Absolutely Small - Michael D. Fayer [44]
FIGURE 8.1. A perfect one-dimensional box. The walls are infinitely high, infinitely thick, infinitely massive, and completely impenetrable. There is no air resistance in the box. In the box, Q, the potential energy is zero, and outside the box, it is infinite. The box has length, L.
Figure 8.2 shows a racquetball bouncing off of the walls in a perfect one-dimensional classical (big) racquetball court. As discussed, the walls are perfect, and there is no air resistance. In addition, the ball is perfect, that is, it is perfectly elastic. When a ball hits a wall, it compresses like a spring and springs back, which causes the ball to bounce off the wall. A real ball is not perfectly elastic. When the ball compresses, not all of the energy that went into compressing the ball goes back into pushing the ball off of the wall. Some of the energy that went into compressing the ball goes into heating the ball. However, here we will take the ball to be perfectly elastic. All of the kinetic energy of the ball that compresses it when it hits the wall goes into pushing the ball off of the wall. Therefore, the speed of the ball just before it hits the wall is equal to the speed of the ball after it bounces off.
FIGURE 8.2. A ball in a perfect one-dimensional racquetball court. There is no air resistance, and the ball is perfect. When the ball strikes the wall at L, it bounces off, hits the wall at 0, and keeps bouncing back and forth. Because the court is perfect, the ball is perfect, and there is no air resistance; once the ball starts bouncing, it keeps bouncing back and forth indefinitely.
In this perfect racquetball court, the ball bounces off of the walls without losing any energy and there is no air resistance or gravity. Therefore, the ball will bounce back and forth between the walls indefinitely. It will hit the wall at position L, bounce off, and then hit the wall at position 0, bounce off, and just continue bouncing back and forth. Because the potential inside the box is zero (see Figure 8.1), there are no forces acting on the ball. Therefore, its energy is purely kinetic energy, ; m is the mass of the ball and V is its velocity. If the ball is hit a little bit harder, it will go a little bit faster, that is, V will be a little bit bigger. Ek will be a little bigger. If the ball is hit somewhat more gently, the ball’s velocity will be a bit smaller, and Ek will be a little smaller. In this perfect racquetball game, the energy can vary continuously. Ek can go up or down by any amount, the amount depending only on how hard you hit the ball.
Another important feature of classical racquetball is that it is possible to stop the ball and put it on the floor. In this situation, the ball has no velocity, V = 0. If V = 0, then Ek = 0. If V = 0, then the momentum is zero because p = mV. So we know the momentum exactly. If the ball is placed on the floor with V = 0, then the position is known. If the position is called x (see Figure 8.2), x can take on values between 0 and L. It cannot have any other values because the ball is inside the court (the box) and can’t get out because of the perfect walls. The ball can be placed at a certain position x on the floor of the court. Therefore, its position is known exactly. This is a characteristic of a macroscopic racquetball court, even a perfect one. It is a classical system, and it is possible to know simultaneously both the momentum, p, and the position, x, precisely.
A racquetball court is 40 feet long (about 12 m), and a ball is 2.25 inches in diameter and weighs 1.4 ounces (about 0.04 kg). Clearly, racquetball is a game describable by classical mechanics. You can watch the ball bounce back and forth by observing it with light without changing it.
PARTICLE IN A BOX—QUANTUM
What are the differences if we now consider quantum racquetball? The court is still perfect, but now its length is 1 nm (10-9 m) rather than 12 m. Furthermore, the particle now has the mass of an electron, 9.1 × 10-31 kg rather than 0.04 kg. This is the quantum particle in a box problem.