Quantum Physics

In another thread I was prodded to produce a Petty style exposition of Quantum Theory. Here's the opening salvo.

Prologue

The leading lights of physics were remarkably smug around the year 1900. They had good reason to be: 50 years earlier heat and temperature became unified with the same laws of motion that had been grandly predicting the movements of everything on earth and in the heavens for the past 200 years. 40 years earlier the Scotshobbit James Clerk Maxwell had pulled together disparate observations of electricity and magnetism, creating a unification that out of the blue explained where light and radio waves came from. 20 years earlier the concepts of statistical mechanics developed by Gibbs and Boltzmann were laying bare the detailed behavior of gasses and chemical reactions, even though the practitioners had no firm proof molecules even existed.

All these achievements were evolutionary rather than revolutionary: nobody had to unlearn old habits of thought to apply the new discoveries, simply learn new skills. So it was that in 1894 Albert Michelson declared that the future of physics lay in measurement to the 6th decimal point, and Lord Kelvin (one of the masterminds who united motion and heat) declared in 1900 that there was nothing new to be discovered in physics, echoing the need for more precise measurement. It should be pointed out that they were talking about far-reaching laws, not details like new elements. It was like saying that even though there were new lands and oceans to explore, the earth would continue to be round, and the sun would still rise like clockwork.

But they were deliberately ignoring the burning fuses of evidence everyone was aware of, deeming them details of small consequence. As early as 1850 the spectra of the elements were known to be more akin to barcodes than rainbows, and this fact was being used to identify elements (and identify helium in the sun before it was discovered on earth). But the current physics could explain discrete wavelengths by fixed oscillators: merely some details to be worked out in the structure of the elements; a spark, not a fire. The shape of the thermal radiation spectrum and the heat capacity of metals could not be explained; but this was also tied to hidden structure of the elements. Small sparks rather than a conflagration. But the sparks you ignore may be the sparkling fuses of bombs, and the brash new generation of physicists whose task it was to wrestle with these small problems ended up proving that the world was neither round nor flat, but topsy-turvy.

2 years after Michelson's statement, he made very careful measurements that showed light always traveled the same speed no matter what direction or speed the earth is moving, and a young Albert Einstein was prodded along the path that led to relativity. A year after Lord Kelvin declared there were no new discoveries, Max Planck explained thermal radiation by assuming energy was quantized rather than continuous. The paper made no stir (it was more a curve fitting technique than a bold new theory), but two years later Einstein repurposed the quantum concept to prove the existence of photons (impossible constructions under Maxwell's theory of light), and it was as if a sledge hammer hit the world of classical physics. That same year he also published Relativity, and the world was demolished.

The twin pillars of 20th century physics, quantum theory and relativity, are notable because they violate our intuition. I plan to provide a guided tour through the basic concepts of quantum theory, moving slowly and keeping the concepts as unmathematical as I can. I'll tackle roughly one topic per week:

1. Waves and superposition: quantum mechanics is often called wave mechanics, and before we get to the weird parts, we have to understand how waves in the everyday world behave. Dry but crucial.

2. History of the development of Quantum Theory: I gave some vague hints above, but will fill it out so that it makes sense.

3. Applying Quantum Theory: this is where the basic tools and predictions of quantum theory (not already described in the history section) are laid out.

4. Quantum Mechanics and Everyday Experience: in case the strangeness of QM is not apparent by now, it will here be related to the world we know.

See you here next weekend.

halfwise wrote:
2 years after Michelson's statement, he made very careful measurements that showed light always traveled the same speed no matter what direction or speed the earth is moving, and a young Albert Einstein was prodded along the path that led to relativity.

Nicely written! I look forward to future installments.

