8. Particle in a one-dimensional infinite depth square well.


To test the effectiveness of the Schrödinger equation in a very simple and well analyzed case, now we will work backward, to show that this equation allows to deduce energy levels and wave functions for a particle confined in a well bounded by insurmountable vertical barriers. 'Insumountable' means that beyond the barriers the potential is infinite. We suppose that between the barriers the potential is null and that the velocity of the particle is perpendicular to the barriers, so the problem is one-dimensional.

We have to resolve the equation (7.9), that is

Eqn001.gif

Since the problem is one-dimensional and the particle states are stationary, we can use a simplified version of (7.9):

Eqn002.gif

A function such that its second derivative with respect to its argument coincides with its opposite is the sine. So we may let

Eqn003.gif

Its derivatives are

Eqn004.gif

The second derivative coincides with (8.1) if

Eqn005.gif

Moreover the function (8.2) must be 0 when x=0 or x=L.

ψ(0) = 0 implies c=0.

ψ(L) = 0 implies

Eqn006.gif

and therefore

Eqn007.gif

These are the energy eigenvalues.

From (8.3) and (8.4) we have

Eqn008.gif

and, using this value of b in (8.2),

Eqn009.gif

To define the parameter a we use the normalization condition (5.13)

Eqn010.gif

Eqn011.gif

from which we have

Eqn012.gif

Ultimately, eigenfunctions are

Eqn013.gif