Particle in a box
If we place a particle in a box with walls that have an infinitely high potential then the probability of finding the particle outside the box drops to zero. Our interpretation of the wave function as a "probability amplitude" such that
Y* Y represents a probability distribution requires that this quantity be equal to 1 between the limits of the box (0 and a). Y and Y*Y are zero outside of these limits. This is illustrated in the Figure below.
Given the boundaries there are now conditions on the wavefunction (boundary conditions). These are
Y(0) = Y(a) = 0. The general solution Y = Aeikx + Be-ikx will satisfy these conditions if A = - B = -i/2. In this case Y = sin(kx) which will be zero for k = np/a. Thus, Y = sin(npx/a) is a solution. Note that n is any integer and thus it is a quantum number. What are the energies?
These energies can also be obtain by inserting the solution into the Schrodinger equation (this is an exercise for the reader). The quantized energies and wave functions of the particle in a box are illustrated below.
Note that the number of nodes in the wave function increases as the energy increases.
Is the wave function normalized? Since we have a bounded area from 0 to a we must integrate
Y*Y from the limit of 0 to the limit of a. Thus, we ask
Substituting in our expression we have
Notice that for a real wave function the star in
Y* has no bearing. It is just the square of the wave function that need be considered.
This evaluates to a/2. Thus, the function is not normalized. How does one normalize a wave function. Let us assume that the normalization constant is N. Then
Y = Nsin(npx/a) and
is our normalization condition. Since we know that the value of the integral is a/2 we can solve for N.
Using this normalization constant we can ask questions about the probability of the particle being found in particular regions of the box.
The particle in a box is a useful example problem that has relevance for spectroscopy. It has been shown that the spectra of linear alkenes can be interpreted in terms of an electron in a box model. For example, the
p – p* transition in hextriene (shown below) can be modeled by counting the p electrons (there are six) using them to fill p energy levels two at a time (we still use Hund’s rule in the model) until we reach the highest occupied energy level.
In this case there are six p electrons and thus we will fill the first three levels.
For a synopsis of the key topics in the particle in a box see below. 