I can't resist one quibble though. Are you familiar with Dayton Miller's critique of the Michelson-Morley experiment and his further research? This article makes an interesting case, even if you choose not to accept it's conclusions:

http://www.orgonelab.org/miller.htm

That does seem to be a very valid and important experiment, and I don't doubt it was largely ignored due to bias towards relativity. But the reasons for this 'bias' were (and continues to be) overwhelming evidence that relativity works. I think the author of this review article damages his case in continuing to refer to this as an 'ether drift' experiment. Note that this phrase starts by assuming an ether, then assumes it is largely not observed because it is being dragged along with the earth, and the experiment observes slight discrepancies from being dragged along perfectly with the earth. This would mean out in space the effect would be huge, which contradicts mountains of observations of relativity at work on the astronomical scale.

If modern physics would go back and take this experiment seriously (and I think they should) they would probably relate it to dark matter, which has been shown to cluster around galaxies. This makes much more sense than assuming a cosmological ether that is being imperfectly dragged along with the earth. I have had doubts about ascribing dark matter to exotic particles that have otherwise been undetected in earthbound experiments, and have felt that observations of distant galaxies are subject to errors. But since I'm not an observational astronomer, I'm not qualified to critique their methods. The Miller experiment may be the only earthbound experiment to indicate the existence of dark matter, if only it were resurrected and brought to the attention of the modern generation of astrophysicists.

Actually Dave, Miller's work has not been ignored. An extremely detailed analysis of Miller's work was published online in 2008, going far beyond Shankland's critique. End result: Miller's positive results were statistically indistinguishable from zero.

http://arxiv.org/pdf/physics/0608238.pdf

Don't know if it made it in a peer-reviewed journal, there's a possibility nobody was interested in an examination of the flaws of a long disregarded experiment.

I expect that Dark Matter will turn out to be so out of our experience or capacity to understand that it never becomes observable in earth bound experiments. Maybe it is just Nothing but Nothing with strange powers, so how can you observe Nothing?

Thanks for this thread Halfwise its fab.

The early 20th century physicists had learned their lesson about the limits of human reasoning so well that by 1945 Richard Feynman's dissertation work (he would later revolutionize quantum mechanics) was dismissed with the phrase "Your idea is crazy, but not crazy enough to be true."

So glad you are undertaking this Halfwise- good opening and already keen for more.
I personally will be particularly interested when you eventually get to macro and micro- how all the weirdness down there makes all the regular stuff at our level, mainly because my own interest in quantum was sparked coming at it from the other side, from the shaman/mystic side and discovering rather suprisingly many of the underlying concepts seem remarkably similar.
Carry on the good work!

I want to know about Spooky Action at a Distance.

I can demonstrate that with nothing more than a gust of wind and my kilt at just the right angle.

Not sure I can compete with Petty's version, but I will cover SAAAD. Unfortunately it's only really spooky if you buy into the logical arguments behind it. Most books for the general audience simply tell you what the interpretation is without convincing you why you should believe it, in my opinion. Not too sure how much better I can do but will try.

So Religion and Science were twins separated at birth!

halfwise wrote:Actually Dave, Miller's work has not been ignored. An extremely detailed analysis of Miller's work was published online in 2008, going far beyond Shankland's critique. End result: Miller's positive results were statistically indistinguishable from zero.

http://arxiv.org/pdf/physics/0608238.pdf

Don't know if it made it in a peer-reviewed journal, there's a possibility nobody was interested in an examination of the flaws of a long disregarded experiment.

Very interesting. Thank you!

I'll confess I only made it through the introduction, conclusions and appendices, and about half the text. It's a little dense reading for a humble farmer.
{I'm personally much more comfortable with models that can be described in terms of fruits and vegetables. }

I lost confidence in your scientific viewpoint (and practical skills) when I saw plenty of carrots but no spaceship, Cap'n Dave - and not one hired carrot-skinner in sight!

David H wrote:
I'll confess I only made it through the introduction, conclusions and appendices, and about half the text. It's a little dense reading for a humble farmer.
{I'm personally much more comfortable with models that can be described in terms of fruits and vegetables. }

What, you think I read the damn thing?

I read the article you sent, but this second more modern one may fall under the category of proof by intimidation. Just don't have time for this kind of thing right now.

