Mike Hosken's

"Hinges and Loops"



  1. Atoms
  2. Bits of Atoms
  3. Unimaginable Models
  4. Quanta
  5. Photons
  6. Quantum Mechanics - Waves
  7. Quantum Mechanics - Probabilities
  8. Conclusions?
  9. Uncertain Coda
Of course the atoms and molecules that go to make up Chapter Four's living things are also bits of the physical universe. What we are to be thinking about in this Chapter are some very broad ideas that concern virtually everything in the known universe, living and non-living, worldly and astronomic, whether obvious to the senses or detectable only by sophisticated instruments.

§5.1 Atoms

Bearing in mind that our subject will soon, in Chapter Six in fact, be the nature of knowledge itself it may be amusing or instructive or both to realise that the first atomic theory that we know anything about was put forward in the fifth century BC. The Greek thinkers Leucippus and Democritus came to believe that everything in the physical world was made up of indestructible and indivisible little chunks that were in constant motion, and with empty space in between. Lots of things might easily be indestructible but what was essential about those little chunks was that they were indivisible - atamos in Greek - so they came to be called atoms.

It was not until 1808 AD that Dalton put forward what is usually thought of as the first fully scientific atomic theory. By that time a lot of different chemical elements had been identified: it was known that those elements could combine in regular proportions by weight to form compounds. Those regularities might be explained by supposing that each element was made up entirely of just one type of indivisible atom, and that different elements had different atoms. Dalton's chemistry had suggested a tidy parallel -

elements unite in fixed ratios to form compounds
atoms combine in fixed ratios to form molecules.

Again, indestructibility could be taken for granted ("the law of conservation of mass" and all that), so the nub of the argument rested on the indivisibility of the smallest bits of each different element. A lump of carbon, say, had to be a sort of digital rather than an analogue concept: the total quantity of nitrogen gas in a jar could be increased or decreased by a whole number of whole atoms, not by just any old arbitrary fractional amount. Admittedly the atoms had to be too tiny to be discernible, since in practice the elements could be measured out to any desired standard of accuracy. A century and a half later F.S. Taylor was able to write that "one could put as many oxygen molecules on a full-stop as one could put full-stops on Kensington Gardens (c. 275 acres)". Perhaps even more imaginatively, the chemist Sidgwick calculated that "If a tumblerful of water is poured into the sea, and in the course of time this becomes uniformly distributed through the sea, the rivers and all the other waters in or surrounding the earth; and if then a tumblerful of water is taken from any sea or river, this will contain about 1,000 of the molecules that were in the original tumbler."

§5.2 Bits of Atoms

Having defined and named "atoms" for their indivisibility, it was not terribly long before science began to consider what bits they might be made up of! Rutherford and Bohr had sown the seeds at the beginning of the twentieth century. We don't need to follow the evolution of the ideas in detail but a thumb-nail sketch as at 1935 or so may be a useful stopping-off point. The modern atom by then consisted of -
  • a central nucleus made up of two kinds of heavy units -
    • protons each of which carried a positive electric charge.
    • neutrons which had no electric charge.
  • relatively light electrons, each of which carried a negative electric charge, circulating round the nucleus, not as a random cloud but in a definite pattern of several orbits. In normal atoms the number of electrons was the same as the number of protons in the nucleus, so that the whole thing was electrically balanced.
Most atoms were evidently fairly complicated: hydrogen was the simplest, uranium the most complex. Mid-range came such examples as -
  • Carbon with a nucleus of 6 protons and usually 6 neutrons, plus 6 electrons in 2 orbits (2 in the inner one, 4 in the outer).
  • Sulphur with a nucleus of 16 protons and usually 16 neutrons, plus 16 electrons in 3 orbits (2 in the inner one, 8 in the next, 6 in the next).
  • Lead with a nucleus of 82 protons and 124 or 125 or 126 neutrons, plus 82 electrons in 6 orbits (2 in the inner one, then 8, 18, 32, 18, and 4 in the outermost).
The little chunks that made up the Leucippus/Democritus atoms were easy enough to visualise. They had to be miniature versions of the material that they made up. The complex picture presented by Rutherford and Bohr was much more challenging. There were not too many difficulties at first sight. Textbooks of the time could show a lumpy sort of nucleus made up of the protons and neutrons clumped together, apparently like frog spawn. The electrons were evidently little "things" of some sort which went round the nucleus just as planets go round the sun. But quite early on in the development of such a solar-system type of model difficulties turned up in matters of detail. In particular, those electrons were difficult to conjure with. If each electron were indeed a "thing" whizzing or even ambling round its "sun" nucleus then energy had to be involved. After all, Newton had long ago sorted out the reasons why the earth and other planets didn't either fly off into deep space or "fall" into the sun: his laws of motion and gravity had explained virtually everything that needed explaining in such matters on earth and in the heavens. But by the time you got down to the size of an atom and started being concerned with negatively charged "things" spinning round an electrically positive "sun" you needed a different sort of model altogether.

