3D Schrodinger Equation of a Wire for Energy Levels - Quantum Physics

[Doopdedoop]

Homework Statement

Given a wire with length a and square base b x b (where a >> b), show that the first 1700 (approximately) levels of the electron in the wire are identical for the one dimensional box, when a = 1m and b = 1mm.

Homework Equations

I know that the allowed energies of a mass M in a 3D rectangular rigid box with sides a, b, and c are:

E1 = ħ^2 * (Pi)^2 / 2M * (nx^2 / a^2 + ny^2 / b^2 + nz^2 / c^2)

The energy of a 1-D box with length a is similar, being

E2 = ħ^2 * (Pi)^2 / 2M * (n^2 / a^2)

The Attempt at a Solution

For the purpose of this problem the energy of my wire is:

E = ħ^2 * (Pi)^2 / 2M * (nx^2 / a^2 + ny^2 / b^2 + nz^2 / b^2)

Now, I know by inspection and common sense that a long and very thing wire can be considered as a one dimensional system, which is used by the 1D Schrödinger equation.

The problem is that I'm not quite sure how it works out mathematically. If a >> b, then doesn't this mean we generally ignore the a component of E1, since the b part basically dominates the energy? But all the same, you can't do that for the equation for the 1D box.

It's all just a bit confusing for me.

 

[Spinnor]

Assuming your formula for E is correct calculate the energy level for

nx, ny, nz = 1700, 1, 1 and nx, ny, nz = 1, 2, 1

they should be roughly equal?