What's the meaning of dimensionality?

I got amazed with this answer and as you can see, I had a little talk with joriki. He suggested me some examples such as the 2DEG. I'm stuck with some aspects on dimensionality, see this line:

A two-dimensional electron gas (2DEG) is a gas of electrons free to move in two dimensions, but tightly confined in the third.

In mathematics, it seems that there could be $n$ dimensions, but how's it on the "real world"? I thought that such $n$-dimensional objects on the real world could exist on the absence or presence of a specific dimension, but at least in this wikipedia article, it seems that this object have the third dimension - at least I conceive "confined" as something different of "abscence". Can you clarify this to me?

I am open to both answers and references on this topic, I just hope that asking both is not being too broad.

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If the wiggle room afforded by a dimension is small enough, it may be a practical choice to think of said dimension as negligible. Many people think of spacetime as having three space dimensions plus time, which is useful even though there may be other tightly wrapped dimensions a la string theory we don't know about. It's simply a matter of simplifying our mental models when possible and appropriate. –  anon Aug 29 '12 at 16:25
In addition to Joriki's suggestion of Flatland you may appreciate Dewdney's Planiverse (there is an annex on 'science and technology in 2D' with a short discussion of QM in 2D) –  Raymond Manzoni Aug 29 '12 at 16:25
@RaymondManzoni Thanks for your sugestion, I was unable to find something "readable", the only thing I could find was something like this. –  Vÿska Aug 29 '12 at 16:27
hum... yes I understand your reaction! ;-) –  Raymond Manzoni Aug 29 '12 at 16:30

Well, this is actually more physics than mathematics. The point is that if you confine quantum objects, you only get discrete energy levels (think atoms), and if the energy isn't high enough to get to the next energy level nothing happens. That's why quantum dots ("zero-dimensional" quantum systems) are also called artificial atoms (and in some sense, atoms are zero-dimensional as long as you don't use too much energy when interacting with them).

Now you can tightly confine an object only in some dimensions, and let them freely move in the others. Then those principles apply for each dimension separately. That is, in the "free" dimensions you can put an arbitrary low energy to get the particles moving, while in the confined dimensions, if you don't put enough energy in, no movement at all is possible. Therefore the quantum system really behaves as if it were lower-dimensional, as long as the energy is low enough. Note, however, that it still has a (small) extension in the confined dimensions, so things happening there still influence the particle. For example, if at some regions the confinement gets more narrow, it acts like a repulsive potential, because the energy goes up if the system is confined in a smaller space.

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If you have a math background, its easy to explain. The first thing physicists do is to go adimensional by scaling their equations. For instance if they do an experiment in a water tank whose dimensions are x=l,y=L,z=h; they first introduce dimensionless variables : a=x/l; b=y/L; c=z/h Doing so allow them to order the terms in their equations and to introduce small parameters used in asymptotic expansion.

For instance, assume the lenght l and width L of the tank are a lot greater than the height h, and assume the process you study is described by : df/dx +df/dy+df/dz=G(x,y,z) Then by going adimensional, you transform your equation into: 1/l*(df/da)+1/L*(df/db)+1/h*(df/dc)=G(l*a,L*b,h*c) The first order expansion in 1/h is two dimensional: df/dc=0 and 1/l*(df/da)+1/L*(df/db)=G(l*a,L*b,h*c)

Practically, physicists sandwich a layer of conducting material between two layers of non conducting materials. The free electrons are then confined in a box whose height is a lot smaller than the other lengths and are at the first order in 1/h free to move in a plane.,

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