§5.3 Unimaginable Models

We must break off at that point to go on a bit of a detour. "Imagination" is of course a word derived from "image". If we can't see or touch something with our physical senses we can nevertheless often make up something which we can handle mentally by building images for ourselves. We do it all the time, in all walks of life, in all aspects of thought.
  • The familiar mental image or an orrery model of the solar system is useful even if it is grossly inaccurate. The whole solar system is in reality a pretty vast volume of space with, here and there more or less in one flat plane, a few minute spots of matter.
  • Images of heaven and hell have exercised the minds and talents of artists and poets since time immemorial, in virtually every culture of the world.
  • Similarly, all cultures handle models of their deities. They are fairly simplistic when first introduced to young children, usually becoming more nebulous as the level of understanding increases - disappearing altogether in the case of Father Christmas!
But I want to consider one case in greater detail and take it as an exemplar. Mention of dimensionality is enough to make many a brave soul quake, so let's just take it steadily and see where it leads us....
  • Think of the ruler on your desk. The number "3" is on there, part-way along - in fact three units distant from one end. So for all practical purposes each number (and each of the fractional marks in between the numbers) has just one dimension - a distance from the end of the ruler. You could use your ruler to draw a line on paper with a fine pen. That line would have length but no width or depth: it would have only one dimension.
  • All surfaces have two dimensions. This star - * - is a measurable distance down (or up) the page and a measurable distance in from the left (or right). Any place on the earth can be allocated its geographical position using the two standardised dimensions of latitude and longitude. Or you can focus in in greater detail by finding on a UK map the figures for Eastings (first dimension) and Northings (second dimension) to give the Ordnance Survey Grid Reference of any feature or location.
  • All volumes have three dimensions. Just look in any furniture catalogue, for instance, to be given the height, width and depth dimensions of cupboards, fridges, tables and what not.
Now in ordinary everyday life the usual thought is that we've got to stop there: we can't add any more dimensions - we can't get any more solid than a solid! But in fact we can, indeed must. The slight snag is that we can't measure the next dimension with a ruler: it's a different sort of dimension altogether but is still nevertheless a dimension. It is TIME, familiar enough on its own but not usually "modelled" or imagined as being like height, width and depth. Let's take a few cases.
  • "Go and meet Auntie Flo's train" is an instruction for which we have to have two bits of detail - where is the station and when is the train due.
  • If I said that the full Grid Reference of my house was, say, XY 456123 that would indeed specify a spot in the UK to within a very few metres. But fully to specify my house you would have to add a time dimension since it has been at that spot only since about 1750 AD: it is unlikely to be still here in 7150 AD.
  • If you were involved with air traffic control the whole essence of the scheme would be the need to know, for every aircraft, its latitude (dimension 1), longitude (dimension 2), altitude (dimension 3), at any time (dimension 4). Miss out any one of the four dimensions or get any one of the four dimensions wrong and disaster could result.
  • Although we do indeed live in a four-dimensional world we rarely have to worry about all four at once: you probably thought I was being a bit pedantic about the time dimension of my house - although you would be rather upset if I sold it to you after it had burned down! But if we are navigating a yacht into an estuary the dimensions we do have to worry about are position (made up of latitude and longitude) and time. We are likely to run aground if our measures of surface position (lat/long) or time (the state of the tide) are wrong. [A purist might argue that the state of the tide is mainly of concern because of the effect on the altitude of sea level, the fourth dimension in this case.]
Having gone from a one-dimensional line to a two-dimensional surface and from a three-dimensional volume to a four-dimensional world we must surely have to stop there. And in terms of images, yes we do. But not in terms of models. We shall start again, using an example readily familiar in principle to computer programmers.
  • If you want to store just one piece of information then you can, on a computer or in your imagination, make a memory box, give it a label - A, perhaps - and pop in your information.
  • If you have got a list of data to store then you can set up what computer aficionados call an array but which we can think of as a chest of drawers. The whole thing can be labelled A again, but now each item, or each drawer containing an item, needs a number too. So a particular memory location or drawer might be known as A(5) for example.
  • But if the information requirement were more complicated then the array may need to be two-dimensional: each memory location needs two dimensions to specify it, like perhaps A(3,8). We can visualise this as being like a set of chests of drawers: so A(3,8) is the eighth drawer in chest number three. Two-dimensional arrays are very useful in handling data: a tabulation, for example, can take the form of many lines each of many columns.
  • You can guess where we go next - to a three-dimensional array which includes perhaps A(6,2,9) which we can think of as corresponding to the 9th file in drawer 2 of filing cabinet number 6.
  • And so on to four dimensions such as memory location A(12,5,3,19), with the 19th word of page 3 of Chapter 5 of volume 12 of your encyclopaedia.
  • But this time we don't have to stop at four dimensions. The computer "language" I usually use is capable of coping with up to 255 dimensions, though it would be unbelievably tedious to try and do anything useful in that way and a more modest example will make the point more sensibly. A hardware stockist will want to keep track of all his goods. Things like screws and bolts and specialist fitments may be in bins on shelves in bays of sections of multiple warehouses on several sites. So stored in data memory location A(2,1,4,3,3,5) might be the current number of 1" stainless steel countersunk wood screws in stock in bin 5 on shelf 3 of bay 3 of section 4 in warehouse 1 on site 2. So we can, after all, handle if not visualise six dimensions. Certainly by being careful with our keyboarding and making use of the computer's display and cross-checking ability it is quite simple to live with as many dimensions as necessary.
Now the whole point of this diversion is that we find that we can handle ideas that we can't visualise. Not many of us are mathematicians but it would certainly be true to say that mathematics can deal with things that are unimaginable. Perhaps the best-known example is i. Not me or I but little i, which in mathematics stands for the square root of minus one. But surely even the worst of us remember from school that -
  • square root means "something" where "something" multiplied by itself gives the number in question, as the square root of 9 is 3 since 3 times 3 = 9.
  • any "minus something" multiplied by a "minus anything" gives a positive or plus answer.
Quite right! So there simply cannot be any number, either positive or negative, which when multiplied by itself gives a negative number.

