I am just trying to get my head round how this models the electron bound to an atom. I don't understand why the potential is zero in the well What physical case corresponds to the condition that V(x)=0 for all values of x?
Yes it was, that is what I thought it would be but then I was wondering why the potential could be zero inside the well as this is meant to represent a bound particle.
Also for a completely free particle, would there be a restriction on the value of n? cheers
If the electron is in a box with impenetrable walls, then it's equivalent to being in an infinite potential well, in this case with V=0 inside. That is, the problem describes an electron confined to a finite region of space with the only forces acting during collisions with the walls.
The infinite square well doesn't really model anything physical. The closest thing that it comes to modeling is a finite quantum well used in semiconductor lasers. However, the square well is basically the simplest test case that you can construct in QM, since it illustrates the quantization of energy levels.
What u might be looking for is the schrödinger equation expressed in radius and angle. You can then make a much more accurate picture as you can use the attraction of the electron to the nucleus as the potenital in the from U(x)= -ke^2 /r. This gives a much more accurate picture of an electron round an atom, as the potential isnt infinite or 0, but increases with distance. Hope this helps.