# Potential Well

## Homework Statement

An electron is confined to a potential well of finite depth and width, 10^-9 cm. The eigenstate of highest energy of this system corresponds to the value $$\xi = 3.2$$.

a. How many bound states does this system have?
b. Estimate the energy of the ground state with respect to the zero energy line at the bottom of the well. Express answer in eV.

## Homework Equations

$$\xi = ka$$ where a is half the width
$$\eta = \kappa a$$

as far as I know k corresponds to the bound state while kappa corresponds to the unbound states

$$\xi tan(\xi) = \kappa$$ for even eigenstates
$$\xi cot(\xi) = -\kappa$$ for odd eigenstates

## The Attempt at a Solution

a. Not exactly sure here.

I figured I'm looking for k but when I solve for it I get a huge number (64 million)
k = xi/a = (3.2 * 2) * 10^9

b. E = 0 at the bottom of the well so then
$$E = \frac{\hbar^2 \xi^2}{2ma^2}$$

correct?