-2 squared = 4 just as surely as 2 squared = 4
-3 squared = 9
-½ squared = ¼
but now we are told that
i squared = -1

My point is that there is no way that any mental image of i can be formed: as luck would have it, mathematicians describe numbers making use of i as being "imaginary". Be that as it may, it is indisputable that imaginary numbers turn out to be not only interesting but useful. And useful not only in creating an imaginary mathematical world but in explaining or at least describing the real world of physics, in generating electric currents for instance (although in electrical work the small letter j is conventionally used in place of i).

So we must again conclude that non-visualisable ideas can be useful and relevant in the context of the real world.

§5.4 Quanta

We can come now, at last, to grapple with the problem of those electrons in their orbits. We have already seen that they cannot be like little electrically charged planets orbiting the oppositely-charged nucleus. Worse than that, when Neils Bohr and his successors came to study the electrons and how they behaved it became clear that in fact they couldn't be "somethings" at all! Not, that is, if we use the word "something" to stand for an everyday bit of ... well, of something. Just for starters, they discovered (but don't ask me how) that electrons could jump from orbit to orbit under certain circumstances. But unlike anything that science had studied up to that date -
  • an electron which disappeared from one orbit reappeared in a different one without having moved across the space in between.
  • energy was released by jumps outward, absorbed by jumps inward. But that energy was always in units of regular amounts. Max Planck had already found that heat-energy radiation was always in similar units which he had called quanta. Energy was evidently a digital concept.
At roughly the same time Einstein threw another spanner into the conceptual works with his deceptively innocent-looking -

Energy = mass times the speed of light squared

To be a mathematical equation the units of energy, mass and speed have to be chosen correctly but our discussion need not bother with such detail. The speed of light is huge: that figure squared is quite tremendous. What the equation means, then, is that if you have a little bit (a modest mass) of suitable matter then it could be possible to transform it into a vast amount of energy. It is not necessary in these days of nuclear energy and atomic bombs to question the validity of the idea.

But it is clearly relevant to our discussion in terms of visualising what matter really is. Is it possible to convert matter into energy, or is it more true to say that matter is just a form of energy? If the second option were true it might help to explain the way those electrons appear and disappear. A brick can't be moved from A to B without going through the space in between, but perhaps a mini-packet of energy might. The way bricks behave is governed by the rules of mechanics: the study of such things as electron behaviour came to be known as quantum mechanics. One of Bohr's remarks provides an illuminating warning: "You cannot interpret a quantum mechanical statement using ordinary common-sense language and ideas".

