Suppose we confine an electron in a microscopic box, then allow the box to get bigger and bigger. What started out as a quantum-mechanical situation should, when the box becomes macroscopic, eventually look like a classical-physics situation. Similarly, a classical situation such as two charged particles revolving about each other should begin to exhibit quantum behavior as the orbit size becomes smaller and smaller.

These examples suggest that there should be some in-between size, or energy, for which the quantum-mechanical solution corresponds in some way to the solution of classical mechanics. Niels Bohr put forward the idea that the average behavior of a quantum system should begin to look like the classical solution in the limit that the quantum number becomes very large—that is, as $n\to \infty$. Because the radius of the Bohr hydrogen atom is $r=n^2{a}_{\text{B}}$, the atom becomes a macroscopic object as $n$ becomes very large. Bohr’s idea, that the quantum world should blend smoothly into the classical world for high quantum numbers, is today known as the correspondence principle.

Our quantum knowledge of a particle in a box is given by its probability density

To what classical quantity can the probability density be compared as $n\to \infty ?$

Interestingly, we can also define a classical probability density $P_{\text{class}}\left(x\right)$. A classical particle follows a well-defined trajectory, but suppose we observe the particle at random times. For example, suppose the box containing a classical particle has a viewing window. The window is normally closed, but at random times, selected by a random-number generator, the window opens for a brief interval $\delta t$ and you can measure the particle’s position. When the window opens, what is the probability that the particle will be in a narrow interval $\delta x$ at position x?

The probability of finding a classical particle within a small interval $\delta x$ is equal to the fraction of its time that it spends passing through $\delta x$. That is, you’re more likely to find the particle in those intervals $\delta x$ where it spends lots of time, less likely to find it in a $\delta x$ where it spends very little time.

Consider a classical particle oscillating back and forth between two turning points with period $T$. The time it spends moving from one turning point to the other is $\frac{1}{2}T$. As it moves between the turning points, it passes once through the interval $\delta x$ at position $x$, taking time $\delta t$ to do so. Consequently, the probability of finding the particle within this interval is

The amount of time needed to pass through $\delta x$ is $\delta t=\delta x\text{/}v\left(x\right)$, where $v\left(x\right)$ is the particle’s velocity at position $x$. Thus the probability of finding the particle in the interval $\delta x$ at position $x$ is

You learned in Chapter 39 that the probability is related to the probability density by

Thus the classical probability density for finding a particle at position $x$ is

where the velocity $v\left(x\right)$ is expressed as a function of $x$. Classically, a particle is more likely to be found where it is moving slowly, less likely to be found where it is moving quickly.

FIGURE 40.11a is the motion diagram of a classical particle in a rigid box of length $L$. The particle’s speed is a constant $v\left(x\right)=v_0$ as it bounces back and forth between the walls. The particle travels distance 2$L$ during one round trip, so the period is $T=2L\text{/}{v}_0$. Consequently, the classical probability density for a particle in a box is

FIGURE 40.11 The classical probability density is indicated by the density of dots in a motion diagram.

$P_{\text{class}}\left(x\right)$ is independent of x, telling us that the particle is equally likely to be found anywhere in the box.

In contrast, FIGURE 40.11b shows a particle with nonuniform speed. A mass on a spring slows down near the turning points, so it spends more time near the ends of the box than in the middle. Consequently the classical probability density for this particle is a maximum at the edges and a minimum at the center. We’ll look at this classical probability density again later in the chapter.

FIGURE 40.12 shows the quantum and the classical probability densities for the $n=1$ and $n=20$ quantum states of a particle in a rigid box. Notice that:

• The quantum probability density oscillates between a minimum of 0 and a maximum of $2\text{/}L$, so it oscillates around the classical probability density $1\text{/}L$.

• For $n=1$, the quantum and classical probability densities are quite different. The ground state of the quantum system will be very nonclassical.

• For $n=20$, on average the quantum particle’s behavior looks very much like that of the classical particle.

FIGURE 40.12 The quantum and classical probability densities for a particle in a box.

As $n$ gets even bigger and the number of oscillations increases, the probability of finding the particle in an interval $\delta x$ will be the same for both the quantum and the classical particles as long as $\delta x$ is large enough to include several oscillations of the wave function. As Bohr predicted, the quantum-mechanical solution “corresponds” to the classical solution in the limit $n\to \infty$.