Orwell wrote:I lost confidence in your scientific viewpoint (and practical skills) when I saw plenty of carrots but no spaceship,

Question my scientific viewpoint if you will, but practically speaking my horse and I are eating very well right now and for the foreseeable future.

David H wrote:

That kind of thing could get your account suspended on some forums, ya know!

So now we start in earnest.  In some sense I feel I must apologize for this long treatise on waves, but I see no way around it: there is no royal road to quantum mechanics.  Many popular books gloss over this part, so when they get to the mysterious parts of quantum mechanics they must assure their readers that these are the logical deductions, who have no other choice but to nod their heads and blindly accept what they are told.  This will be the most tedious section, but my hope is that it is not tedious at all: waves are where it's at - embrace them!

NOTE: I have noticed that many of the diagrams have disappeared. It may be due to a limited capacity in servimg, at some point I may come back and try to restore some images.


Part 1: The Waves at the Base of Reality

You cannot master the quantum until you have mentally digested the most basic structural unit of all reality: waves and their interactions. For those who have had a year of college physics most of this will be familiar.  But even for those who haven’t, nothing here should feel strange, for we all have watched the ripples on a pond as they lace through each other – all that is asked is to highlight the simple rules that govern wave society. No longer can we watch passively; from now on we must cultivate awareness.

But first a vocabulary lesson, let’s knock a little dust off and look at a wave:


Note that the wavelength is measured between any part of a wave and the same part of the next.  Amplitude is the height of the wave. We should also note that as all waves in a medium tend to move at nearly the same speed, as the frequency of oscillation goes up, the wavelength must go down as more waves are crowded into the same distance. That should be enough to get us started, on to the observations!

In the pond, you see waves criss-crossing; what happens when two waves meet?  Physicists use the terms ‘constructive and destructive interference’, which are just complicated terms for some common sense ideas, as illustrated below.  In these examples we are limited by the 2 dimensional geometry of the page, so will show waves meeting head-on, though you should know from idyllic afternoons watching ripples in the pond that waves are generally meeting at all angles.  Here we simplify down to little wave pulses, either crests or troughs moving across the smooth surface indicated by a dotted line.  We show what the situation looks like when the waves approach, the overlap upon encounter, and the departure.



There’s something important to note here: when waves meet, there is a superposition of waves and crests and whatnot, and the result momentarily may not look like either wave.  Two crests add to make a double sized crest, and if a crest and a trough are the same size, they can cancel out (in general you’d only get partial cancellation if the waves don’t have the same amplitude).  But despite all the skullduggery that may happen as they meet, even if it instantaneously looks like they’ve destroyed each other in an orgy of total destructive interference and the game is up, the waves actually pass right through each other and continue on blithely as though nothing at all had happened.  This is called linear superposition, and is a general trait of nearly all waves in the real world.  Some types of waves DO affect each other, but these are not nice waves, and we shan’t have anything to do with them.

Waves may not affect the motion of other waves, but they do bounce off boundaries.  When you attempt to confine a wave, very interesting things happen due to interference effects.  A good way to see this is to tie a rope to a doorknob or something else solid, pull it nearly tight, and with some wrist action start flicking some waves towards the ‘node’ at the door.  If you send a single crest, you’ll see that it is reflected upside down.  You’d think this would cancel out a continuous wave until you look at the situation more closely:


What we see happening is the part of the wave that would be transmitted behind the wall is opposite to that arriving at the wall.  If flipped upside down, it will interfere constructively and we’ll get a nice wave action going.  But this is only possible if we get everything in sync.  

