Boundary Smoothness

Our first conclusion could be that electrons cannot be in the metal. That works if we only consider energy, but we must also consider the wave behavior of the electrons. Remember that the probability interpretation requires a smooth connection at the boundary. Wave functions such as the one in Figure 2 give zero probability inside the metal but fail the smoothness test.

 

Figure 2: This wave function is not smooth at the boundary.

 

While the wave properties are important, energy conservation must also be taken into account. Suppose electrons interact with a large region where they do not have enough energy to go. If we go far into that region, we do not expect to find electrons (Figure 3).

Figure 3: Energy considerations tell us that electrons will not be found on the far right for this situation.

So, the wave function in the metal must satisfy two independent criteria:


(i) It must be consistent with a zero probability of finding the electron far into the metal (energy).
(ii) It must connect smoothly to with the wave function on the left (wave behavior).

A clue to dealing with both of these criteria is given by the behavior of light waves. In special circumstances, light can exist where we think it should not be --- but only for a very short distance. When light penetrates into these regions, it is no longer an oscillating wave. Instead, the light's magnitude decreases rapidly. It connects smoothly with the oscillating wave but rapidly decreases its value to essentially zero. Figure 4 shows such a situation.

 

Figure 4: The wave function of the electron in empty space and the metal, when the electron's total energy is less than its potential energy in the metal.

The graph in Figure 4 also meets our criteria for the electrons. The wave functions in the metal and empty space connect smoothly. It decreases to zero to insure a zero probability of finding the electron in that region.

Generalizing from the wave function in empty space and the metal, as sketched in Figure 4 we can arrive at the following recipe for sketching the wave function:

A decreasing wave function does not necessarily have to decrease from a positive value to zero, it could also "decrease" from a negative value toward zero. For instance, another valid wave function to corresponding to the above physical situation is shown in Figure 5.


Figure 5: Another possible wave function for an electron that enters a metal where its potential energy is greater than its total energy

 

Make a sketch of the probability densities for the wave functions in Figure 4 and 5.
Describe the sketches you made below:

How do the probability densities for the two wave functions compare?

How do you interpret the probability densities in terms of the electron's location for these two wave functions?