A finite potential well has depth Uo = 3.00 eV. What is the penetration distance for an electron with energy 2.50 eV? I am pretty sure that the...

The wave function of a particle near the barrier (or the "wall" of the finite potential well) is

 , where

 . Here,  is the depth of the well, and m and E are the mass and the energy of the particle, respectively. The constant  determines the penetration distance (depth), which equals  . This is a distance over which the wave function becomes 1/e of its initial value.

In the given problem,...

The wave function of a particle near the barrier (or the "wall" of the finite potential well) is

 , where

 . Here,  is the depth of the well, and m and E are the mass and the energy of the particle, respectively. The constant  determines the penetration distance (depth), which equals  . This is a distance over which the wave function becomes 1/e of its initial value.

In the given problem, the particle is an electron with the mass

and the energy E = 2.5 eV. 

The penetration depth is then

= 

This is the same as 0.28 nm, which approximately equals your answer. The discrepancy might be due to my rounding the Planck's constant (I used 4.14*10^(-15) eV*s instead of 4.136*10^(-15) eV*s.)