Interpreting quantum mechanics

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Quantum mechanics is one of the weirdest things to come out of the twentieth century1. It is so weird that Niels Bohr, one of the founding fathers of the theory, eventually concluded that it was fundamentally incomprehensible, while Richard Feynman (one of its greatest teachers) famously said, "If you think you
understand quantum mechanics, you don't understand quantum mechanics". It doesn't get much better than that.

Unfortunately we are stuck with this theory. Like it or loathe it, and a surprising number of people cannot make up their minds, it -- or something fundamentally similar -- is necessary to explain a number of important experiments done at the turn of the century, and without it almost no electronic device would work properly. Electron microscopes work because electrons, under certain circumstances, can be treated exactly like light particles -- or is it that electrons are in fact waves, just like light? What do we even mean by "particle" or "wave"? These sort of questions plague any attempt to understand -- well, anything, really.

Waves and Particles

First, a bit of background.

When scientists talk about something apparently commonplace like force or energy or quarks, they usually mean something highly specific and precise. "Force" is something which changes motion, "energy" is the ability to do work, and "quarks" are those six pints of beer in Finnegans Wake2. This precision is so that easily-understood concepts, like beer, can be talked about rapidly by those who need to clarify less well-defined ground -- and indeed, usually, the newer the science, the less precise or standard the terms. But particles and waves, with which this section deals, are both very old birds indeed.

Dots ...

A particle is usually interpreted to be a point-like object possessing mass4 and charge, and a number of other properties like position and momentum. Position anyone can figure out, and momentum is a quality which expresses, roughly, how fast and in what direction the particle is moving. The greater the particle's mass, the greater the momentum. Momentum is a quality which possesses direction -- which is to say that if two people weighing the same are both travelling at, say, five miles per hour, one North, one South, they have very different momenta indeed. The direction of travel is just as important as the speed.

So basically particles can be viewed as lots of little heavy dots moving in a straight line until something pushes them.

... and squiggles

A wave, on the other hand, is a lump of energy travelling through a medium. This is a rather imprecise definition, and indeed the theory of waves was rather imprecise for many years. To illustrate this more clearly (but not, unfortunately, more precisely -- that is a whole other article) imagine a bar made of infinitely5 bouncy foam rubber. As you hit one end, energy is stored in the piece of foam rubber nearest you, which is moving away from you. It compresses and nudges the adjacent piece of rubber, which now also has energy stored in it. However, the compressed rubber will resist the moving rubber -- and the force exerted on the original end lump stops it.

The next to end lump in is now moving and will nudge the next lump in turn, and so on, and it is easy6 to see that the energy will travel all the way down the rubber bar. This type of wave is called a compression wave. More generally, anywhere where something (usually energy in some form) can travel from one area to another and things like the concentration of the energy affect its rate of travel, waves will occur.

An important thing about waves: They can interfere with one another. The classic diagram for showing this has two sets of concentric circles one next to the other. Where two circles cross, or where two blank patches in between the circles cross, is a constructive interference fringe: the waves "add together" to make a wave twice as high. Wherever a line crosses a blank space in between is a destructive interference fringe: the waves "cancel out". You can try this out with water waves in a bath7.

Light entertainment

Light, as everyone knows, is a wave. Or a particle.

Squiggles ...

The reason for thinking that light is a wave is very simple: you can interfere two light waves in exactly the same way as with water waves. This is best illustrated by the famous Young's double slit experiment, which works as follows.

A single light beam of one frequency (ideally, in this modern age, a laser ... but Young managed without) is aimed at a double slit. The slits are separated by some tiny distance, and each slit is less wide than a single wave of the light. When the beam emerges at the other side of each slit, an effect known as diffraction causes the beams to separate into arcs of circles, and the diagram already mentioned with the concentric circles and the pattern they make shows exactly how these waves will affect one another. The double beam is played against a far wall and a pattern of light and dark bands -- corresponding to the constructive and destructive fringes of interference -- is seen.

So light is a wave.

... or dots?

Newton actually had some very interesting reasons for thinking light was a particle, but there is a much easier way to tell, thanks to the marvel of the solar cell, which was first studied in this way by Einstein8.

A solar cell is basically a metal plate with light shining upon it. Light knocks electrons from the metal loose, and this can be measured as a current -- the more light, the greater the current. All clear so far.

However, Einstein found that the if the light hitting the plate was below a certain frequency, there would be no current at all. Above this frequency, the current would be proportional to the intensity of the light. But below the critical frequency, no matter how intense the light beam, no electrons would be loosened.

Not one.