It would be nice if I could describe for you an experiment or demonstration in the tradition of "Blue Peter" or of "Take Nobody's Word For It" which would actually prove that the world is indeed quantum mechanical at the small scale as well as being Newtonian mechanical at the scale of houses and machinery. But a do-it-yourself atomic reactor would be a bit ambitious: a cyclotron plus cloud chamber rather expensive. In any case, ready supplies of electrons are usually associated with rather risky high voltages.

§5.5 Photons

But another quantum concept is the photon. This has many parallels with the electron but is the digital unit of light energy rather than electrical energy. Nothing, but nothing, could be more "everyday" and "common-sense" than light, but unfortunately photon effects can be detected only at very very low levels indeed. It is thought that some cells in the retina at the back of the human eye can react to photons arriving one at a time. Certainly the modern sophisticated high-tech video night cameras and astronomical amplifier telescopes can do so, at a price.

But the best I can do is to describe the classic quantum mechanical double slit experiment. Although this is simple enough in principle it's not the kind of thing that can be set up at home with a torch and bits of cooking foil, again because of the scale of the things being investigated. But in terms of visualising what happens it may be easiest to think in domestic terms. So imagine if you will a small but powerful light source, such as a super torch bulb (but with no reflector to complicate matters). Unfortunately it has to be a miniature version of the orange sodium street lights in order to give the right sort of single colour (single wavelength) light. An inch or two in front of it is a wall of metal foil which has one very thin slit cut carefully in it. In fact the slit must not be more than about a thousandth of a millimetre wide, although the length doesn't matter particularly. Light passing through the slit falls onto a screen and can be observed or measured or photographed. The result is not a bright bar on the screen due to the light going straight from the lamp through the slit and continuing straight on to the screen. What happens is that the slit itself acts as if it were a smaller light source. This happens because the light "squeezing" through the slit gets bent round at all sorts of angles so that it hits the screen almost anywhere. The intensity is greatest just opposite the slit but there is no shadow edge: instead, decreasing amounts of light reach the screen to either side. On a massively greater scale, sea waves behave in the same way if they can pass through a quite small opening through a sea wall, spreading out into an open harbour beyond.

OK so far. But now we add another slit, again a millionth of a metre wide, parallel to the first one, something between one and two tenths of a millimetre away from it. If we temporarily screen off the first slit we can check that light passing through the second slit behaves in exactly the same way as it did with the first slit.

But with both slits open together we do not simply get the first-slit-pattern and the second-slit-pattern superimposed on the screen. Instead, what happens is that bands of light and dark are formed parallel to the slits. Now these are easy enough to explain in terms of waves. Recall that all the waves we are using are of the same wavelength - our monochromatic sodium light. So in some places on the screen the waves reaching the screen via slit 1 will be reinforced by waves which have arrived via slit 2: a bright band appears in such places. Conversely, slit 1 "up" waves will in other places be exactly counteracted by slit 2 "down" waves so that nothing reaches the screen and a dark strip results. Again, water waves would behave in just the same way: the classic school demonstration of wave reinforcement and interference used water in a two-compartment glass tank with two narrow gaps in the dividing wall.

So there we are. A totally self-consistent experiment and explanation. No problem! Until, that is, we try thinking about it in a quantum mechanical context. We now know, apparently, that light is in the nature of a stream of discrete, digital photons. Thinking first about the single slit, we may be left to wonder why and/or how a lot of the photons that do "squeeze" through the slit (at the speed of light, remember - 186,000 miles per second!) - to wonder what it is that diverts their course instead of letting them simply go straight on and hit the screen as a bright image of the slit. Curious, though perhaps not conceptually challenging.

