Though it wasn’t fully recognized during Planck’s time, his quantum hypotheses can be generalized beyond oscillating charges in blackbodies to any oscillating system, be it a little girl on a backyard swing, a sloshing wave of seawater, or a retired thrill-seeker dangling from a bungee cord. In contrast to our artificial “quantum staircase” from the previous section, these systems are quantized by the laws of quantum physics.
But, you may interject, scientists have dealt with springs and waves and pendulums for centuries. Why didn’t they ever notice quantization before? Let’s examine the first example to see.
Suppose that a 10-kilogram girl is seated comfortably on a swing that hangs 3 meters below a tree branch. You give her a few pushes so that at the highest point her swing is at an angle of about 30 degrees from vertical. A simple application of Newton’s laws tells us that, when at the highest point, her gravitational potential energy is about 40 joules relative to her starting point.
If you were to stop pushing, her swinging will begin to slow, her maximum height will gradually diminish, and she will eventually come to a halt. The reason is that there is friction at the point where the swing’s rope meets the tree branch above her, plus air all around that drags like friction as she moves through it. When she returns to the lowest point, her gravitational potential energy will return to zero.
Now, Accordingly to Planck’s hypotheses, her energy cannot drop smoothly and continuously from 40 joules to zero. It must drop in finite, discrete bursts. Surely you (or at least she!) would notice, right? Fortunately Planck gave us all the tools we need to provide the answer.
He told us that the minimum quantum of energy for an oscillating system is given by the system’s frequency times Planck’s constant, the value of which he published in his seminal paper. Since Newton’s law tells us that the girl is swinging at approximately 0.3 Hz, we can easily calculate the minimum quantum of energy as 0.3 Hz times Planck’s constant. Here goes: 2 × 10-34 joules. To emphasize this important point, let’s write that out in decimal form: 0.0000000000000000000000000000000002 joules!
Herein lies our answer. Since the minimum quantum of energy is so incredibly small compared to the little girl’s gravitational potential energy, she will never notice the small, discrete bursts as her swing comes to a halt. The quantum of energy is simply too small. For all intents and purposes, her decrease in energy will appear to be entirely continuous.
In fact, for the little girl to begin to feel the quantum effects, she would need to be about 1,000,000,000,000,000 times smaller and the swing would need to be about 1,000,000,000,000,000 times shorter. This example illustrates that quantum physics reigns not in the world of little girls, but in the world of tiny, microscopic entities.
QUANTUM QUOTE
It has shattered the foundations of our ideas not only in the realm of classical science but also in our everyday ways of thinking.
—Niels Bohr on the impact of Planck’s quantum hypotheses (Die Naturwissenschaften, vol. 26, 1938)
To explore this branch of physics, then, we are going to need some tiny tools. Fortunately, Mother Nature has given us a few that, though very small, are just within our grasp. The first of these is light, which oscillates with a wavelength of about 1 × 10-7 meters. The other is the atom, which, as we saw has an approximate diameter of about 1 × 10-10 meters. Given the utility of these to the study of quantum physics, we will study them each in turn, this time from a quantum-physics perspective.
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