The Schrödinger equation plays the role of Newton's laws and conservation of energy in classical mechanics — it predicts the future behavior of a dynamic system. It is a wave equation in terms of the wavefunction which predicts analytically and precisely the probability of events or outcome. The detailed outcome is not strictly determined, but given a large number of events, the Schrödinger equation will predict the distribution of results.
In the last episode we learned that Schrödinger and Heisenberg simultaneously discovered the equations that describe motion for tiny particles such as electrons and atoms. Although there was some initial confusion, Schrödinger quickly proved they were equivalent. In Heisenberg’s formulation operators incorporate a dependency on time, but the state vectors are time-independent.
It stands in contrast to the Schrödinger picture in which the operators are constant, instead, and the states evolve in time. The two pictures only differ by a basis change with respect to time-dependency, which is the difference between active and passive transformations. In other words, they were just two different views of the same phenomenon.
Heisenberg did, however, have a point about the physical aspect of Schrödinger’s theory. What was waving in Schrödinger’s matter wave? The mathematical representation of the wave is called the “wavefunction.” In some very real sense, the wavefunction of an object is the object. In quantum theory there is no atom in addition to the wavefunction of the atom. There is just the wavefunction.
But what, exactly, is Schrödinger’s wavefunction physically? At first, Schrödinger didn’t know, and when he speculated, he was wrong. So, let’s just look at some wavefunctions that the equations tell us exist. That’s what Schrödinger did.
The essentials of quantum mechanics can be seen with the wavefunction of a simple little thing moving along in a straight line. It could be an electron or an atom, for example. To be general, we usually refer to an “object.”
A couple of years before Schrödinger’s vacation inspiration, Compton showed that photons bounced off electrons as if they were each tiny billiard balls. On the other hand, to display interference, each and every photon or electron had to be a widely spread-out thing. Each photon, for example, had to go through both slits in the double-slit experiment. How can any object be both compact and spread-out too? Well, a wave can be either compact or spread out. (But, of course, it can’t be both at the same time.)
The waveform of a moving atom might look much like ripples, or a series of waves, a “wave packet,” moving on water. A wave equation, the one for water waves or matter waves, can describe a spread-out packet with many crests, or a compact packet with only a few crests, or even a single crest moving along.
For big things, object much larger than atoms, Schrödinger’s equations just turns into Newton’s universal equations of motion. Schrodinger’s equation governs not only the behavior of electrons and atoms, but also the behavior of everything made of atoms — molecules, baseballs, and planets given an initial wavefunction, it tells what the wavefunction will be like later. It’s the new universal law of motion. Newton’s equation is just an approximation for big things.
Schrödinger’s equation says a moving object is a moving packet of waves. But what is waving? Think of these analogies — Schrödinger no doubt did:
At a stormy place in the ocean, the waves are big. Let’s call that a region of large “waviness.” The boom of a drum, on its way to you from a distant drummer, is where the air pressure waviness is large, where the sound is. The bright patch where the sunlight hits the wall, the region of large electric field waviness, is where the light is. Waviness somehow tells where something is. It might seem reasonable to carry this notion over to the quantum case.
The waviness of a packet of quantum waves is large where the amplitude of the waves is large. Perhaps that is where the object is. (In quantum theory, the technical expression for the waviness is the “absolute square of the wavefunction.” By squaring we make the “negative” troughs add to the “positive” crests instead of subtracting since any number, when squared, is now positive.) The square of the wavefunction finds common use in quantum theory as a probability.
Not only is there a wavefunction for a moving atom, but we know parts within the atom move too. The electrons are in orbits around the nucleus. Early on, Schrödinger calculated the wavefunction off the single electron within the hydrogen atom and duplicated Bohr’s results for the experimentally observed hydrogen spectrum — without needing Bohr’s arbitrary assumptions. He was elated. He thought he had gotten rid of quantum jumps. He was wrong!
Modern descriptions of the orbits of electrons often refer to an “electron cloud.” That is, rather than a specific electron in orbit, it is more like a cloud. And it is a cloud … a “probability cloud.” You can visualize the waviness as clumps of fog. The fog is densest where the waviness is largest. Pictures such as these provide chemists with insight into how atoms and molecules bind with each other.
The wavefunction, being the object itself, actually includes everything knowable about an object, the velocity of an atom or its rate of spin, for example. So I’ve suggested that the waviness perhaps tells where the object is. It’s not quite like that … but close. But what exactly is the waviness?
Well, I see we’ve run out of time. You’ll just have to wait for the next chapter to learn of the modern interpretation of the wavefunction … just what it is!