If you look at the diagram, the wave has to go to zero at the wall, but has to be at a crest  or trough where your arm causes the motion.  The zero point in between a crest and trough is called a node.  The distance between a node and the nearest crest or trough is ¼ wavelength, and the distance between successive crests or troughs is ½ a wavelength.  So you have to produce a wave that has at least a quarter wavelength between you and the wall; or ¾ wavelength, or 1 ¼ wavelength etc.  If you do this, you get a ‘standing wave’, but if you don’t shake it at the right frequency, it will fizzle out.  When constrained, a wave has to be tuned to the situation.  This is called resonance: when you sing in the shower, some notes will be louder because their wavelengths fit in the room and so the echoes overlap constructively.  It’s fun to find the point of resonance – just slide the note up and down until it locks in.

This is the basis of musical instruments, as Pythagoras discovered two and a half thousand years ago.  A guitar or violin string is constrained to nodes at both ends (you pluck or bow it somewhere in the middle instead of shaking the end).  In this case you need multiples of half a wavelength to fit in the boundaries you made for it; half a wavelength produces the fundamental, a full wavelength the octave, 3/2 an octave and a fifth, and so on.  The same effect pertains to wind instruments as well, though the boundaries enforce the locations of crests or troughs rather than nodes.  As far as we know, Pythagoras’ discovery that musical harmony conformed to whole number ratios was the first time in human thought that an abstract mathematical law had been applied to the physical world. (Astronomical observations were of specific numbers, not abstract principles). The realization that the universe is mathematical in nature must be listed as one of the most profound moments in human history.


This characteristic of confined waves to take on specific states is not only the key to music, but to the quantized energy levels of atoms.  When the ancients compared the mathematical ratios of instruments to the music of the spheres, they were exactly right, but looking in the wrong direction.  They should have looked to inner space rather than outer space.

Off course waves aren’t always approaching each other head-on, they may come together in nearly parallel but converging paths and interfere when they meet:


These two waves look like when they meet, the troughs and crests will line up.  We say that they will be in phase and so will experience constructive interference.  Things will get very splashy where their paths cross!  If one of the waves was flipped upside down so that crest would meet trough, we say the waves of out of phase, and will interfere destructively.  You’d see ripples approaching each other, but an eery calm will prevail at the point of rondevouz.  We’ll see pictures of this later happening for water waves.

A very interesting and crucial idea is that any shape of wave, such as the plane waves shown below on the left can be made out of a superposition of circular waves on the right.

So how is this done?  Actually, we’ve all done it.  Let’s say you drop a chip of wood in the water: it will fall and bob up and down, generating a set of circular waves.  Now bond a bunch of chips together to make a pencil, and drop that in the water so it lands flat.  It’s as though every chip you stuck together makes it’s own circular wave, and they all add up to make the plane wave  (rectangular or oblong may be a better description) that propagates out from the pencil:


We see strong waves going up and down on the page from the sides of the pencil, and much weaker somewhat circular waves going out from the ends of the pencil.  Try it in your sink and see.

This idea of wave fronts being built up of circular waves from point sources is crucial to an understanding of how waves interact with material objects.  Let’s start by looking at how a plane wave behaves when it encounters an opaque wall with a slit in it:



Remember that plane waves can be constructed of a bunch of circular waves. I have suppressed the point source nature of the waves to the left of the wall for clarity.  But the slit ‘edits out’ all but a few of the point sources, so the circular components become important.  We see when the slit is about as wide as the wavelength or larger, we get a nearly plane wave propagating through the slit, with maybe some rounded ends.  But when the slit is smaller than the wavelength, we get a fairly rounded wavefront spreading out from the slit.  This bending of light around edges is called diffraction.  


The picture above shows photographs of this situation done with water.  Instead of changing the slit size, the wavelength was changed by adjusting the frequency of vibration.  But the same rule holds: when the slit is larger than the wavelength (left) you get a beam of waves with only a little spreading.  When the slit approaches the size of the wavelength or is smaller (right) the transmitted wave spreads out, becoming circular.

The inverse situation behaves similarly: if you put an object in the path of a wave that has a size smaller than the wavelength, the wave will wrap around it.  Very small objects compared to the wavelength barely seem to affect the waves: large objects will block the wave but have some bending around the edges.  When standing in the ocean, your body has little effect on the long waves, but will block and diffract the ripples.

