9. Particle in a one-dimensional finite depth square well.

Now we will consider a one-dimensional field in which the potential V is everywhere 0 except on the interval [0;L] where the potential energy of a particle is negative with constant value U0 < 0. Intervals like this are said one-dimensional potential wells with depth U0.


In Classical Mechanics a particle with positions 0<x<L and total energy Eqn000.gif lesser than 0 but greater than U0 can not escape from the well because outside it its kinetic energy K = Eqn000.gif-0 would be negative.


The Quantum Mechanics approach is the following:

Since the wave function must be continuous and differentiable in the entire field, for x=0 or x= L the two solutions and their derivatives must have the same values.

Moreover, since the Quantum Mechanics approach must asymptotically converge to Classical Mechanics, x→±∞ ⇒ ψ → 0.

For x=0 the two solutions and their derivatives must be equal. So



The same happens for x=L. So


From (9.6) and (9.8) we have


Finally, using the values of c and kI from (9.3) and (9.7), we have


By solving this equation with respect to Eqn000.gif for each value of n we can obtain the energy levels inside the well. There is no analytical way to calculate thise solutions, but they can be approximated at will with a numerical method beginning from n=1 and raising it while negative solutions are found.

Here there is a Javascript function that calculates energy levels of an electron in a one-dimensional well, given its width, measured in nanometers, and its depth measured in volts.

JS uses the simple bisection algorithm. The code is in the HTML source of this page.