diifcult to understand for me as I just have started my study of quantum mechanics...

 

[Amok]

what I could gather from wikipedia was that hilbert spaces are the generalization of all the spaces(whether with infinite or finite dimensions...)
But still what are their applications in quantum was pretty diifcult to understand for me as I just have started my study of quantum mechanics...

You can say that all wavefunctions are vectors (state vectors), that live in a Hilbert space. An operator in QM will map a Hilbert space to the same Hilbert space. Just like a function can map R to R or R³ to R³. The eigenfunctions of a QM operator form the basis for a hilbert space.

from wiki:

The Rules of Quantum Mechanics are fundamental; they assert that the state space of a system is a Hilbert space and that observables of that system are Hermitian operators acting on that space; they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the correspondence principle, which states that the predictions of quantum mechanics reduce to those of classical physics when a system moves to higher energies or, equivalently, larger quantum numbers (i.e. whereas a single particle exhibits a degree of randomness, in systems incorporating millions of particles averaging takes over and, at the high energy limit, the statistical probability of random behaviour approaches zero).

Actually, it is kind of weird.

 

[giant_bog]

Hilbert spaces are used so you can treat functions like vectors.
If you have a function F(x) defined between, say, a and b, well, kF(x) is also a function; and kF(x) + mG(x) is also a function; and there's a 0 function and an additive inverse and so on.

So the set of all functions defined between a and b is a vector space. Big whoop, right?

It gets interesting when you start asking things like, "What's the dimension of this space? What are its basis vectors?"

For basis vectors, consider Fourier series. A function F can be decomposed into the sum \sum_{-\infty}^{\infty} a_n e^{inx}. The a_n are therefore it components in the Fourier basis. And it's apparently a complex, infinite dimensional space.

This is useful for QM because vectors (or kets) in this space are manipulated with operators and many of these operators have the property that they don't commute with each other: ie, AB|\psi\rangle \neq BA|\psi\rangle. This is exactly what you want in QM, because QM requires the Heisenberg Uncertainty principle, [X,P] = i\hbar.

If you use Hilbert space, this relationship is built in.

 

[schumi1991`]

but isn't it necessary that for a function to be written in Fourier has to be periodic... ?

 

[giant_bog]

but isn't it necessary that for a function to be written in Fourier has to be periodic... ?

Not necessarily. A continuous Fourier transform, \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-i\omega x}f(x) dx = F(\omega) for example, will give you the Fourier transform of a function over the real line (or in Hilbert space terms, the components of the function in the Fourier basis).

But let's not get sidetracked. The point I'm making about Hilbert spaces is that it's a different way of thinking about what a function really is. Our intuitive notion of a function is that it's a wavy line on a piece of graph paper. In the Hilbert space viewpoint, it's a single point in an infinite dimensional vector space.

You want its spectrum? Express it in the Fourier basis. You want its values at some position? That's just its components in the Dirac delta basis

f(x_0) = \int \delta(x-x_0) f(x) dx

You want its derivative there? Use the delta-prime basis

f'(x_0) = \int \delta'(x-x_0) f(x) dx

In quantum mechanics, every observable is represented by a Hermitian operator acting on kets in Hilbert space. And every Hermitian operator comes fully equipped with its very own set of orthonormal eigenvectors. These eigenvectors are perfectly suited to be a basis. The basis vectors represent the states the system can be in and the components of the ket in this basis represent the probability amplitudes that the system is measured in this state.

There you go. That's quantum mechanics. That's all there is to it: the rest is just details :)

 

[Matterwave]

I would advise that you don't get too bogged down with the terminology. Don't be intimidated by the fact that it's called "Hilbert space". Learn how to use the mathematics and Rules of Quantum Mechanics to solve real problems. That's the important part here.

 

[Amok]

Not necessarily.

Wait, wait, wait. Yes necessarily. He was asking about the Fourier series.