Kinetic energy formula for electron in lowest energy state

[Kognito]

Homework Statement

Use the relationship kinetic energy E = p^2/2m to show that the energy $E_{0}$ of an electron of mass m in its lowest energy state is given by $E_{0}$ = h^2/8mL^2

Homework Equations

E = p^2/2m

$E_{0}$ = h^2/8mL^2

The Attempt at a Solution

I've stared at this for far too long, googled it to death as well as checked through all my course materials and I can't seem to get started with it. Any ideas at all would be welcome.

Thank you in advance,
Kognito

 

[Pagan Harpoon]

What is L here? Is the electron in a 3d box and L is the side length? Does the problem say what boundary conditions it wants you to apply at the box edges?

 

[Kognito]

My apologies, I missed that off the end of the sentence (too busy messing with equation formatting controls).

But yes, L represents a box within which the electron resides. The question doesn't specifically mention boundary conditions though the previous part of the question said to draw the electron in its lowest energy state, able to move freely along the length of the box, as represented by a standing wave.

 

[Pagan Harpoon]

There's two things you can do... one is write down the Schrödinger equation and then solve it.

The other is to say that the electron's wavefunction has to have nodes at the box edges, so write down an expression for its deBroglie wavelength and apply the appropriate constraints to it.