More than this: it was discovered that each individual electron emitted had energy proportional to the difference between the light frequency and the discovered minimum frequency. The method by which this was discovered was, again, a whole other article.

However, classical theory (which did predict interaction between light and electric charge) stated that the light energy should be absorbed continuously by electrons until they had enough energy to escape the atom -- rather like a large rocket engine feeding power to a spacecraft until it escapes the earth's pull. What seemed to happen instead was that each electron was receiving a discrete amount of energy all at once. To warp the spacecraft analogy, imagine a rocket booster which fires only once, and which has to be completely expended. If the booster is small, the spacecraft will reach a higher orbit, but not escape. If the booster is larger, the spacecraft will escape, and the speed at which it travels will be affected directly by the size of the booster. But the booster cannot be switched off, so the spacecraft will always attain the same speed for a given booster size.

It was as though the incoming light, instead of feeding energy to the electrons until they escaped the atoms, was feeding them rocket boosters of a fixed size, only one of which could be used at once9 -- explaining why the low-energy boosters never launched the electrons off the metal plate. The energy of the boosters was directly proportional to the light frequency, giving rise to the now famous equation:

E = hf

which relates energy E to frequency f with a number, h -- Planck's constant. The most important conclusion for this discussion, however, was that light comes in discrete chunks of energy -- each "booster" being a light particle -- called photons, or, originally, quanta.

Quantum Mechanics



Quantum mechanics, as the name suggests, is a theory devoted to the laws of motion that these quanta have. As we have already established, light, among other things, appears to behave as a wave, but comes in discrete chunks of fixed size, as though a particle. Much effort has gone into figuring out which one it is and why it behaves so strangely. To go further into the mysteries of quantum mechanics it is necessary to understand a bit about wavefunctions.

The strange theory of squots and diggles

A wavefunction is a mathematical relationship between position, time, and the probability of certain events. The wavefunction is a solution to the Schrdinger equation -- which means to say that if the function is put into the equation and a position and time value is put into the function, the resulting equation will be true10.

1Even weirder than Frank Zappa.2This is totally irrelevant, but Murray Gell-Mann named "Quarks" from the part in Finnegans Wake where someone brings "three quarks for master Mark", referring to six pints statute of stout, I suppose. The point is that in a proton or neutron, quarks come in triplets -- three per particle -- just like the "quarks" in Finnegans Wake. There is some controversy as to whether it should be pronounced "quoorks" to rhyme with "quarts" (of beer) or "quarrks" to rhyme with "Mark". I prefer the latter but many disagree, and this goes to show that not everything3 in science is as clear-cut and standard as you or I might think.3Least of all Murray gell-Mann, whom many regard as a genius simply because he was reading Finnegan's Wake for fun in the first place.4Mass is usually measured by weighing, and for this reason mass and weight are often confused, so to clarify: mass measures how much matter an object contains and weight measures how much force downward an object exerts under gravity.5Scientists will often bandy about terms like "infinitely" or "frictionless" or "perfect" as though they meant something. Try to humour them.6Another thing scientists will do is talk airily about what is easy. What they mean is, once you've understood it, it's easy; until you've understood it, it may, however, be almost impossible.7Have a towel handy, as all good hitch-hikers should.8This is what Einstein got his Nobel prize for, not relativity!9Actually, an electron can absorb the energy from two photons and thus escape from an atom, but the probability of this is surprisingly low.10If you aren't familiar with equations, try this one:

y = ax + b

The solution to this equation is the series of pairs of numbers, x and y, where y is b more than a multiplied by x. Differential equations are much more complex, but essentially, they deal instead with the rate of change of a quantity -- how fast a car is moving is the rate of change of its position, acceleration is the rate of change of its speed and so on. A solution to a differential equation is usually a set of functions, each of which is a set of sets of numbers11 -- just like the solution to the equation above. And so on. The Schrdinger equation is a differential equation, and each wavefunction is a set of positions and times and a value for each one.
11Each "point" in a function is a set of numbers. Examples would be position and colour on a television screen (six numbers: frame number, left, down, and red, green and blue intensities), or wind speed over time (seven numbers: position in latitude, longitude and altitude are common, followed by time, followed by north, east and up-down components of the wind speed). The set of sets is generated usually by running through all the possible values of one or more of the quantities. Thus in a television show we run through each frame and position on screen and measure the colour. Likewise for wind speed we would take each position and time and measure the speed. This is, of course, impossible in reality, but in a computer simulation of the weather it is exactly what is really done -- and the description of measuring a television picture is more or less how real television pictures are actually taken. Often functions ... hang on, perhaps an article is needed ....

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