So far if you don’t have any way to see the waves you could convince yourself diffraction was all due to bullets, and as the slit got smaller it was more likely the bullets would bounce off the edges and you get a sort of spray pattern:


This was Newton’s view, published about 15 years after his Principia that launched the scientific era.  Since he had done so much with the mechanical view of the universe, extending this approach to light was only natural.   Given his monumental reputation, nearly everyone followed his lead.  

The proof that light was in fact a wave waited a century until someone figured out how to make two beams of light interfere.  The problem was that natural light was a mishmash of many wavelengths, and if the light came from two different sources, the sources would have random phases: the waves would be out of step with each other – sometimes ahead, sometimes behind.  It was Thomas Young who figured out that light had to be filtered into one color, and that if all light came from the same point source, it would always be in step when split into separate beams.

This experiment is so important conceptually that I will hit it with several diagrams.  In the diagram below the crests are shown in red, and the troughs in yellow.  The fact that the waves behind the wall are circular instead of plane doesn’t make a real difference, the important thing is that the wave behind the slits starts all over again as it passes through the slits, and they are in phase because they come from the same source.  Red arrows show a line where all the crests line up with crests or troughs line up with troughs; black arrows where all the crests meet the troughs.  The result is alternating bands of bright and dark that can only be due to wave interference.

This is a rather abstract diagram, let’s make it more real by putting a couple of vibrating styluses in water: the result will be the same interference pattern we’d get with the light setup above.  It’s nice to see the waves that we can’t see with the light.  You can see that at the bottom there would be places where the waves slosh up and down really big, and places where they cancel out.  If this was light, where the waves are big and splashy is where we’d see bright spots, and in between would be dark.  At the bottom is an actual picture taken when light is passed through two slits and projected onto a screen.


So this is all very well, and demonstrates quite nicely that light is a wave, but what good is it?  Turns out it’s fabulously good: this is exactly the technique used to measure wavelengths, which can be used to measure the spectra of light coming off of heated objects, which can be used to detect composition.  It’s used in CSI; it’s used by astronomers looking at stars - it’s a fundamental tool.  What’s more, it’s all around you and very easy to see.  It’s so important I just had to demonstrate using household items.

So I took a pot of water, and I took a plastic fork and broke off the middle two tines.  I could see an effect when vibrating it with my hands, but it wasn’t quite uniform enough for taking pictures.  So I attached a #cough# adjustable speed oscillatory device to the fork, and could see an effect.  Wasn’t jumping out at me the way I liked until I got the bright spot of a light bulb right above it, then the interference fringes jumped right out.  Unfortunately our eye and the camera are not the same device, I think the eye performs some averaging. I played around with contrast and stuff but it never looked good.  I could have put in serious time to make the lighting work, but hey, I love you guys, but not THAT much.  So though the pictures I took don’t really show what I wanted, I can’t resist putting them here anyway to encourage others to play around themselves.  


What you see here is no vibration (top left), slow vibration (top right), medium vibration (bottom left) and fast vibration (bottom right).  I recommend you zoom up on them.

One thing I think the pictures do show: as the vibration rate goes up, which means the wavelength goes down, the interference pattern becomes tighter and tighter.   Eventually you wouldn’t be able to tell it was there and would have no discernable evidence that wave phenomena were taking place.  This idea will become important when we look at the relationship between the quantum and the classical world.

You don’t even have to use two slits or point sources to see interference fringes.  If you look back at the single slit diagrams, you will see a slit looks like a number of point sources: it’s a highly compressed multiple slit experiment!  I won’t go through the arguments in detail but it shouldn’t feel unreasonable for a small slit to produce interference fringes.  You’ve all seen them without realizing it - put your fingers close together and look through them at a light source: you’ll see hair thin dark lines.  Interference at work.

