[yesort]

Hello,

I was just wondering whether the wavefunction ψ can take on any dimension? My textbook uses the Schrodinger Eq to derive solutions for a particle in a 1D box, but can it also yield solutions for higher dimensions?

Thanks!

[Borek]

No idea about "any", but 2D and 3D definitely. Most work is done in 3D.

[Corribus]

Bear in mind that all problems are problems of 3 spatial dimensions because we live in a 3D universe. But in some system geometries, due to symmetry or whatever, we can treat the wavefunction as separable and then only worry about 1 or 2 of those dimensions. As an example, long conjugated molecules are three-dimensional structures, so properly speaking we should treat them with three dimensional wavefunctions. But the long dimension is so distinct from the other two that we can approximate the system as one dimensional and worry only about a one-dimensional wavefunction to achieve some relatively good, although approximate, results for how an important subset of the electrons in that system behave. It happens that most of the properties of interest arise out of changes between states defined almost exclusively by wavefunctions along the long axis, so approximating the system as "one dimension" works really well here. We cannot do this in a spherical atom, however, in which all three dimensions have almost equal important by virtual of the system's geometry.

This kind of dimensional reduction is common in physics due to how much it can simply the math. We often go so far as to actually design experimental systems so that 1D mathematical treatments can be used. If we want to know heat transfer coefficients in a new material, for example, the common approach would be to fashion the material into a long, thin geometry like a wire or bar. This way, you can use the 1D heat equation, which has much easier solutions than its multidimensional counterpart. In fact, some multidimensional differential equations aren't even analytically solvable, so reducing the dimensionality of your experimental system may be your only way to get an exact value for parameters of interest.

[wildfyr]

You should write an intro to chemistry textbook Corribus. Or something.

[Corribus]

lol, thanks.

Actually, I wrote something a little misleading there. Even in the atom due to its nice spherical symmetry and separation of variables we can separatethe wavefunction into (the product of) lower dimension counterparts - radial parts and angular parts. These can be solved independently and analytically for certain systems. Whether or not solutions to these lower dimension counterparts in isolation provide enough information depends a lot on what you're looking for.