The extension of a particle’s wave functions into the classically forbidden region is an important difference between classical and quantum physics. Let’s take a closer look at the wave function in the region $x\ge L$ of Figure 40.13a. The potential energy in the classically forbidden region is $U_0;$ thus the Schrödinger equation for $x\ge L$ is

We’re assuming a confined particle, with $E$ less than $U_0$, so $E-U_0$ is negative. It will be useful to reverse the order of the two energies and write

where

is a positive constant. As a homework problem, you can show that the units of $\eta$ are meters.

The Schrödinger equation of Equation 40.34 is one we can solve by guessing. We simply need to think of two functions whose second derivatives are a positive constant times the functions themselves. Two such functions, as you can quickly confirm, are $e^{x\text{/}\eta }$ and $e^{-x\text{/}\eta }$. Thus, according to Equation 40.8, the general solution of the Schrödinger equation for $x\ge L$ is

One requirement of the wave function is that $\psi \to 0$ as $x\to \infty$. The function $e^{x\text{/}\eta }$ diverges as $x\to \infty$, so the only way to satisfy this requirement is to set $A=0$. Thus

This is an exponentially decaying function. Notice that all the wave functions in Figure 40.14a look like exponential decays for $x>L$.

The wave function must also be continuous. Suppose the oscillating wave function within the potential well $(x\le L)$ has the value ${\psi }_{\text{edge}}$ when it reaches the classical boundary at $x=L$. To be continuous, the wave function of Equation 40.37 has to match this value at $x=L$. That is,

This boundary condition at $x=L$ is sufficient to determine that the constant $B$ is

If we use the Equation 40.39 result for $B$ in Equation 40.37, we find that the wave function in the classically forbidden region of a finite potential well is

In other words, the wave function oscillates until it reaches the classical turning point at $x=L,$ then it decays exponentially within the classically forbidden region. A similar analysis could be done for $x\le 0$.

FIGURE 40.15 shows the wave function in the classically forbidden region. You can see that the wave function at $x=L+\eta$ has decreased to

Although an exponential decay does not have a sharp ending point, the parameter $\eta$ measures “about how far” the wave function extends past the classical turning point before the probability of finding the particle has decreased nearly to zero. This distance is called the penetration distance:

FIGURE 40.15 The wave function in the classically forbidden region.

A classical particle reverses direction at the $x=L$ turning point. But atomic particles are not classical. Because of wave–particle duality, an atomic particle is “fuzzy” with no well-defined edge. Thus an atomic particle can spread a distance of roughly $\eta$ into the classically forbidden region.

The penetration distance is unimaginably small for any macroscopic mass, but it can be significant for atomic particles. Notice that the penetration distance depends inversely on the quantity $U_0-E$, the distance of the energy level below the top of the potential well. You can see in Figure 40.14a that $\eta$ is much larger for the $n=4$ state, near the top of the potential well, than for the $n=1$ state.