But what of the two-slit situation and the pattern it produces? Naïvely we assume that each photon reaching the screen must have gone through either slit 1 or slit 2. Now we know that there are light and dark stripes: by definition a dark stripe is where no photons are arriving. Let's consider some arbitrary spot on the screen which is in the middle of a dark stripe: we'll call that dark spot D.
When slit 1 only is open photons do reach D.
When slit 2 only is open photons do reach D.
When BOTH slits are open NO photons reach D.
Now that takes some explaining! Think of one specimen photon setting off from the lamp. It is one of the fortunate ones that happens to be headed straight for one of the two open slits, let's say slit 1. So far as slit 1 alone is concerned our photon can (among other options) take on a course to point D and contribute to some brightness there. But since slit 2 is also open it is not allowed to. There are only a few possible explanations.
  • The photon "knows" when there are two slits open even though it can only go through one of them. But a photon is not a conscious being. In any case, it would have to be able to "see" the other slit, which is impossible since it is itself moving at the speed of light and no enquiry could get to the second slit and back with an answer in the time available: both quantum mechanics and Einsteinian relativity would prohibit it.
  • The photon splits into two parts, half going through each slit and re-uniting on the final leg to the screen. A moment's thought about matters of detail shows the impossibility of such a mechanism.
  • The photon itself passes through both slits. But, well, wasn't each photon supposed to be a single, unitary, digital something-or-other? So that can't be right either.
Before sinking any further into the quagmire perhaps we should bring some other, related thoughts together.
  • Electrons don't seem to be "somethings" at all.
  • Photons don't behave like "somethings".
  • "You cannot interpret a quantum mechanical statement using ordinary common-sense language and ideas".
Evidently we must take Bohr's advice and abandon common sense. Atoms do not behave like little billiard balls. Electrons are not planets in miniature. Single "things" that can take two routes at once cannot really be "things" at all.

Now this is not to deny the usefulness of common sense, nor of Newtonian mechanics, classical physics and so on. But instead of concentrating on the negative aspects - on ideas that must be abandoned in the contexts we are considering - we must move on to the more positive contributions made by quantum mechanics.

§5.6 Quantum Mechanics - Waves

And we had better begin with another quotation from Neils Bohr. "Anybody who is not absolutely shocked by the statements of quantum theory simply hasn't understood the theory."

My own warning is this: having abandoned ordinary common-sense language and ideas we are left only with mathematics. So any attempt to translate in reverse, from highly abstruse mathematics back into more universally understandable words, is foredoomed to inaccuracy, distortion and omission. Nevertheless we must, if we are to leave as few hinges and loops as possible, do our best. So let's start by thinking about this difficult wave/particle duality of light - of the photon.
  • It is absolutely certain that photons are particles: they can be counted, using ordinary numbers 1, 2, 3 and so on, when a few of them react with an appropriately sensitive TV camera acting as a photon detector.
  • It is absolutely certain that photons are waves: there can be no other explanation for the striped interference pattern on the screen behind the double slit.
Countless other demonstrations have confirmed this impossible duality over the years. There is one fairly easy way round the difficulty but I don't find it very satisfactory, personally. The argument goes like this:
  1. The particulate nature of light is apparent only when we measure it with particle-sensitive equipment.
  2. The wave nature of light is apparent only when we measure it with wave-sensitive equipment.
  3. So the problem is not the dual nature of light itself but the dual possibilities of measuring it. There are two different modes in which the us/light interface can operate. When we don't measure it light itself has no properties. Philosophically speaking, anything with no properties cannot be said to exist. So light itself does not have two conflicting sets of properties: it is merely that we observers can endue it with different properties according to our choice of method of observation.
Is this like saying that a human being has weight (being a physical object) and an IQ (being an intellectual entity)? Probably not, since even intellectual entities need physical embodiment in order to function. Weight and IQ are not contradictory: rather are they in some way complementary.

Well it just so happens that in 1923 the French Prince Louis de Broglie was able to show mathematically that all particles have, carry, or correspond to complementary waves of particular length and frequency. It tied in nicely with the earlier work of Planck and Einstein. I suppose one could argue that it didn't really explain anything since even de Broglie himself didn't clarify what "correspond to" meant.

But at least it was supposed to be a universal effect. The reason nobody had come across it before - the reason we don't find ourselves vibrating as we walk down the street or even lie in bed - is that the numbers are so tiny. Again we don't need the detail, but it may help to satisfy curiosity by stating that there is a number involved which has a decimal point and then thirty-two zeros before the significant digits are reached. It would be likely to affect a whole human being only if the effect were a thousand million million million million million times as big. But it is an inverse relationship: the smaller the particle the greater the wavelength. So down there at the subatomic scale the wavelength could be greater than the diameter of the particle.

