Continuity…

Quantum mechanics undermines one of our most deeply held intuitions about the physical world, which is that nearly all changes that happen in the world occur in a continuous manner. The numbers we attach to physical quantities like velocity, acceleration, momentum and energy are all subject to change over time. When these changes happen, we typically understand that change to be smooth and continuous, not instantaneous and “jumpy”.

For example, if you are driving down the road at 50 mph and speed up to 60 mph, it seems obvious that while you are speeding up, for an instant anyway, you will briefly be traveling 55mph. Or 55.001 mph. Or 59.99999 mph. In other words, your speed changes smoothly from 50 to 60, and as it changes, it passes through every possible speed in between. The equations that govern your motion are referred to as “continuous”, and this term has a very precise mathematical definition in this context. But even if you are ignorant of the underlying calculus, I think it’s probably obvious to you that this is the way things must happen.

If you aren’t sure you agree, think about it this way – if your speed changed discontinuously at any point, even a little bit, from 55.000001 to 55.000002, then that would mean it changed by a finite amount – .000001mph – in zero time. Which would make your acceleration – your change in velocity over time – infinite! (Or at least mathematically “undefined: .000001 mph/0 seconds.) Wouldn’t that take an infinite amount of force? It doesn’t really make any sense to talk about discontinuous changes in the context of the classical mechanics of Galileo and Newton.

What we have said about velocity goes for all of the basic concepts of mechanical physics – momentum, energy, angular velocity, and so on. When these quantities change, they do so in a continuous way, governed by continuous equations. This sort of continuous change of variables is an underlying assumption of the calculus devised by Newton to describe the motions of physical objects in our universe.

Quantum mechanics, on the other hand, allows for some events occurring at the atomic and sub-atomic level to unfold very differently. Like any macroscopic object, an electron in an atom can change its energy. But this change does not happen continuously like the energy of an accelerating car. An electron might have 5 units of energy, and then absorb a photon of light, putting it in a state with 7 units of energy. But at no time in the process does the electron ever have 6 units of energy. In fact, it may be that the electron isn’t even “allowed” to have 6 units of energy while it’s a part of that atomic system. The changes in its energy are simply discontinuous.

This kind of “quantum jump” is the norm for subatomic particles. Electrons can absorb and emit discrete amounts of energy, change the direction of their spin by 180º, or even pass through solid barriers without ever occupying the intermediary states between. What are we to make of this?

We shouldn’t make too much of it, I think. In fact, I think we probably should have expected it. After all, there was a time when many people thought MATTER was continuous. That you could have a drop of water, or half a drop of water, or 1/4 of a drop of water… and if you kept halving that water over and over again – as many times as you wish – you would always still have water. It would still be “wet”. It would still be transparent. You could still float something in it. It would have all of the properties of water, no matter how small a piece of it you had.

This view of matter eventually fell out of vogue in favor of atomic theories of matter, which are based on the idea that there are some smallest “chunks” of matter that are indivisible. Or at least, if you were to divide them, the nature of the material would be changed. For example, if you keep dividing a drop of water in half, and half, and half again, eventually you will reach the level of an object which, when you cut it, would cease to be water. And presumably if you kept trying to divide the resulting material, you’d wind up with some smallest bits of matter which themselves were IMPOSSIBLE to cut. The etymology of the word atom is a from the Greek “a” (not) + “tomos” (cutting). That which is uncuttable.

Of course we know today that there is no “atom of water”. Water is a chemical compound made of hydrogen and oxygen. And even those hydrogen and oxygen “a-toms” are not the uncuttable primary bits of matter postulated by the Greek atomists or by Newton. We know that you can split an atom into component parts – electrons, protons, and neutrons. We even believe the protons and neutrons are further divisible into particles called quarks. But we still believe that there are some smallest indivisible fundamental particles in nature. Maybe electrons and quarks are examples of those, or maybe we are wrong again and there is some smaller chunk to be found. That doesn’t matter. What matters is that science has grown to accept a picture of nature in which matter is not continuous or infinitely divisible.

Quantum mechanics asks us to accept something similar about quantities like energy. Sometimes changes in energy are discontinuous, instantaneous, and “jumpy”. This isn’t the way things work at the everyday macroscopic scale, but so what? You don’t encounter objects are are fundamentally indivisible at the everyday macroscopic scale either. If the subatomic realm consists of tiny discontinuous chunks of matter, then why shouldn’t they interact by exchanging tiny discontinuous chunks of energy? Shouldn’t this behavior have been not just accepted, but expected? Maybe in retrospect. But it took physicists quite a while to accommodate the idea.