What is Configuration Space in Quantum Mechanics?

[Naty1]

In a section describing "Problems Raised by Statistical Interpretation" of the Schrodinger wave equation, Albert Messiah (QUANTUM MECHANICS) says this:

...the wave equation is described in configuration space not in ordinary space; it can not therefore be identified with the concrete wave we are discussing...which is the representation of a quantum system by its wave function.

Huh?

I assume he is referring to psi (r,t) which he originally introduced almost 100 pages earlier this way:

The simplist system is that of a particle, for instance an electron, in an external force field. The wave which is associated with it at each instant t is a function of psi(r,t) of the position coordinates of that particle.

This seems "concrete" enough.

How to interpret the first quote?
(Wikipedia did not help. Nor a search here in the forums. )
Thanks.

 

[xts]

If you find Wiki explanation as not clear enough, try this:

http://lesswrong.com/lw/pi/classical_configuration_spaces/

In case of single electron you may take 3-D geometrical space as equivalent to configuration space in position representation.

 

[Fredrik]

It's not correct to say that "a configuration space is a way of visualizing the state of an entire system as a single point in a higher-dimensional space". A point in the configuration space represents all the positions of all the component parts, but that's only half of what you need to specify a "state". You need the momenta too.

In the fancy version of Lagrangian mechanics, the configuration space is a smooth manifold, and the phase space (the set of states) is its cotangent bundle.

I agree that \mathbb R^3 is the configuration space of a classical particle.

I don't understand the first quote either.

 

[nnnm4]

A wave function for a system of N particles is a function of 3N arguments. It is simply saying for this reason, the quantum mechanical wave functions shouldn't be considered as existing in real space, but rather an abstract space. In quantum mechanics we don't use a configuration space of position and momenta, we use the wave function which contains all the information to specify the state.

 

[Naty1]

Fredrik: "I don't understand the first quote either. "
ok, thanks, I feel a little better...

The "less wrong" link above is easy enough to understand, and but says the following:

...Quantum configuration space isn't quite like classical configuration space...

(and near the end)... Quantum physics inherently takes place in a configuration space. You can't take it out.

This is just why the Wikipedia article wasn't clear to me...How are 'quantum' and 'classical' configuration space different?? Are they referring to the statistical nature of quantum variables?? For example, what's so different if they both use cartesian coordinates??


I am now coming up against the same issue with HILBERT space further along in the same QUANTUM MECHANICS book: The author says

The wave function capable of representing a given quantum system belong to a function space...The wave functions of wave mechanics are the square integrable functions of configuration space...the function space defined is a Hilbert space.

And he goes on with a list of characteristics of Hilbert space...like those shown in Wikipedia...ok so what...How are they different than Eucledean space...?
Wikipedia says:

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions

so it SEEMS that Hilbert IS higher dimensional Euclidean space??...unfortunately under Euclidean space Wikp[edia says:

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions.

So once again I can't figure out what's different...sounds like the same thing.

I'm not looking for a rigorous mathematical analysis; what I am interested in is how all these spaces are different from each other,and from say plain old Euclidean space, and what effects, if any, they might have regarding the physical observables represented in quantum mechanics. For example, one must apparently be able to impose boundary conditions in all these spaces because it's tough to get quantized states if you don't.

My gut suggests all the different spaces can't be all THAT different because if QM predicts something in an entiely different dimensional entity that is not applicable to what we observe in our space, what use is it if we can't observe and verify?

 

[nnnm4]

Euclidean space is a type of Hilbert space. Consider a space of sines and cosines as well as their linear combinations defined on some interval [0,L]. With the definition of an inner product of these elements, this is a Hilbert space, but it certainly is not a Euclidean space.

 

[jjustinn]

I'm not looking for a rigorous mathematical analysis; what I am interested in is how all these spaces are different from each other,and from say plain old Euclidean space, and what effects, if any, they might have regarding the physical observables represented in quantum mechanics. For example, one must apparently be able to impose boundary conditions in all these spaces because it's tough to get quantized states if you don't.

I can feel your frustration with the author defining the wavefunction as psi(r, t), then 100 pages later saying that it can't be visualized in a Euclidean space...

The trick is the wavefunction as first defined is a single-particle wavefunction, which is indeed just the same as Euclidean space. Where configuration space gets different (and useful) is when you have more particles, and / or varying degrees of freedom.

Basically, a configuration space is a compact way of representing all of the information about the positions of (the configuration of) a system of particles; if you have two particles that are constrained to move on the surface of a sphere, you can represent their positions in a four-dimensional configuration space: (theta1, phi1, theta2, phi2) -- thetaN / phiN giving the latitude / longitude of the Nth particle.

Maybe an easier way to visualize how configuration space differs from Euclidean space is to take a system with just two degrees of freedom -- say two particles constrained to move on the X axis: (x1, x2). Now the configuration space of those two particles IS a rectangle -- but to talk about a "rectangle" or a "disc" in that space obviously doesn't mean much.

To take it a step further, usually in mechanics you start with configuration space -- which is just the positions of particles -- then move to "phase space", where you have not only the positions, but the momenta of each particle. So each classical free particle gets 6 components: (x, y, z, px, py, pz). Same idea, but just more dimensions. Obviously here the geometrical / Euclidean picture is even less useful -- since some of the "coordinates" have nothing to do with position at all.

The Hilbert space of quantum mechanics is basically just the same thing -- but in quantum mechanics, you can't disentangle a particle out from the rest of the system, so each "state vector" in the space (like the (x1, x2) or (x, y, z, px, py, pz) above), instead of just containing all the degrees of freedom of one particle, has to account for all the degrees of freedom of ALL particles, so each vector has an infinite number of degrees of freedom -- though in practical situations, you're usually just dealing with a finite number of those degrees of freedom.

If that last part doesn't make sense, don't lose sleep over it -- if you can at least get your head around the idea of a phase space, you should be fine...I don't think anything you'll be coming across in the near future will depend on you having the pure mathematical distinctions between phase space and Hilbert space nailed down (though obviously it can't hurt).

 

[Naty1]

jjustinn: good explanation...gives me the inutitive feel I was seeking.

Since posting, I've come across references, most wihout definitions, in the same text to a lot more "spaces" and I'm coming to the conclusion most of them are for mathematical convenience...