Since waves are thus universal perhaps we ought to think about their various kinds:
  • Musical stringed instruments make use of the fact that a string under tension can be made to vibrate. There is a nice simple relationship between the length of the vibrating string and the note or frequency produced. This may be most obvious in the case of the harp, but the violinist, for instance, alters the length of the vibrating section of the string to change the note. [I was amazed to find that harpists do that too: they have pedals which twist pegs inside the framework of the instrument to increase lengths for flat notes and shorten them for sharps.]
  • So it is quite clear just what it is that is vibrating: it is the material of the string itself that goes to and fro at the behest of the violin bow, the harpist's fingers or the hammers on a piano. That motion is passed on to the intervening air to reach our ears as music. With many wind instruments the whole thing starts in the air, with the frequency set by the effective length of tubing involved.
  • Ocean waves are a different matter. We shall ignore breaking waves and concentrate on the smooth variety which have a much purer wave form. When such a wave travels across the sea it is only the energy that actually travels: the water molecules involved simply get pushed around in a vertical circle as the wave passes. Curiously perhaps, unlike light waves and sound waves, the speed with which the wave travels is greater the greater the wavelength, the lower the frequency.
    Manifestly, we must be a lot more careful in thinking of the water wave as a "thing". It is in fact simply an ordered pattern of up-and-down, to-and-fro movements of the host material. The "things" go round in circular motions: it is only the pattern and the energy that travel.
  • But the converse situation can apply in a river. Again we should ignore the breaking waves of white water, but in any rapidly flowing river there are usually numbers of standing waves - stationary waves - perhaps marking the positions of rocks below the surface or the rejoining of two or more local currents. In that case the "things", the water molecules, are travelling but the wave pattern stays more or less fixed.
Good heavens! Have we stumbled across something insightful here? With the ocean wave what we thought was the "thing" - the wave - turns out to be just a pattern of energy and not a thing in its own right at all. Could that be what an electron is like? Is each electron just a pattern of energy? Is each orbit like a sea surface? Mmmm: it bears thinking about!

§5.7 Quantum Mechanics - Probabilities

It may be profitable now to go back and have another look at those photons going through the slits of our double slit experiment to reach (or not to reach) the screen behind. If we were thinking at the common sense level, visualising photons as golf balls, we could perhaps describe what was happening in the following terms:
  • The lamp gives out golf ball photons in all directions, but we are concerned only with those that are going in the general direction of the foil wall. (If we had a golf ball howitzer we could aim it at the slits: that would be like having a monochromatic laser beam, providing a coherent stream of photons.)
  • It is probable that some of the golf balls will indeed go through slit 1. We might be able to predict this mathematically, or perhaps find the numbers by experiment and observation. The convention would then be that if all the golf balls went through the slit we should say that the probability value was 1: we should actually expect that only a smallish proportion would in fact go through slit 1 so the probability of any particular ball doing so would be a small fraction, say 0.1 (one tenth, one in ten) for the sake of argument.
  • Now let's think about the chances of one of those photon golf balls hitting a particular stripe that we had marked out on the screen. If our sampling stripe were quite narrow then the probability of any particular slit-1 photon hitting it would be quite small: let's say the probability value was 0.05, one in twenty.
  • Putting the two parts of the journey together, any particular golf ball that we followed from the lamp would have a probability of 0.1 that it would get through slit 1 and then (of those that got through) a probability of 0.05 of hitting the test stripe. So the total probability of "our" photon reaching the stripe would be -

    0.1 × 0.05 = 0.005
    five chances in a thousand.

  • Similarly of course, if the same applied to slit 2, and assuming our sample stripe was almost equidistant from slits 1 and 2, then the probability of scoring a hit within the test strip via slit 2 might be, say -

    0.1 × 0.06 = 0.006
    six chances in a thousand.

  • So if we have a 0.005 chance of a ball hitting the stripe via slit 1 and a 0.006 chance of a ball hitting the stripe via slit 2 then we clearly have a total probability of scoring of -

    0.005 + 0.006 = 0.011
    eleven chances in a thousand.

But we already know that we mustn't think in common sense terms: photons may possibly be particles but they most certainly are not like miniature golf balls. So what modifications have got to be made to make the corresponding quantum mechanical picture? Unfortunately the answer can lie only in rather complicated mathematics, beyond the capabilities of the average author (or the present one at any rate), not to mention the average reader. One clue, though, may be intellectually accessible at the not-too-technical level. It has to do with i, the square root of minus one that we met earlier. Apparently the way the photons behave, in contradistinction to the golf balls, is best described mathematically using complex numbers - numbers which have ordinary parts and imaginary, i-based, parts. If we visualise the imaginary part of the measure being displayed as a distance up the page, and the real part rightwards across the page, as is the mathematician's habit, then the complex number probability for any particular photon lies somewhere less than one unit from the bottom left-hand corner of the page. And that distance from the bottom left-hand corner is known as the modulus of the complex number probability.

