We have come to the shape of the wave function where the electron has a negative kinetic energy by logic and analogy. If we liked heavy-duty mathematics we could get the same result using Schrödinger's Equation. Solving the equation also tells us the dependence of the rate of decrease of the wave function on the difference between the total energy and kinetic energy. But, you can probably use logic and a little intuition to get the general idea.
Consider the two situations in Figure 6:
In which of the two cases in Figure 6 will the wave function in the metal decrease more rapidly? Explain your answer.
Your intuition probably served you well. The larger the difference between the potential and total energies, the more rapidly the wave function decreases.
Using Schrödinger's Equation we can make the conclusion quantitative. We define a decrease length as a distance from the boundary to where the wave function has dropped by a specified fraction. The decrease length is
In a distance equal to the decrease length the value of the wave function drops to about 0.4 of its value at the boundary (See Figure 7). In two times the decrease length the value is (0.4) x (0.4) = .16 of its value at the boundary. And so forth.
This equation works for any object. It states that any object has a probability of being in areas where it does not have the energy to be. The physics of large objects tell us that this cannot happen, but wave nature of matter says that it must.
To see how we can reconcile the physics of Newton with that of Schrödinger, consider the gnat discussed in Module C tutorial "Electrons behave as waves". It has a mass of 0.001 kg and a speed of 0.10 m/s. Suppose that it hits a wall with a potential energy of 1 Joule. (Planck's constant = 6.63 ´ 10-34 J·s)
What is the decrease length for the gnat? (An approximate answer is good enough.)
As you can see, the length is so small that Newton would never have noticed. Even with today's technology we could not measure such a small distance.
The equation can be used for other objects. As with the de Broglie wavelength the magnitude only becomes useful when the mass is very small. For electrons the decrease length becomes
The energies must be in electron volts and the result is in nanometers.
Calculate the decrease length of an electron Cases A and B in Figure 8.
Sketch approximately the wave function and probability density for each case. Describe your sketches below: