Figure 40.4, the potential-energy diagram for a particle in a rigid box, is an example of a potential well, so named because the graph of the potential-energy “hole” looks like a well from which you might draw water. The rigid box was an infinite potential well. There was no chance that a particle inside could escape the infinitely high walls.

A more realistic model of a confined particle is the finite potential well shown in FIGURE 40.13a . A particle with total energy $E<U_0$ is confined within the well, bouncing back and forth between turning points at $x=0$ and $x=L$. The regions $x<0$ and $x>L$ are classically forbidden regions for a particle with $E<U_0$. However, the particle will escape the well if it manages to acquire energy $E>U_0$.

FIGURE 40.13 A finite potential well of width $L$ and depth $U_0$.

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For example, consider an electron confined within a metal or semiconductor. An electron with energy less than the work function moves freely until it reaches the edge, where it reflects to stay within the solid. But the electron can escape if it somehow—such as absorbing energy from a photon—acquires an energy larger than the work function. Similarly, a neutron is confined within the nucleus by the nuclear force, but it can escape the nucleus if it has enough energy. The electron, the neutron, and many other particles that are confined can be modeled as a particle in a finite potential well, so it is one of the most important models in quantum mechanics.

FIGURE 40.13b is the same potential well, simply redrawn to place the zero of potential energy—which, you will recall, is arbitrary—at the level of the “energy plateau.” Both have width $L$ and depth $U_0$ so both have the same wave functions and the same energy levels relative to the bottom of the well. Which one we use is a matter of convenience.

Although it is possible to solve the Schrödinger equation exactly for the finite potential well, the result is cumbersome and not especially illuminating. Instead, we’ll present the results of numerical calculations. The derivation of the wave functions and energy levels is not as important as understanding and interpreting the results.

As a first example, consider an electron in a 2.0-nm-wide potential well of depth $U_0=1.0\text{eV}$. These are reasonable parameters for an electron in a semiconductor device. FIGURE 40.14a is a graphical presentation of the allowed energies and wave functions. For comparison, FIGURE 40.14b shows the first three energy levels and wave functions for a rigid box $(U_0\to \infty )$ with the same 2.0 nm width.

FIGURE 40.14 Energy levels and wave functions for a finite potential well. For comparison, the energies and wave functions are shown for a rigid box of equal width.

The quantum-mechanical solution for a particle in a finite potential well has some important properties:

• The particle’s energy is quantized. A particle in the potential well must be in one of the stationary states with quantum numbers $n=1\text{,}2\text{,}3\text{,}\dots \text{.}$

• There are only a finite number of bound states—four in this example, although the number will be different in other examples. These wave functions represent electrons confined to, or bound in, the potential well. There are no stationary states with $E>U_0$ because such a particle would not remain in the well.

• The wave functions are qualitatively similar to those of a particle in a rigid box, but the energies are somewhat lower. This is because the wave functions are slightly more spread out horizontally. A slightly longer de Broglie wavelength corresponds to a lower velocity and thus a lower energy.

• Most interesting, perhaps, is that the wave functions of Figure 40.14a extend into the classically forbidden regions. It is as though a tennis ball penetrated partly through the racket’s strings before bouncing back, but without breaking the strings.