Now we come to the crunch. Apparently when two such moduli are added the resulting probability amplitude may be greater or less than the two figures that are being added. In other words, when the slit-1 probability is added to the slit-2 probability it may be found that the probability amplitude is -

less or even zero
(corresponding to interference and a dark stripe)
(corresponding to reinforcement and a bright stripe).

The reason has something to do with the fact that the addition procedure, simple adding in the case of the golf balls, involves squaring the moduli and then taking the complex square root of the complex sum. So when only one slit is open the photons and golf balls behave similarly. But when both are open the probabilities interact in totally different ways at the two scales.

Assuming that the scientists are right with their mathematics - that they are no mere "rude (quantum) mechanicals" - that would seem to fit nicely. But the argument can be taken on to another stage. It follows from what has been said so far that every part of the two-slit experiment screen can be given a probability amplitude. This figure would be related to the probability of a photon arriving at that spot, and hence after a period of exposure, to the brightness of the image at that spot. But in just the same way all the other relevant points in space can be allocated probability amplitudes - in theory at least. (Actually a point in space, having zero volume, can't have an amplitude at all. The figures are best regarded as amplitude densities.) Clearly, one mm in front of the screen surface the probability amplitude densities will be pretty much the same as those of the screen itself, whereas the back surface of the foil, away from the slits, will have very low if not zero values. But can we talk sensibly about the probability amplitudes within the slits themselves? The answer, apparently, is yes we can. In fact the quantum state of a particle is the sum total of all the complex-number probability weightings within the space open to it. The mathematical function that summarises all those numbers is called the wave function of the particle.

And that's as near as we're going to get to a common sense answer to the question about which slit a particular photon passes through. The wave function of every photon interacts with both slits. So to that extent, yes, the photon particle does indeed get from A to B by two (at least) routes simultaneously. But it would probably be better not to ask the question at all. If it is indeed true that "To succeed you must first ask the right questions" (see "Introduction and Synopsis"), then it seems as if quantum mechanical questions need to be posed in complex mathematical form.

Nevertheless, common sense still may be insistent. "Why don't we put a photon detector in each slit and then fire just one photon at the set-up and see what happens?" There are two reasons for a negative answer here, one fairly commonsensical and one very much more quantum mechanical.
  • It is a universal principle that you can't measure things without interfering with them. In order to get some "thing" to affect some measuring instrument you have got to alter something or other: you can never measure things in the act of doing something else (unless your data happen to be an incidental by-product of the main activity). This applies to everything from electrons to beef cattle and beyond: streams of electrons are measured by seeing how much they bend under a known force, and beef cattle have to be walked through a weigh-crush (losing some weight in the process). Similarly, it is simply not possible for our photon to register a hit in slit 1, say, and yet still continue its journey unaffected.
  • The wave function of our photon specifies that it is free, subject to certain probabilities, to go towards slit 1 and/or slit 2. But if it reaches any form of measuring device in, say for a change, slit 2 then the wave function collapses and the probability of its also appearing at slit 1 immediately becomes zero. It cannot materialise (if that's the right word) in both places: having reached the measuring device its probability of being there is in fact certainty (probability 1) so the probability of its being anywhere else must be zero.
This does actually embody one of the more serious mysteries of quantum mechanics since the principle applies regardless of distances. What we are saying is, in effect, that as soon as one or other of the detectors fitted in each slit (or anywhere else for that matter) is triggered it immediately causes the wave function in respect of everywhere else to evaporate. Now that means - or seems to mean, at least - that a message has to reach all the other parts at the same instant as the triggering: to do so requires the message to be transmitted faster than the speed of light, which Einstein has shown to be impossible.

So in fact the presence of measuring devices prevents the photons behaving normally. If the only way we can find out about quantum-sized components is by interfering with them it follows, according to some philosophers, that the world is totally unknowable. And it is worth realising that we ourselves are living measuring instruments whenever we make use of our senses: a photon that has registered in an eye has had its wave function destroyed by achieving certainty.

§5.8 Conclusions?

Standing back and trying to assess this rather complex Chapter, can we engineer any further insights? Some people have certainly tried to. For example, the waves we thought about in §5.6 were only four-dimensional. An ocean wave, for instance, had height as well as a latitude/longitude position and was travelling with time. But quite a lot of scientists are of the opinion that at the level of quantum phenomena there are almost certainly at least two, possibly three more dimensions, taking account of the complex, i-based number measurements involved. So the question arises, what forms might waves take when they exist in six or seven dimensions?

Another interpretation is even more extreme in this way. A single electron may be thought of as needing only the "normal" three spatial dimensions. But with anything more complex than a hydrogen atom there are many electrons - 6 for carbon, 16 with sulphur, a veritable cloud of 82 for lead, and so on - and each can be regarded as occupying its own three-dimensional space. So lead exists in 82 × 3 = 246 dimensions! (- plus one for time = 247.) We had earlier been mystified by the way electrons could jump from orbit to orbit without passing through the space in between. Perhaps we were "looking" in the wrong dimensions.

In any case, if the electron is not a "thing" at all but more closely resembles a pattern then it is fairly easy to think of one pattern being replaced by another without need of any jump across from one bit of four-dimensional space-time to another. We can't say that that's what happens, but it could at least be conceptually acceptable.

Another conceptual mounting block might be the standing wave. Whereas musical notes and ocean swells travel through time the standing waves in a river are virtually stationary: they are static in their three space dimensions and persist uniformly through their time dimension - at least until drought, flood or boat intervene. So could an electron actually BE a standing probability wave? Certainly some people have thought that it might.

§5.9 Uncertain Coda

It follows from all this discussion that quantum phenomena can never be predicted, in detail. Probability densities can be worked out for various systems apparently, and those figures can be used to make effective and useful predictions of the way streams of electrons or beams of photons will behave, in the mass. But there is no way that anything can be said about individual electrons or photons. The problem arises from the measurement difficulty. On the normal, Newtonian scale it is a fairly simple matter to find the mass, position and velocity of, say, a billiard ball. You would have to measure the mass first and then assume it doesn't change once it's in motion, but that should be a valid assumption in Newtonian physics. Then you could perhaps use radar to find the position and velocity, without your measuring process affecting the ball in any Newtonianly significant way. It is then a fairly simple bit of mathematics to work out what it will do next.

But we now know we can't do that with quantum concepts. What it boils down to is that if we do find the position of a photon we have, in the very process, stopped it dead in its tracks: it can no longer carry on as it might have done had we not measured it. Measurement or even detection causes collapse of the wave function. So we certainly can't take two successive position readings a known time apart and thus work out the velocity. There are other ways, non-"destructive" ways, of finding the momentum to high degrees of accuracy. But not at the same time as finding position.

So we can -
get a rough idea of both position and momentum
we can get an accurate position fix and know nothing of the particle's momentum
we can measure the momentum accurately and know nothing of its position

- but no other combination: we can't for example, have a rough idea of position and an exact measure of momentum. This is a necessary and unavoidable attribute of quantum objects and not just a limitation of current experimental technique.

Now it may seem that a mountain is being made out of a molehill here. Position can be measured when necessary: when momentum is of interest it can be found. Fine: don't be greedy! But the normal basis of science (I use the word "normal" advisedly) is to find rules and then see how they apply by predicting and testing outcomes: effects must be linked to causes. But without both position and momentum it is impossible to predict what any particular particle will do.

Going back to our double slit experiment, we have so far concluded that we cannot follow any particular photon from source, through slit(s), to screen. Nevertheless, we do know that quite a lot of photons are destined to reach what will turn out to be the bright stripes on the screen, with lesser numbers at the intermediate strips and very few along the lines of the dark stripes. "Normal" science would insist that at the moment each individual photon leaves the source its destiny is fixed (even though we may not be clever enough, yet, to be able to predict it). Quantum mechanics insists, on the contrary, that such certainty is simply not available in principle, that wave functions are probabilistic not mechanical, that "God plays dice" in Einstein's telling phrase (though he himself was loath to believe it). In honour of the originator of the relevant mathematics, this is usually known as Heisenberg's Uncertainty Principle.

So if we try to follow one particle in our minds we are left with no reason why that particular particle went just there. Worryingly, there is no cause for that particular outcome. Again, it is not the case that we are simply not clever enough to find the cause: there IS no cause to find. This seems to be so very significant in terms of our over-all enquiry that I take the trouble to cite a corroboration -
Quantum mechanics, by its essence, entails "the necessity of a final renunciation of the classical ideal of causality and a radical revision of our attitude toward the problem of physical reality".

Atomic Theory and Human Knowledge
NEILS BOHR (John Wiley, 1958)

Let's leave it there for now.

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