Schrodinger's equation and the finite well(conceptual)

[hover]

Hello,

I have a question about Schrödinger's equation and the finite well. It isn't so much as a math question but rather how to interpret the problem. I'll use the picture on the right from here for reference and for simplicity, I'll stick to one dimension. When I think of this problem, I literally picture a particle(example, an electron) trapped inside a wooden box. The edges of this box, have a finite width. The picture on the right however seems to show a "box" that has edges of an infinite width. Now I understand that the wave function must decay to zero as x approaches (+/-)infinity but both of these ideas contradict themselves. This box must have an infinite width so the wave function decays to zero at an infinity but then we aren't dealing with a real wooden box since no box has edges of infinite width. Is there a way to model a real box with a finite width? Also, I know that the areas outside the "box" are areas of potential energy and I know that these areas are classically forbidden regions for particles to be but I am slightly lost as what this potential energy is suppose to represent.

Thanks for reading and trying to clear this up,
Hover

 

[Jano L.]

Hover,
finite walls complicate the calculation of the wave function only slightly, it can be done. Look at the first picture at

https://en.wikipedia.org/wiki/Quantum_tunnelling

J.

 

[ShamelessGit]

Thinking of the walls of the box as being infinitely thick but penetrable is a bad idea. An infinite potential well would be more like a wall that was infinitely hard and sturdy, but it could be paper thin. The spatial derivative of the energy is force, so in a picture in which it shows vertical walls for the potential energy, that would also mean infinite force at the edge of the box repelling the particle (because the slope is infinite).