I can’t resist a historical anecdote.  When Young published his two slit experiment with light, the mental pull towards light being particles was so strong that what should have been an open and shut case got stuck in the re-opened state.  The French Academy of Sciences announced a competition for irrefutable proof that light was either a wave or a particle.  They only received one serious entry, from Augustin Fresnel.  He had been dabbling with wave constructions, and predicted that light waves would result in rings being formed inside the shadow of a spherical object.  Just like the double slit experiment this would require a single point source of single color light for the waves to be in step, so was not a simple observation and he had very weak experimental evidence.  He was not enough of a mathematician to put his ideas on a firm mathematical footing, but the famous mathematician Poisson was on the panel of judges and carried out a full calculation.  The result was that not only would rings appear in the shadow, but a bright spot of light should appear directly behind the sphere!  This might be considered a case of reducto ad absurdum putting the wave theory out to pasture for another generation.  Fortunately the panel was nothing if not meticulous, and arranged for the experiment to be performed.  To everyone’s amazement the inconceivable bright spot appeared, and all opposition to the wave theory of light crumbled.

One last concept will complete the survey of wave properties: what affects the amount of energy contained in a wave?  The simplest physical analogy here is the wave in a rope.  The wave may travel horizontally, but all the actual physical motion of the mass in the rope is up and down.  Recall that kinetic energy goes as the square of the speed (mv^2)/2, so we need to relate the speed to the wave properties of amplitude and frequency.  To make things clear, we distinguish the speed of the mass in the rope v from the propagation speed of the wave c.


Speed is distance/time, and the distance a segment of the rope moves as the wave passes is clearly proportional to the amplitude A.  The time it takes to move is the period of the wave T, so  V ~ A/T  (we are using proportionalities here instead of equalities because the speed will not be constant etc etc so we just want to understand the tendency).  But frequency f is by definition 1/period: if the period is half a second the frequency is 2 vibrations per second.  So we get  V ~ Af.  This means the energy goes as V^2 ~ (A^2)x(f^2).  Even though this was derived for the special case of a rope with mass, the result applies to all waves, even massless electromagnetic waves.

So the final summary of wave properties, the crib sheet for quantum mechanics:

• Waves pass through each other, but can superpose to either add or cancel.  Particles can only add (two bullets can never add to zero).  This is called interference.  Interference patterns can be used to measure wave properties.
• When confined, waves can only take on certain wavelengths fixed by the confinement geometry in order to produce standing waves.  This is called resonance.
•  If an object or slit is much larger than the wavelength, the waves will behave like particles and move in straight beams.  When the slit is smaller than the wavelength the waves will spread out from it, or when encountering a small object this same spreading means the waves wrap around it and are not blocked by the object.
• Energy of a wave goes as (Amplitude*Frequency)^2.

------------

That's all for now.  Next week, we look at the development of Quantum Mechanics.

Wave animation:




{{I'm here to help!}}

Come to think of it, there were animations I could point to, I was thinking of this in book form. Oh well, there'll be other opportunities.

Enjoying this wave thingy, its as dry as a camels hoof, but I am prepared to go there if Halfwise guides me with his hairy hand.

It's much better if you do some of the experiments: looking through the space between fingers near each other, point sources in water, etc. After this the experiments aren't so doable, but I'll try to show color pictures.

Dry yes but informative. Thanks Halfwise I look forward to the next one. My only question so far is about light- does it not also act like a particle when no ones looking or some other quantum silliness? I sem to remember something about that in relation to the slit experiment.

yes, that is coming. You won't get this strangeness 'explained' until the 3rd post - but the idea will come forth to the perplexity of everyone involved (I'm speaking historically) next week.

to elaborate, you haven't seen any quantum stuff yet, waves are all classical. In the history installment I'll bring you to the point where the physicists had figured out how to get the numbers that matched reality, but still were trying to figure out what it all meant - the readers should be as confused as they were. In the theory and applications installment I'll put in place the mental structure they created to deal with it....and it requires an acceptance that the world is not what we thought. The last installment may or may not be necessary, but will tie up loose ends in the relationship between the quantum world and us.